Jamel Dammak

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.

Boolean sum of graphs and reconstruction up to complementation

Abstract. Let V be a set of cardinality v (possibly infinite). Two graphs G and with vertex set V are isomorphic up to complementation if is isomorphic to G or to the complement of G. Let k be a non-negative integer. The graphs G and are k-hypomorphic up to complementation if for every k-element subset K of V, the induced subgraphs and are isomorphic up to complementation. A graph G is k-reconstructible up to complementation if every graph which is k-hypomorphic to G up to complementation is in fact isomorphic to G up to complementation. We prove that a graph G has this property provided that . Moreover, under these conditions, if or , then G and are the only graphs k-hypomorphic to G up to complementation. A description of pairs of graphs with the same 3-homogeneous subsets is a key ingredient in our proof.

Graphic topology on tournaments

Abstract Alexandroff spaces are the topological spaces in which the intersection of arbitrary many open sets is open. Let T be an indecomposable tournament. In this paper, first, we associate a trivial topology to T. Then we define another topology on T, called the graphic topology of T, and we show that it is an Alexandroff topology. Our motivation is to investigate some properties of this topology.