Split-perfect graphs: Characterizations and algorithmic use

Abstract

Two graphs G and H with the same vertex set V are P4- isomorphic if every four vertices {a, b, c, d} ⊂ V induce a chordless path (denoted by P4) in G if and only if they induce a P4 in H. We call a graph split-perfect if it is P4-isomorphic to a split graph (i.e. a graph being partitionable into a clique and a stable set). This paper characterizes the new class of split-perfect graphs using the concepts of homogeneous sets and p-connected graphs, and leads to a linear time recognition algorithm for split-perfect graphs, as well as linear time algorithms for classical optimization problems on split-perfect graphs based on the primeval decomposition of graphs. These results considerably extend previous ones on smaller classes such as P4-sparse graphs, P4-lite graphs, P4-laden graphs, and (7,3)-graphs. Moreover, split-perfect graphs form a new subclass of brittle graphs containing the superbrittle graphs for which a new characterization is obtained leading to linear time recognition.