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

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

([], [[[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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

(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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).

([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).