Competition graphs of degree bounded digraphs

If each vertex of an acyclic digraph has indegree at most i and outdegree at most j, then it is called an (i,j) digraph, which was introduced by Hefner et al. (1991). Whereas Hefner et al. characterized (i,j) digraphs whose competition graphs are interval, characterizing the competition graphs of (i,j) digraphs is not an easy task. In this paper, we introduce the concept of (i,j) digraphs, which relax the acyclicity condition of (i,j) digraphs, and study their competition graphs. By doing so, we obtain quite meaningful results. Firstly, we give a necessary and sufficient condition for a loopless graph being an (i,j) competition graph for some positive integers i and j. Then we study on an (i,j) competition graph being chordal and present a forbidden subdigraph characterization. Finally, we study the family of (i,j) competition graphs, denoted by G(i,j), and identify the set containment relation on {G(i,j) : i, j >= 1}.