# Full-Vertex Connectivity, Acyclicity, Edge Counts and Unweighted Boundaries

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

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

batch-20261014-advance-20261104-r06-spanning-tree-foundations · 2026-10-14 · 100 questions

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

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

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



## 1. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 3], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 1, 2, 3], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 2, 3], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0001/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 2. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 2, 3, 4], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 3, 4], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0002/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 3. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 2, 3, 4], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 3, 4], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0003/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 4. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 3, 4, 5], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 3, 4, 5], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0004/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 5. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 1, 3, 4], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 3, 4], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0005/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 6. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 2, 4, 5], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 4, 5], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0006/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 7. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 2, 4, 5], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 4, 5], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0007/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 8. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 3, 5, 6], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 3, 5, 6], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0008/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 9. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 1, 3, 4], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 3, 4], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0009/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 10. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 2, 4, 5], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 4, 5], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0010/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 11. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 2, 4, 5], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 4, 5], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0011/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 12. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 3, 5, 6], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 3, 5, 6], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0012/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 13. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 1, 4, 5], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 4, 5], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0013/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 14. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 2, 5, 6], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 5, 6], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0014/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 15. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 2, 5, 6], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 5, 6], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0015/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 16. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 6, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 3, 6, 7], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 3, 6, 7], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0016/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 17. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 1, 2, 4], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 2, 4], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0017/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 18. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 2, 3, 5], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 3, 5], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0018/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 19. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 2, 3, 5], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 3, 5], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0019/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 20. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 3, 4, 6], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 3, 4, 6], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0020/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 21. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 1, 3, 5], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 3, 5], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0021/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 22. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 2, 4, 6], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 4, 6], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0022/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 23. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 2, 4, 6], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 2, 4, 6], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0023/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 24. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 3, 5, 7], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 3, 5, 7], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0024/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 25. A Spanning Tree Uses the Entire Parent Vertex Set

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree).

View answer and explanation([0, 1, 3, 5], Yes, Yes, Yes, Yes)

Read the selected parent edges and report all connectivity and acyclicity obligations separately. A disconnected candidate parent has no spanning tree; a failed selection does not refute existence for another selection in a connected parent. The exact result, retaining the requested order, is ([0, 1, 3, 5], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0025/)

Spanning tree · qf-mod35-spantree-audit · candidate





## 26. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 3], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0026/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 27. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0027/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 28. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0028/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 29. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0029/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 30. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0030/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 31. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0031/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 32. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0032/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 33. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0033/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 34. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0034/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 35. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0035/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 36. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0036/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 37. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0037/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 38. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0038/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 39. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0039/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 40. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0040/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 41. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 6, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0041/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 42. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0042/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 43. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0043/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 44. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0044/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 45. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0045/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 46. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0046/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 47. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0047/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 48. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0048/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 49. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0049/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 50. Counting n−1 Edges Alone Does Not Establish a Tree

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).

View answer and explanation(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)

Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0050/)

Spanning tree · qf-mod35-spantree-n-minus-one-certificate · candidate





## 51. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(1)={0,…,0}, canonical sorted edges=[], and distinct parent-edge selection=[], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(1, [], Yes, Yes)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (1, [], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0051/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 52. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 3], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 1, 2, 3], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 1, 2, 3], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0052/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 53. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 2, 3, 4], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 3, 4], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0053/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 54. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 2, 3, 4], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 3, 4], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0054/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 55. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 3, 4, 5], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 3, 4, 5], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0055/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 56. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 1, 3, 4], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 1, 3, 4], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0056/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 57. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 2, 4, 5], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 4, 5], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0057/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 58. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 2, 4, 5], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 4, 5], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0058/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 59. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 3, 5, 6], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 3, 5, 6], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0059/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 60. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 1, 3, 4], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 1, 3, 4], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0060/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 61. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 2, 4, 5], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 4, 5], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0061/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 62. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 2, 4, 5], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 4, 5], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0062/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 63. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 3, 5, 6], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 3, 5, 6], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0063/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 64. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 1, 4, 5], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 1, 4, 5], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0064/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 65. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 2, 5, 6], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 5, 6], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0065/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 66. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 2, 5, 6], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 5, 6], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0066/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 67. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 6, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 3, 6, 7], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 3, 6, 7], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0067/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 68. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 1, 2, 4], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 1, 2, 4], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0068/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 69. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 2, 3, 5], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 3, 5], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0069/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 70. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 2, 3, 5], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 3, 5], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0070/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 71. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 3, 4, 6], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 3, 4, 6], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0071/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 72. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 1, 3, 5], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 1, 3, 5], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0072/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 73. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 2, 4, 6], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 4, 6], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0073/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 74. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 2, 4, 6], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 2, 4, 6], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0074/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 75. Recognize the Single-Vertex Empty-Edge Tree without Assuming It

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_indices, spanning_tree, one_vertex_empty_selection).

View answer and explanation(5, [0, 3, 5, 7], Yes, No)

The final flag checks whether the actual carrier has one vertex and the selection is empty. That case is a valid spanning tree. For a larger carrier the flag is No even when a nonempty selection forms a spanning tree; it is not a mislabeled single-vertex theorem. The exact result, retaining the requested order, is (5, [0, 3, 5, 7], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0075/)

Spanning tree · qf-mod35-spantree-one-vertex · candidate





## 76. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 3], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0076/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 77. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0077/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 78. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0078/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 79. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0079/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 80. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0080/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 81. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0081/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 82. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0082/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 83. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0083/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 84. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0084/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 85. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0085/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 86. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0086/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 87. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0087/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 88. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 1, 4, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0088/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 89. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0089/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 90. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 2, 5, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0090/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 91. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]], and distinct parent-edge selection=[0, 3, 6, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0091/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 92. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 2, 4], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0092/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 93. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0093/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 94. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0094/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 95. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0095/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 96. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0096/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 97. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0097/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 98. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 4, 6], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0098/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 99. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 3, 5, 7], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0099/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate





## 100. An Unweighted Tree Audit Withholds a Weighted-Minimum Claim

Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [1, 2], [1, 4], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 1, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (spanning_tree, weighted_minimum_claim, boundary).

The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights.

View answer and explanation(Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim)

Report actual spanning-tree validity, then No for the absent weighted-minimum claim. This is not a statement that the tree cannot be minimum under some separately supplied weights. No weighted optimization problem was specified. The exact result, retaining the requested order, is (Yes, No, unweighted_spanning_tree_not_minimum_spanning_tree_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r06-spanning-tree-0100/)

Spanning tree · qf-mod35-spantree-non-mst-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261104-r06-spanning-tree-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261104-r06-spanning-tree-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261104-r06-spanning-tree-foundations/validation-report.json)