Imed Boudabbous

SUBTOURNAMENTS ISOMORPHIC TO W 5 OF AN INDECOMPOSABLE TOURNAMENT

Abstract. WeconsideratournamentT =(V,A). ForeachsubsetX ofV isassociatedthesubtournamentT(X)=(X,A∩(X×X))ofT inducedby X. We saythat atournament T ′ embeds intoa tournament T whenT ′ is isomorphic to a subtournament of T. Otherwise, we say that TomitsT ′ . Asubset X ofV isaclanofT providedthat fora,b∈ X andx∈ V\X,(a,x)∈ Aifandonlyif(b,x)∈ A. Forexample,∅,{x}(x∈ V)andV areclansofT,calledtrivialclans. Atournamentisindecomposableifall its clans aretrivial. In 2003, B.J. Latka characterized the class Tof indecomposable tournaments omitting a certain tournament W 5 on 5vertices. In the case of an indecomposable tournament T, we willstudythe set W 5 (T) of vertices x∈ V for which there exists a subset X of Vsuchthatx∈ X andT(X)isisomorphictoW 5 . Weprovethefollowing:foranyindecomposable tournament T,ifT 6∈ T,then |W 5 (T)|≥|V| −2and |W 5 (T)|≥|V | −1if|V| iseven. Bygivingexamples,wealsoverifythatthisstatement isoptimal. 1. IntroductionA tournament T = (V (T),A(T)) (or (V,A)) consists of a finite set V ofvertices together with a set A of ordered pairs of distinct vertices, called arcs,such that for all x 6= y ∈ V , (x,y) ∈ A if and only if (y,x) 6∈A. The order ofT, denoted by |T |, is the cardinality of V (T). Given a tournament T = (V,A),with each subset X of V is associated the subtournament T(X) = (X,A ∩(X × X)) of T induced by X. For X ⊆ V (resp. x ∈ V ), the subtournamentT(V \X) (resp. T(V \{x})) is denoted by T −X (resp. T −x). Let T = (V,A)and T

Ultracritical and hypercritical binary structures

A binary structure is an arc-coloured complete digraph, without loops, and with exactly two coloured arcs (u,v) and (v,u) between distinct vertices u and v. Graphs, digraphs and partial orders are all examples of binary structures. Let B be a binary structure. With each subset W of the vertex set V(B) of B we associate the binary substructure B[W] of B induced by W. A subset C of V(B) is a clan of B if for any c,d@?C and v@?V(B)@?C, the arcs (c,v) and (d,v) share the same colour and similarly for (v,c) and (v,d). For instance, the vertex set V(B), the empty set and any singleton subset of V(B) are clans of B. They are called the trivial clans of B. A binary structure is primitive if all its clans are trivial. With a primitive and infinite binary structure B we associate a criticality digraph (in the sense of [11]) defined on V(B) as follows. Given v w@?V(B), (v,w) is an arc of the criticality digraph of B if v belongs to a non-trivial clan of B[V(B)@?{w}]. A primitive and infinite binary structure B is finitely critical if B[V(B)@?F] is not primitive for each finite and non-empty subset F of V(B). A finitely critical binary structure B is hypercritical if for every v@?V(B), B[V(B)@?{v}] admits a non-trivial clan C such that |V(B)@?C|>=3 which contains every non-trivial clan of B[V(B)@?{v}]. A hypercritical binary structure is ultracritical whenever its criticality digraph is connected. The ultracritical binary structures are studied from their criticality digraphs. Then a characterization of the non-ultracritical but hypercritical binary structures is obtained, using the generalized quotient construction originally introduced in [1].