# Complete Edge Coverage, Supersets and Minimal-versus-Minimum Boundaries

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261015-advance-20261114-r07-vertex-cover-foundations · 2026-10-15 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3]] and unique candidate subset=[v1, v2, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3], [[v0, v1], [v1, v2], [v2, v3]], [v1, v2, v3], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3], [[v0, v1], [v1, v2], [v2, v3]], [v1, v2, v3], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0001/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 2. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4]] and unique candidate subset=[v0, v2, v3, v4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4], [[v0, v1], [v1, v2], [v2, v3], [v3, v4]], [v0, v2, v3, v4], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4], [[v0, v1], [v1, v2], [v2, v3], [v3, v4]], [v0, v2, v3, v4], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0002/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 3. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]] and unique candidate subset=[v0, v1, v3, v4, v5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v3, v4, v5], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v3, v4, v5], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0003/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 4. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]] and unique candidate subset=[v0, v1, v2, v4, v5, v6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [v0, v1, v2, v4, v5, v6], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [v0, v1, v2, v4, v5, v6], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0004/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 5. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]] and unique candidate subset=[v0, v1, v2, v3, v5, v6, v7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [v0, v1, v2, v3, v5, v6, v7], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [v0, v1, v2, v3, v5, v6, v7], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0005/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 6. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7, v8]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0006/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 7. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v7, v8, v9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0007/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 8. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v8, v9, v10]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [[[v0, v1], Yes], [[v1, v2], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [[[v0, v1], Yes], [[v1, v2], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0008/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 9. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v9, v10, v11]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v9], [[[v0, v1], Yes], [[v1, v2], Yes], [[v10, v11], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v9], [[[v0, v1], Yes], [[v1, v2], Yes], [[v10, v11], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0009/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 10. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v0]] and unique candidate subset=[v0, v2, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3], [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], [v0, v2, v3], [[[v0, v1], Yes], [[v0, v3], Yes], [[v1, v2], Yes], [[v2, v3], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3], [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], [v0, v2, v3], [[[v0, v1], Yes], [[v0, v3], Yes], [[v1, v2], Yes], [[v2, v3], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0010/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 11. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v0]] and unique candidate subset=[v1, v2, v3, v4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4], [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], [v1, v2, v3, v4], [[[v0, v1], Yes], [[v0, v4], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4], [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], [v1, v2, v3, v4], [[[v0, v1], Yes], [[v0, v4], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0011/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 12. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v0]] and unique candidate subset=[v0, v1, v2, v3, v4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v2, v3, v4], [[[v0, v1], Yes], [[v0, v5], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v2, v3, v4], [[[v0, v1], Yes], [[v0, v5], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0012/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 13. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [v0, v1, v2, v3, v4, v6], [[[v0, v1], Yes], [[v0, v6], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [v0, v1, v2, v3, v4, v6], [[[v0, v1], Yes], [[v0, v6], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0013/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 14. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [v0, v1, v2, v3, v4, v6, v7], [[[v0, v1], Yes], [[v0, v7], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [v0, v1, v2, v3, v4, v6, v7], [[[v0, v1], Yes], [[v0, v7], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0014/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 15. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7, v8]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], Yes], [[v0, v8], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], Yes], [[v0, v8], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0015/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 16. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7, v8, v9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v9], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v9], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0016/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 17. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7, v8, v9, v10]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v10], Yes], [[v1, v2], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v10], Yes], [[v1, v2], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0017/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 18. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11], [v11, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7, v8, v9, v10, v11]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v11], Yes], [[v1, v2], Yes], [[v10, v11], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v11], Yes], [[v1, v2], Yes], [[v10, v11], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0018/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 19. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3]] and unique candidate subset=[v0, v1, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3], [[v0, v1], [v0, v2], [v0, v3]], [v0, v1, v3], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3], [[v0, v1], [v0, v2], [v0, v3]], [v0, v1, v3], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0019/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 20. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4]] and unique candidate subset=[v0, v1, v2, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4]], [v0, v1, v2, v3], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4]], [v0, v1, v2, v3], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0020/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 21. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]] and unique candidate subset=[v0, v1, v3, v4, v5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]], [v0, v1, v3, v4, v5], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]], [v0, v1, v3, v4, v5], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0021/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 22. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]] and unique candidate subset=[v1, v2, v3, v4, v5, v6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]], [v1, v2, v3, v4, v5, v6], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]], [v1, v2, v3, v4, v5, v6], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0022/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 23. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [v0, v1, v2, v3, v4, v5, v7], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes], [[v0, v7], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [v0, v1, v2, v3, v4, v5, v7], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes], [[v0, v7], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0023/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 24. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7, v8]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes], [[v0, v7], Yes], [[v0, v8], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes], [[v0, v7], Yes], [[v0, v8], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0024/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 25. Every Undirected Edge Needs at Least One Selected Endpoint

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]] and unique candidate subset=[v0, v1, v2, v3, v5, v6, v7, v8, v9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).

View answer and explanation([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes], [[v0, v7], Yes], [[v0, v8], Yes], [[v0, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)

List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes], [[v0, v7], Yes], [[v0, v8], Yes], [[v0, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0025/)

Vertex cover · qf-mod45-vcover-complete-edge-audit · candidate





## 26. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0026/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 27. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0027/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 28. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0028/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 29. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0029/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 30. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0030/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 31. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0031/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 32. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0032/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 33. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v1, v2], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0033/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 34. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v1, v2], No], [[v10, v11], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v10, v11], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0034/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 35. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v3], No], [[v1, v2], No], [[v2, v3], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v3], No], [[v1, v2], No], [[v2, v3], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0035/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 36. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v4], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v4], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0036/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 37. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v5], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v5], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0037/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 38. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v6], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v6], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0038/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 39. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v7], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v7], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0039/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 40. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v8], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v8], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0040/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 41. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v9], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v9], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0041/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 42. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v10], No], [[v1, v2], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v10], No], [[v1, v2], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0042/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 43. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11], [v11, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v11], No], [[v1, v2], No], [[v10, v11], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v11], No], [[v1, v2], No], [[v10, v11], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0043/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 44. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0044/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 45. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0045/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 46. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0046/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 47. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0047/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 48. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0048/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 49. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No], [[v0, v8], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No], [[v0, v8], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0049/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 50. One Edge Whose Both Endpoints Are Missing Refutes the Candidate

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).

View answer and explanation([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No], [[v0, v8], No], [[v0, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)

Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No], [[v0, v8], No], [[v0, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0050/)

Vertex cover · qf-mod45-vcover-uncovered-edge-witness · candidate





## 51. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1], loop-free simple undirected edges=[] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v1], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(Yes, Yes, [v0, v1], [], Yes, [v1], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (Yes, Yes, [v0, v1], [], Yes, [v1], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0051/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 52. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v1, v2]] and unique candidate subset=[v0, v1, v2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v3], [v0, v1, v2], Yes, [v0, v1, v2, v3], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v3], [v0, v1, v2], Yes, [v0, v1, v2, v3], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0052/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 53. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3]] and unique candidate subset=[v0, v1, v2, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v4], [v0, v1, v2, v3], Yes, [v0, v1, v2, v3, v4], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v4], [v0, v1, v2, v3], Yes, [v0, v1, v2, v3, v4], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0053/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 54. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4]] and unique candidate subset=[v0, v1, v2, v3, v4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v5], [v0, v1, v2, v3, v4], Yes, [v0, v1, v2, v3, v4, v5], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v5], [v0, v1, v2, v3, v4], Yes, [v0, v1, v2, v3, v4, v5], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0054/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 55. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]] and unique candidate subset=[v0, v1, v2, v3, v4, v5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v6], [v0, v1, v2, v3, v4, v5], Yes, [v0, v1, v2, v3, v4, v5, v6], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v6], [v0, v1, v2, v3, v4, v5], Yes, [v0, v1, v2, v3, v4, v5, v6], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0055/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 56. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v7], [v0, v1, v2, v3, v4, v5, v6], Yes, [v0, v1, v2, v3, v4, v5, v6, v7], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v7], [v0, v1, v2, v3, v4, v5, v6], Yes, [v0, v1, v2, v3, v4, v5, v6, v7], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0056/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 57. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v8], [v0, v1, v2, v3, v4, v5, v6, v7], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v8], [v0, v1, v2, v3, v4, v5, v6, v7], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0057/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 58. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0058/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 59. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0059/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 60. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0060/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 61. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2], loop-free simple undirected edges=[[v0, v1]] and unique candidate subset=[v0, v1]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v2], [v0, v1], Yes, [v0, v1, v2], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v2], [v0, v1], Yes, [v0, v1, v2], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0061/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 62. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v0, v2]] and unique candidate subset=[v0, v1, v2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v3], [v0, v1, v2], Yes, [v0, v1, v2, v3], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v3], [v0, v1, v2], Yes, [v0, v1, v2, v3], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0062/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 63. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3]] and unique candidate subset=[v0, v1, v2, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v4], [v0, v1, v2, v3], Yes, [v0, v1, v2, v3, v4], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v4], [v0, v1, v2, v3], Yes, [v0, v1, v2, v3, v4], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0063/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 64. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4]] and unique candidate subset=[v0, v1, v2, v3, v4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v5], [v0, v1, v2, v3, v4], Yes, [v0, v1, v2, v3, v4, v5], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v5], [v0, v1, v2, v3, v4], Yes, [v0, v1, v2, v3, v4, v5], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0064/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 65. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]] and unique candidate subset=[v0, v1, v2, v3, v4, v5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v6], [v0, v1, v2, v3, v4, v5], Yes, [v0, v1, v2, v3, v4, v5, v6], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v6], [v0, v1, v2, v3, v4, v5], Yes, [v0, v1, v2, v3, v4, v5, v6], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0065/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 66. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v7], [v0, v1, v2, v3, v4, v5, v6], Yes, [v0, v1, v2, v3, v4, v5, v6, v7], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v7], [v0, v1, v2, v3, v4, v5, v6], Yes, [v0, v1, v2, v3, v4, v5, v6, v7], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0066/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 67. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v8], [v0, v1, v2, v3, v4, v5, v6, v7], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v8], [v0, v1, v2, v3, v4, v5, v6, v7], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0067/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 68. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0068/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 69. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0069/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 70. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9], [v0, v10]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0070/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 71. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2], loop-free simple undirected edges=[] and unique candidate subset=[v0, v1]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(Yes, Yes, [v0, v1, v2], [v0, v1], Yes, [v0, v1, v2], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (Yes, Yes, [v0, v1, v2], [v0, v1], Yes, [v0, v1, v2], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0071/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 72. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v0]] and unique candidate subset=[v0, v1, v2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v3], [v0, v1, v2], Yes, [v0, v1, v2, v3], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v3], [v0, v1, v2], Yes, [v0, v1, v2, v3], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0072/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 73. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v0]] and unique candidate subset=[v0, v1, v2, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v4], [v0, v1, v2, v3], Yes, [v0, v1, v2, v3, v4], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v4], [v0, v1, v2, v3], Yes, [v0, v1, v2, v3, v4], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0073/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 74. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v0]] and unique candidate subset=[v0, v1, v2, v3, v4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v5], [v0, v1, v2, v3, v4], Yes, [v0, v1, v2, v3, v4, v5], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v5], [v0, v1, v2, v3, v4], Yes, [v0, v1, v2, v3, v4, v5], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0074/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 75. Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks

Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).

View answer and explanation(No, null, [v6], [v0, v1, v2, v3, v4, v5], Yes, [v0, v1, v2, v3, v4, v5, v6], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)

Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v6], [v0, v1, v2, v3, v4, v5], Yes, [v0, v1, v2, v3, v4, v5, v6], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0075/)

Vertex cover · qf-mod45-vcover-empty-isolated-superset · candidate





## 76. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1]] and unique candidate subset=[l0_0, l0_1]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1], [[l0_0, [l0_1], No], [l0_1, [l0_0], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1], [[l0_0, [l0_1], No], [l0_1, [l0_0], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0076/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 77. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2]] and unique candidate subset=[l0_0, l0_1, l0_2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2], [[l0_0, [l0_1, l0_2], No], [l0_1, [l0_0, l0_2], No], [l0_2, [l0_0, l0_1], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2], [[l0_0, [l0_1, l0_2], No], [l0_1, [l0_0, l0_2], No], [l0_2, [l0_0, l0_1], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0077/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 78. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l0_3], [[l0_0, [l0_1, l0_2, l0_3], No], [l0_1, [l0_0, l0_2, l0_3], No], [l0_2, [l0_0, l0_1, l0_3], No], [l0_3, [l0_0, l0_1, l0_2], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3], [[l0_0, [l0_1, l0_2, l0_3], No], [l0_1, [l0_0, l0_2, l0_3], No], [l0_2, [l0_0, l0_1, l0_3], No], [l0_3, [l0_0, l0_1, l0_2], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0078/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 79. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l0_3, l0_4], [[l0_0, [l0_1, l0_2, l0_3, l0_4], No], [l0_1, [l0_0, l0_2, l0_3, l0_4], No], [l0_2, [l0_0, l0_1, l0_3, l0_4], No], [l0_3, [l0_0, l0_1, l0_2, l0_4], No], [l0_4, [l0_0, l0_1, l0_2, l0_3], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4], [[l0_0, [l0_1, l0_2, l0_3, l0_4], No], [l0_1, [l0_0, l0_2, l0_3, l0_4], No], [l0_2, [l0_0, l0_1, l0_3, l0_4], No], [l0_3, [l0_0, l0_1, l0_2, l0_4], No], [l0_4, [l0_0, l0_1, l0_2, l0_3], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0079/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 80. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0080/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 81. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0081/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 82. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6], [c0, l0_7]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0082/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 83. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6], [c0, l0_7], [c0, l0_8]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8], No], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8], No], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0083/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 84. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6], [c0, l0_7], [c0, l0_8], [c0, l0_9]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], No], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], No], [l0_9, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], No], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], No], [l0_9, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0084/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 85. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9, l0_10], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6], [c0, l0_7], [c0, l0_8], [c0, l0_9], [c0, l0_10]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9, l0_10]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_1, [l0_0, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_10, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_2, [l0_0, l0_1, l0_10, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_3, [l0_0, l0_1, l0_10, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_4, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_5, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], No], [l0_6, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], No], [l0_7, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], No], [l0_8, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], No], [l0_9, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_1, [l0_0, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_10, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_2, [l0_0, l0_1, l0_10, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_3, [l0_0, l0_1, l0_10, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_4, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_5, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], No], [l0_6, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], No], [l0_7, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], No], [l0_8, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], No], [l0_9, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0085/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 86. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l1_0, l1_1], [[l0_0, [l0_1, l1_0, l1_1], No], [l0_1, [l0_0, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l1_1], No], [l1_1, [l0_0, l0_1, l1_0], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1], [[l0_0, [l0_1, l1_0, l1_1], No], [l0_1, [l0_0, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l1_1], No], [l1_1, [l0_0, l0_1, l1_0], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0086/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 87. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1, l1_2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l1_0, l1_1], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l1_0, l1_1], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0087/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 88. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0088/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 89. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0089/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 90. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0090/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 91. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0091/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 92. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6], [c1, l1_7]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6, l1_7], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6, l1_7], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6, l1_7], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_7], No], [l1_7, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6, l1_7], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6, l1_7], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6, l1_7], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_7], No], [l1_7, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0092/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 93. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1]] and unique candidate subset=[l0_0, l0_1, l0_2, l1_0, l1_1]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l1_0, l1_1], No], [l0_1, [l0_0, l0_2, l1_0, l1_1], No], [l0_2, [l0_0, l0_1, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l0_2, l1_1], No], [l1_1, [l0_0, l0_1, l0_2, l1_0], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l1_0, l1_1], No], [l0_1, [l0_0, l0_2, l1_0, l1_1], No], [l0_2, [l0_0, l0_1, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l0_2, l1_1], No], [l1_1, [l0_0, l0_1, l0_2, l1_0], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0093/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 94. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2]] and unique candidate subset=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0094/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 95. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3]] and unique candidate subset=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0095/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 96. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3, l1_4], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]] and unique candidate subset=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0096/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 97. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5]] and unique candidate subset=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5], No], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5], No], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0097/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 98. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6]] and unique candidate subset=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0098/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 99. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, c1, l1_0, l1_1], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l1_0, l1_1]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0099/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate





## 100. Inclusion-Minimal Covers Need Not Have Minimum Cardinality

Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, c1, l1_0, l1_1, l1_2], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1], [c1, l1_2]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).

minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum.

View answer and explanation([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1, l1_2], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)

Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1, l1_2], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r07-vertex-cover-0100/)

Vertex cover · qf-mod45-vcover-minimal-vs-minimum-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261114-r07-vertex-cover-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261114-r07-vertex-cover-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261114-r07-vertex-cover-foundations/validation-report.json)