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

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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