Given a tournament T=(V,A), a subset X of $V$ is an interval of T provided that for every 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 critical tournament is an indecomposable tournament T of cardinality $\geq 5$ such that for any vertex x of T, the tournament T-x is decomposable. The critical tournaments are of odd cardinality and for all $n \geq 2$ there are exactly three critical tournaments on 2n+1 vertices denoted by $T_{2n+1}$, $U_{2n+1}$ and $W_{2n+1}$. The tournaments $T_{5}$, $U_{5}$ and $W_{5}$ are the unique indecomposable tournaments on 5 vertices. We say that a tournament T embeds into a tournament T' when T is isomorphic to a subtournament of T'. A diamond is a tournament on 4 vertices admitting only one interval of cardinality 3. We prove the following theorem: if a diamond and $T_{5}$ embed into an indecomposable tournament T, then $W_{5}$ and $U_{5}$ embed into T. To conclude, we prove the following: given an indecomposable tournament T, with $\mid\!V(T)\!\mid \geq 7$, T is critical if and only if the indecomposable subtournaments on 7 vertices of T are isomorphic to one and only one of the tournaments $T_{7}$, $U_{7}$ and $W_{7}$.

Indecomposable tournaments and their indecomposable subtournaments on 5 and 7 vertices.

In 1990, Thomas Schwartz proposed the conjecture that every nonempty tournament has a unique minimal τ -retentive set (τ stands for tournament equilibrium set). A weak variant of Schwartz's Conjecture was recently proposed by Felix Brandt. However, both conjectures were disproved very recently by two counterexamples. In this paper, we prove sufficient conditions for infinite classes of tournaments that satisfy Schwartz's Conjecture and Brandt's Conjecture. Moreover, we prove that τ can be calculated in polynomial time in several infinite classes of tournaments. Furthermore, our results reveal some structures that are forbidden in every counterexample to Schwartz's Conjecture.

/pdf/when-does-schwartz-conjecture-hold-47vyjju6s8.pdf

When does Schwartz conjecture hold

Decomposability and co-modular indices of tournaments

In this thesis we prove a variety of theorems on tournaments. A \emph{prime} tournament is a tournament $G$ such that there is no $X \subseteq V(G)$, $1 < |X| < |V(G)|$, such that for every vertex $v \in V(G) \minus X$, either $v \ra x$ for all $x \in X$ or $x \ra v$ for all $x \in X$. First, we prove that given a prime tournament $G$ which is not in one of three special families of tournaments, for any prime subtournament $H$ of $G$ with $5 \le |V(H)| < |V(G)|$ there exists a prime subtournament of $G$ with $|V(H)| + 1$ vertices that has a subtournament isomorphic to $H$. We next prove that for any two cyclic triangles $C$, $C^\prime$ in a prime tournament $G$, there is a sequence of cyclic triangles $C_1,...,C_n$ such that $C_1 = C$, $C_n = C^\prime$, and $C_i$ shares an edge with $C_{i+1}$ for all $1 \le i \le n-1$. Next, we consider what we call \emph{matching tournaments}, tournaments whose vertices can be ordered in a horizontal line so that every vertex is the head or tail of at most one edge that points right-to-left. We determine the conditions under which a tournament can have two different orderings satisfying the above conditions. We also prove that there are infinitely many minimal tournaments that are not matching tournaments. Finally, we consider the tournaments $K_n$ and $K_n^\ast$, which are obtained from the transitive tournament with $n$ vertices by reversing the edge from the second vertex to the last vertex and from the first vertex to the second-to-last vertex, respectively. We prove a structure theorem describing tournaments which exclude $K_n$ and $K_n^\ast$ as subtournaments.

Various Theorems on Tournaments

Let U5 be the tournament with vertices v1, ', v5 such that v2i¾?v1, and vii¾?vj if j-ii¾?1, 2mod5 and {i,j}i¾?{1,2}. In this article, we describe the tournaments that do not have U5 as a subtournament. Specifically, we show that if a tournament G is "prime"-that is, if there is no subset Xi¾?VG, 1<|X|<|VG|, such that for all v∈VGi¾?X, either vi¾?x for all x∈X or xi¾?v for all x∈X-then G is U5-free if and only if either G is a specific tournament Tn or VG can be partitioned into sets X, Y, Z such that Xi¾?Y, Yi¾?Z, and Zi¾?X are transitive. From the prime U5-free tournaments we can construct all the U5-free tournaments. We use the theorem to show that every U5-free tournament with n vertices has a transitive subtournament with at least nlog32 vertices, and that this bound is tight.

Structure Theorem for U5-free Tournaments

Indecomposable tournaments and their indecomposable subtournaments on 5 and 7 vertices

Sous-tournois isomorphes à dans un tournoi indécomposable

Morphologie des tournois (−1)-critiques

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