  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 $\{\mathcal{G}_{\langle i,j
\rangle}\colon\, i,j \ge 1\}$.
