Two {4,n-3}-isomorphic n-vertex digraphs are hereditarily isomorphic

Abstract

Let D and D be two digraphs with the same vertex set V, and let F be a set of positive integers. The digraphs D and D are hereditarily isomorphic whenever the (induced) subdigraphs D[X] and D [X] are isomorphic for each nonempty vertex subset X. They are F-isomorphic if the subdigraphs D[X] and D [X] are isomorphic for each vertex subset X with | X |∈ F. In this paper, we prove that if D and D are two {4, n − 3}-isomorphic n-vertex digraphs, where n ≥ 9, then D and D are hereditarily isomorphic. As a corollary, we obtain that given integers k and n with 4 ≤ k ≤ n − 6, if D and D are two {n − k}-isomorphic n-vertex digraphs, then D and D are hereditarily isomorphic. To the memory of my dear master Gérard LOPEZ who taught me and gave me the passion of reconstruction. With all my gratitude and admiration. 1. Introduction All digraphs mentioned here are finite, and have no loops and no multiple edges. Thus a digraph (or directed graph) D consists of a nonempty and finite set V(D) of vertices with a collection E(D) of ordered pairs of distinct vertices, called the set of edges of D. Such a digraph is denoted by (V(D), E(D)). We recall the basic notions of the reconstruction problem in the theory of relations what we apply to the case of digraphs. Consider two digraphs D and D on the same vertex set V with |V| = n ≥ 1, and let k be a positive integer. The digraphs D and D are hereditarily isomorphic if the subdigraphs D[X] and D [X], induced on X, are isomorphic for each nonempty subset X of V. They are k-isomorphic whenever for every k-element vertex subset X, the subdigraphs D[X] and D [X] are isomorphic. They are (≤ k)-isomorphic if they are k-isomorphic for every positive integer k with k ≤ k. The digraphs D and D are (−k)-isomorphic whenever either k ≥ n or D and D are (n − k)-isomorphic with k < n. Let F be a set of non zero integers. The digraphs D and D are F-isomorphic whenever D and D are p-isomorphic for every p ∈ F. The digraph D is F-reconstructible if every digraph F-isomorphic to D is 2010 Mathematics Subject Classification. 05C20;05C60