$$(-1)$$ ( - 1 ) -Hypomorphic Graphs with the Same 3-Element Homogeneous Subsets

Abstract

A 3-element homogeneous subset of a graph G is a 3-element subset of the vertex set of G which forms a clique or an independent set. We prove that Ulam Reconstruction Conjecture is true for pairs of graphs having the same 3-element homogeneous subsets. This new approach in graph reconstruction is motivated by a result, obtained in 2011 by Pouzet, Si Kaddour and Trotignon, giving a description of the boolean sum $$G\dot{+} G'$$ of two graphs G and $$G'$$ having the same 3-element homogeneous subsets.