Intersection graph of maximal stars

Abstract

A biclique of a graph G is an induced complete bipartite subgraph of G such that neither part is empty. A star is a biclique of G such that one part has exactly one vertex. The star graph of G is the intersection graph of the maximal stars of G. A graph H is star-critical if its star graph is different from the star graph of any of its proper induced subgraphs. We begin by showing that a star-critical pre-image of an n-vertex star graph has at most On2vertices. We then describe a Krausz-type characterization for star graphs. We combine these results to show that the problem of recognizing star graphs is in NP. We also present some properties of the class. In particular, we show that they are biconnected, that every edge belongs to at least one triangle, characterize the structures the pre-image must have in order to generate degree two vertices in its star graph, and bound the diameter of the star graph with respect to the diameter of its pre-image. Finally, we prove a monotonicity theorem, which we apply in order to list every star graph on at most eight vertices.