bibkey: belkhechineboudabbous2010indecomposable authors: Houmem Belkhechine and Imed Boudabbous year: 2010 title: Indecomposable tournaments and their indecomposable subtournaments on 5 and 7 vertices doi: null url: https://arxiv.org/abs/1007.3049v1 claim: The paper recalls the classical odd-order W family of critical tournaments and uses related deletion and surviving-pair arguments in its analysis of indecomposable subtournaments. strata_touched:

• D5/S3/ConceptDynamics/GraphIrregularity/LinkIrregularTournamentExistence license: citation-only triage: anchor

Indecomposable tournaments and small subtournaments

Page 3 defines the classical odd tournament family W_(2n+1) as a transitive chain together with one vertex whose incident orientations alternate. The arguments culminating on page 11 use related deletion and surviving-pair structure to locate distinguished vertices in critical tournaments.

At odd orders at least seven, the repository construction uses this classical W family; at even orders at least eight, it adds one sink to the preceding odd-order W. Both have the same alternating-pivot relation. In the repository proof, after deleting any chain vertex, further deletion of the pivot leaves a transitive tournament. Further deletion of any other vertex leaves an intact triangle using the pivot and one of three disjoint adjacent chain pairs. Hence the pivot is unique in each chain-deleted card under this further-deletion test; a card isomorphism fixes it, then fixes the ranks of the surviving transitive chain. The smaller deleted rank has opposite pivot edge parity in the two cards. The pivot-deleted card itself is transitive, unlike each chain-deleted card. The cited paper does not state this link-irregularity conclusion.

Verified locator

• arXiv abstract and version record: https://arxiv.org/abs/1007.3049v1
• PDF: https://arxiv.org/pdf/1007.3049v1