A Characterization of Graphs with Interval Two-Step Graphs

Dedicated by the other authors to Professor John Maybee on the occasion of his 65th birthday. Abstract. One of the intriguing open problems on competition graphs is determining what digraphs have interval competition graphs. This problem originated in the work of Cohen 5, 6] on food webs. In this paper we consider this problem for the class of loopless symmetric digraphs. The competition graph of a symmetric digraph D is the two-step graph of the underlying graph H of D, denoted S 2 (H). The two-step graph is also known as the neighborhood graph, and has been studied recently by Brigham and Dutton 4] and Boland, Brigham and Dutton 1, 2]. This work was motivated by a paper of Raychaudhuri and Roberts 20] where they investigated symmetric digraphs with a loop at each vertex. Under these assumptions, the competition graph is the square of the underlying graph H without loops. Here we will rst consider forbidden subgraph characterizations of graphs with interval two-step graphs. Second, we will characterize a large class of graphs with interval two-step graphs using the Gilmore-Hooman characterization of interval graphs.