Bipartite bithreshold graphs

Abstract

A bithreshold graph G is the intersection of two threshold graphs T 1 and T 2 on the same vertex set such that every stable set of G is also a stable set of T 1 or T 2. The complements of bithreshold graphs form an important subclass of the class of graphs of threshold dimension two. The complexity of recognizing the latter class remains open while the former subclass has a known O( n 4)-recognition algorithm. We show that the vertex set of a bithreshold graph partitions into a clique and a set inducing a bipartite graph. We characterize the class of bipartite bithreshold graphs as the union of five classes of graphs with explicit description and also by eleven forbidden induced subgraphs. We use this to obtain an O( n 2)-recognition algorithm for bipartite bithreshold graphs.