A characterization of graphs with interval two-step graphs Original Research Article

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 on food webs. We consider it for the class of loopless symmetric digraphs. The competition graph of a symmetric digraph D is the two-step graph of the underlying grap H of D, denoted S2(H). The two-step graph is also know as the neighborhood graph, and has been studied recently by Brigham and Dutton and by Boland, Brigham and Dutton. This work was motivated by a paper of Raychaudhuri and Roberts 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 first consider forbidden subgraph characterizations of graphs with interval two-step graphs. Second, we characterize a large class of graphs with interval two-step graphs using the Gilmore-Hoffman characterization of interval graphs.