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~{\it et al.}~(1991). Whereas Hefner~{\it 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 $\langle i,j \rangle$ 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 $\langle i,j \rangle$ competition graph for some positive integers $i$ and $j$. Then we study on an $\langle i,j \rangle$ competition graph being chordal and present a forbidden subdigraph characterization. Finally, we study the family of $\langle i,j \rangle$ competition graphs, denoted by $\mathcal{G}_{\langle i,j \rangle}$, and identify the set containment relation on $\{\mathcal{G}_{\langle i,j \rangle}\colon\, i,j \ge 1\}$.