Houmem Belkhechine

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

Criticality of switching classes of reversible 2-structures labeled by an Abelian group

Abstract Let V be a finite vertex set and let (𝔸, +) be a finite abelian group. An 𝔸-labeled and reversible 2-structure defined on V is a function g : (V × V) \ {(v, v) : v ∈ V } → 𝔸 such that for distinct u, v ∈ V, g(u, v) = −g(v, u). The set of 𝔸-labeled and reversible 2-structures defined on V is denoted by ℒ(V, 𝔸). Given g ∈ ℒ(V, 𝔸), a subset X of V is a clan of g if for any x, y ∈ X and v ∈ V \ X, g(x, v) = g(y, v). For example, ∅, V and {v} (for v ∈ V) are clans of g, called trivial. An element g of ℒ(V, 𝔸) is primitive if |V | ≥ 3 and all the clans of g are trivial. The set of the functions from V to 𝔸 is denoted by 𝒮(V, 𝔸). Given g ∈ ℒ(V, 𝔸), with each s ∈ 𝒮(V, 𝔸) is associated the switch gs of g by s defined as follows: given distinct x, y ∈ V, gs(x, y) = s(x) + g(x, y) − s(y). The switching class of g is {gs : s ∈ 𝒮(V, 𝔸)}. Given a switching class 𝔖 ⊆ ℒ(V, 𝔸) and X ⊆ V, {g↾(X × X)\{(x,x):x∈X} : g ∈ 𝔖} is a switching class, denoted by 𝔖[X]. Given a switching class 𝔖 ⊆ ℒ(V, 𝔸), a subset X of V is a clan of 𝔖 if X is a clan of some g ∈ 𝔖. For instance, every X ⊆ V such that min(|X|, |V \ X|) ≤ 1 is a clan of 𝔖, called trivial. A switching class 𝔖 ⊆ ℒ(V, 𝔸) is primitive if |V | ≥ 4 and all the clans of 𝔖 are trivial. Given a primitive switching class 𝔖 ⊆ ℒ(V, 𝔸), 𝔖 is critical if for each v ∈ V, 𝔖 − v is not primitive. First, we translate the main results on the primitivity of 𝔸-labeled and reversible 2-structures in terms of switching classes. For instance, we prove the following. For a primitive switching class 𝔖 ⊆ ℒ(V, 𝔸) such that |V | ≥ 8, there exist u, v ∈ V such that u ≠ v and 𝔖[V \ {u, v}] is primitive. Second, we characterize the critical switching classes by using some of the critical digraphs described in [Y. Boudabous and P. Ille, Indecomposability graph and critical vertices of an indecomposable graph, Discrete Math. 309 (2009) 2839–2846].