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- W2082674173 abstract "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∈ VX,(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" @default.
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- W2082674173 date "2012-11-01" @default.
- W2082674173 modified "2023-10-01" @default.
- W2082674173 title "SUBTOURNAMENTS ISOMORPHIC TO W5OF AN INDECOMPOSABLE TOURNAMENT" @default.
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- W2082674173 doi "https://doi.org/10.4134/jkms.2012.49.6.1259" @default.
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