La 5-reconstructibilité et L’indécomposabilité Des Relations Binaires

Abstract

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.

The minimal non-(⩽k)-reconstructible relations

“…. , t 2n ) all of whose neutral edges are full, let R be a relation obtained from S by almost-dilating t 1 …”

“…The class MD k includes in particular the class of the indecomposable ( k)-deformable relations. The determination of the class MD 5 is obtained in [1] and we characterize here classes MD k for k = 2. The determination of MD 4 follows basically from the properties of the ( 4)-hypomorphy [13].…”

Les tournois (⩽k)-demi-reconstructibles pour k⩽6

“…Lopez [9] a établi la ( 6)-reconstruction des tournois, de plus il a montré que la valeur 6 est optimale. Ensuite, le problème de la ( k)-reconstructibilité pour k ∈ {3, 4, 5} a été étudié dans [1,3]. En particulier, pour k = 3, Y. Boudabbous et G. Lopez [3] ont obtenu le résultat suivant :…”

“…Lopez [9] established that the tournaments are ( 6)-reconstructible and showed that the value 6 is the best possible. The ( k)-reconstructibility problem for k ∈ {3, 4, 5} was studied in [1,3]. In particular, for k = 3, Y. Boudabbous and G. Lopez [3] showed that a tournament is ( 3)-reconstructible if and only if all its intervals are self-dual.…”

Les graphes 2-reconstructibles indécomposables

“…G. Lopez a prouvé [6] que tout graphe ayant au moins 7 sommets est 6-reconstructible. La classification des graphes k-reconstructible pour k ∈ {4, 5} a été obtenu par Y. Boudabbous [1] et pour k = 3 par Y. Boudabbous et G. Lopez [2].…”