Endpoint-Disjoint Matchings, Maximality and Larger-Witness Distinctions
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batch-20261014-advance-20261101-r05-graph-matching-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.
1. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 1], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 1], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 1], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
2. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 2], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
3. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 2], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
4. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 3], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
5. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 2], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
6. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 3], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
7. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 3], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
8. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 4], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 4], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
9. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 2], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
10. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 3], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
11. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 3], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
12. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 4], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 4], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
13. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 3], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
14. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 4], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 4], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
15. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 4], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 4], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
16. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 5], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 5], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 5], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
17. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 1], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 1], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 1], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
18. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 2], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
19. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 2], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
20. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 3], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
21. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 2], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
22. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 3], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
23. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 3], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
24. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 4], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 4], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
25. Selected Edges Form a Matching Only When Endpoints Are Disjoint
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.
View answer and explanation
([0, 2], [[0, 1], [2, 3]], [], Yes)
Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-audit · candidate
26. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 1], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
27. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
28. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
29. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
30. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
31. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
32. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
33. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
34. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
35. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
36. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
37. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
38. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
39. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
40. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
41. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 5], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
42. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 1], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
43. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
44. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
45. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
46. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
47. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
48. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
49. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
50. List the Whole Matched and Unmatched Vertex Sets
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.
View answer and explanation
([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)
Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-matched-vertices · candidate
51. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 1], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
52. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
53. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
54. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
55. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
56. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
57. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
58. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
59. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
60. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
61. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
62. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
63. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
64. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
65. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
66. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 5], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [], Yes, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
67. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 1], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [2], No, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [2], No, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
68. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [3], No, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [3], No, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
69. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [3], No, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [3], No, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
70. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [4], No, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [4], No, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
71. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [3], No, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [3], No, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
72. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [4], No, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [4], No, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
73. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [4], No, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [4], No, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
74. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [5], No, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [5], No, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
75. A Maximal Matching May Still Have Smaller Cardinality than Another Matching
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.
View answer and explanation
(Yes, [3], No, maximal_not_necessarily_maximum)
A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [3], No, maximal_not_necessarily_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximality-audit · candidate
76. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 1], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 1], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 1], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
77. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
78. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
79. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
80. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
81. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
82. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
83. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 4], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 4], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 4], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
84. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
85. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[1, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [1, 2], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [1, 2], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
86. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
87. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[1, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [1, 3], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [1, 3], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
88. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
89. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[1, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [1, 3], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [1, 3], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
90. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 4], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 4], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 4], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
91. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[1, 4], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [1, 4], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [1, 4], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
92. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 1], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 1], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 1], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
93. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
94. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
95. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
96. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
97. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
98. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
99. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 4], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 4], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 4], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
100. A Supplied Larger Valid Matching Refutes Maximum Cardinality
Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.
View answer and explanation
([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum)
Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).
Reviewed step-by-step derivation
Graph matching · qf-mod32-matching-maximum-distinction · candidate
Canonical question JSONL · English JSONL · Validation report