J. Korean Math. Soc. 48 (2011), No. 3, pp. 599–626 DOI 10.4134/JKMS.2011.48.3.599 HEREDITARY HEMIMORPHY OF {-k}-HEMIMORPHIC TOURNAMENTS FOR k ≥ 5 Moncef Bouaziz, Youssef Boudabbous, and Nadia El Amri Abstract. Let T =(V,A) be a tournament. With every subset X of V is associated the subtournament T [X]=(X, A ∩ (X × X)) of T , induced by X. The dual of T , denoted by T * , is the tournament obtained from T by reversing all its arcs. Given a tournament T ′ =(V,A ′ ) and a non- negative integer k, T and T ′ are {-k}-hemimorphic provided that for all X ⊂ V , with |X| = k, T [V -X] and T ′ [V -X] or T * [V -X] and T ′ [V -X] are isomorphic. The tournaments T and T ′ are said to be hereditarily hemimorphic if for all subset X of V , the subtournaments T [X] and T ′ [X] are hemimorphic. The purpose of this paper is to establish the hereditary hemimorphy of the {-k}-hemimorphic tournaments on at least k +7 vertices, for every k ≥ 5. 1. Introduction A tournament T =(V,A) (or (V (T ),A(T ))) consists of a finite vertex set V with an arc set A of ordered pairs of distinct vertices, satisfying: for x, y ∈ V , with x ̸= y,(x, y) ∈ A if and only if (y,x) ̸∈ A, in this case we write x → y. Given two sets of vertices A and B, write A → B to mean that there is an arc from any element of A to any element of B. For singletons, just write a → B for {a}→ B and with A → b as well. The cardinality of T is that of V . This car- dinality |V | is also denoted by |T |. For each x ∈ V , we denote by N + (x) (resp. N - (x)) the set {y ∈ V :(x, y) ∈ A} (resp. {y ∈ V :(y,x) ∈ A}). The score of a vertex x (in T ), denoted by s T (x), is the cardinality of N + (x). The dual of T is the tournament T * =(V,A * ) defined by: for all x, y ∈ V ,(y,x) ∈ A * if and only if (x, y) ∈ A. With every subset X of V is associated the subtournament T [X]=(X, A ∩ (X × X)) of T induced by X. The subtournament T [V - X] is also denoted by T - X. For each x ∈ V , the subtournament T -{x} is denoted by T - x. Say that a set W of vertices satisfies a property if the subtourna- ment T [W ] enjoys it. A transitive tournament or a total order is a tournament T such that for all x, y, z ∈ V (T ), if (x, y) ∈ A(T ) and (y,z) ∈ A(T ), then Received February 10, 2010. 2010 Mathematics Subject Classification. 05C20, 05C60. Key words and phrases. tournament, isomorphy, hereditary isomorphy, hemimorphy, hereditary hemimorphy. c ⃝2011 The Korean Mathematical Society 599