Characterization of balanced graphs within claw-free graphs

Abstract

A graph is balanced when its clique matrix is balanced. Bonomo, Durán, Lin and Szwarcfiter (2006) proved that a graph is balanced if and only if it contains no induced subgraphs known as extended odd suns. However, a characterization of balanced graphs by minimal forbidden induced subgraphs is not known. In this work, we find such a characterization when restricted to the class of claw-free graphs. As a consequence, we prove that there is an O(m2+ n)-time algorithm that, given any graph, either decides that it is balanced or gives a certificate of the fact that it is not claw-free balanced.