Les tournois (-1)-critiques

TL;DR

This paper characterizes (-1)-critical tournaments, a class of indecomposable tournaments with exactly one non-critical vertex, extending the understanding of tournament structures beyond critical tournaments.

Contribution

It introduces and characterizes (-1)-critical tournaments, detailing their existence and enumeration for odd cardinalities greater than or equal to 7.

Findings

Existence of 3m-15 (-1)-critical tournaments for each odd m ≥ 7.

Characterization of tournaments with exactly one non-critical vertex.

Extension of critical tournament theory to (-1)-critical tournaments.

Abstract

Given a tournament T=(V,A), a subset X of V is an interval of T provided that for any a, b\in X and x\in V-X, (a,x) \in A if and only if (b,x)\in A. For example, \emptyset, \{x\} (x\in V) and V are intervals of T, called trivial intervals. A tournament, all the intervals of which are trivial, is indecomposable; otherwise, it is decomposable. A vertex x of an indecomposable tournament is critical if T-x is decomposable. In 1993, J.H. Schmerl and W.T. Trotter characterized the tournaments, all the vertices of which are critical, called critical tournaments. The cardinality of these tournaments is odd. Given an odd integer m \geq 5, there exist three critical tournaments of cardinality . and there are exactly three critical tournaments for each such a cardinality. In this article, we characterize the tournaments which admit a single non critical vertex, that we call (-1)-critical tournaments. The cardinality of these tournaments is odd. Given an odd integer m \geq 7, there exist 3m-15 (-1)-critical tournaments of cardinality m.