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If each vertex of an acyclic digraph has indegree at most $i$ and outdegreeat most $j$, then it is called an $(i,j)$ digraph, which was introduced byHefner~{\it et al.}~(1991). Whereas Hefner~{\it et al.} characterized $(i,j)$digraphs whose competition graphs are interval, characterizing the competitiongraphs of $(i,j)$ digraphs is not an easy task. In this paper, we introduce theconcept of $\langle i,j \rangle$ digraphs, which relax the acyclicity conditionof $(i,j)$ digraphs, and study their competition graphs. By doing so, we obtainquite meaningful results. Firstly, we give a necessary and sufficient conditionfor a loopless graph being an $\langle i,j \rangle$ competition graph for somepositive integers $i$ and $j$. Then we study on an $\langle i,j \rangle$competition graph being chordal and present a forbidden subdigraphcharacterization. Finally, we study the family of $\langle i,j \rangle$competition graphs, denoted by $\mathcal{G}_{\langle i,j \rangle}$, andidentify the set containment relation on $\{\mathcal{G}_{\langle i,j\rangle}\colon\, i,j \ge 1\}$.

Competition graphs of degree bounded digraphs