Youssef Boudabbous

Indecomposability and duality of tournaments

Abstract Let T =( V , A ) be a tournament. A subset X of V is an interval of T provided that for a , b ∈ X and for x ∈ V − X , ( a , x )∈ A if and only if ( b , x )∈ A . For example, ∅,{ x }, where x ∈ V , and V are intervals of T, called trivial intervals. A tournament is said to be indecomposable if all of its intervals are trivial. In another respect, with each tournament T =( V , A ) is associated the dual tournament T ★ =( V , A ★ ) defined as: for x , y ∈ V ,( x , y )∈ A ★ if ( y , x )∈ A . A tournament T is said to be self-dual if T and T ★ are isomorphic. The paper characterizes the finite tournaments T =( V , A ) fulfilling: for every proper subset X of V, if the subtournament T ( X ) of T is indecomposable, then T ( X ) is self-dual. The corollary obtained is: given a finite and indecomposable tournament T =( V , A ), if T is not self-dual, then there is a subset X of V such that 6⩽| X |⩽10 and such that T ( X ) is indecomposable without being self-dual. An analogous examination is made in the case of infinite tournaments. The paper concludes with an introduction of a new mode of reconstruction of tournaments from their proper and indecomposable subtournaments.

La 5-reconstructibilité et L'indécomposabilité Des Relations Binaires

A binary relation is (?k)-reconstructible, if it is determined up to isomorphism by its restriction to subsets of at most k elements. In 8], Lopez has shown that finite binary relations are (? 6)-reconstructible. To prove that the value 6 of its result, is optimal, Lopez 3], associates to all finite binary relation, an infinity of finite extensions, that are not (? 5)-reconstructible. These extensions are obtained from the relations given, by creation of intervals. Rosenberg has then asked if all finite binary relations, not (? 5)-reconstructible, were obtained by the same process. In this paper, we give an affirmative answer to the question, by characterizing finite binary relations that are not (? 5)-reconstructible. We deduce the 5-reconstructibility of finite indecomposable binary relations, of at least 9 elements. We extend then this last result to the binary multirelations.

Sur la détermination d'une relation binaire à partir d'informations locales

This paper deals with the problematic aspect of the reconstruction of binary relations: it includes all the questions raising the possibility or impossibility to determine a structure by gathering given substructures. It is the continuation of three studies: the first made by G. Lopez [9] in 1972 about the determination of a binary relation through the types of isomorphism of its restrictions, the second made by K. B. Reid and C. Thomassen [15] in 1976 about the strongly self-complementary tournaments (every subtournament is self-complementary), the third made by C. Thomassen [16] in 1989 about the cycle space of a tournament. In the second section, we use the notion of class of difference (which was introduced in [9]) to extend a study made in [16] to binary relations. Then, in the third section, we improve the result of this last study in the case of the tournaments. After noticing that the result of [15] inferred itself naturally from the approach developed in [9], we extend, in the fourth section, the study made in [15] to binary relations.

Cut-primitive directed graphs versus clan-primitive directed graphs

Abstract Given a directed graph G = (V,A), a subset X of V is a clan of G provided that for a, b ∈ X and x ∈ V \ X, (a, x) ∈ A if and only if (b, x) ∈ A, and similarly for (x, a) and (x, b). For instance, ∅, V and {x}, where x ∈ V, are clans of G, called trivial. A directed graph is clan-primitive if all its clans are trivial. Given a directed graph G = (V,A), a subset X of V is a cut of G if X and V \ X are clans of G. For example, ∅ and V are cuts of G, called trivial. A directed graph is cut-primitive if all its cuts are trivial. Ehrenfeucht and Rozenberg [Theoret. Comput. Sci. 70: 343–358, 1990] proved: Given a clan-primitive directed graph G = (V,A), if X is a subset of V such that |X| ≥ 3, |V \ X| ≥ 2 and G[X] is clan-primitive, then there are x ≠ y ∈ V \ X such that G[X ∪ {x, y}] is clan-primitive. We show: Given a clan-primitive directed graph G = (V,A), if X is a proper subset of V such that |X| ≥ 3 and G[X] is cut-primitive, then there are x, y ∈ V \ X such that G[X ∪ {x, y}] is cut-primitive and {x}, {y} are maximal proper clans of G[X ∪ {x, y}].

Reconstructible and Half-Reconstructible Tournaments: Application to Their Groups of Hemimorphisms

Let T and T1 be tournaments with n elements, E a basis for T, E′ a basis for T′, and k ≥ 3 an integer. The dual of T is the tournament T” of basis E defined by T(x, y) = T(y, x) for all x, y e E. A hemimorphism from T onto T′ is an isomorphism from T onto T” or onto T. A k-hemimorphism from T onto T′ is a bijection f from E to E′ such that for any subset X of E of order k the restrictions T/X and T1/f(X) are hemimorphic. The set of hemimorphisms of T onto itself has group structure, this group is called the group of hemimorphisms of T. In this work, we study the restrictions to n – 2 elements of a tournament with n elements. In particular, we prove: Let k ≥ 3 be an integer, T a tournament with n elements, where n ≥ k + 5. Then the following statements are equivalent: (i) All restrictions of T to subsets with n – 2 elements are k-hemimorphic. (ii) All restrictions of T to subsets with n – 2 elements are 3-hemimorphic. (iii) All restrictions of T to subsets with n – 2 elements are hemimorphic. (iv) All restrictions of T to subsets with n – 2 elements are isomorphic, (v) Either T is a strict total order, or the group of hemimorphisms of T is 2-homogeneous.

The {-2,-1}-selfdual and decomposable tournaments

Abstract We only consider finite tournaments. The dual of a tournament is obtained by reversing all the arcs. A tournament is selfdual if it is isomorphic to its dual. Given a tournament T, a subset X of V (T) is a module of T if each vertex outside X dominates all the elements of X or is dominated by all the elements of X. A tournament T is decomposable if it admits a module X such that 1 < |X| < |V (T)|. We characterize the decomposable tournaments whose subtournaments obtained by removing one or two vertices are selfdual. We deduce the following result. Let T be a non decomposable tournament. If the subtournaments of T obtained by removing two or three vertices are selfdual, then the subtournaments of T obtained by removing a single vertex are not decomposable. Lastly, we provide two applications to tournaments reconstruction.