Competition-common enemy graphs of degree-bounded digraphs

The competition-common enemy graph (CCE graph) of a digraph $D$ is the graph with the vertex set $V(D)$ and an edge $uv$ if and only if $u$ and $v$ have a common predator and a common prey in $D$. If each vertex of a digraph $D$ has indegree at most $i$ and outdegree at most $j$, then $D$ is called an $⟨i,j⟩$ digraph. In this paper, we fully characterize the CCE graphs of $⟨2,2⟩$ digraphs. Then we investigate the CCE graphs of acyclic $⟨2,2⟩$ digraphs, and prove that any CCE graph of an acyclic $⟨2,2⟩$ digraph with at most seven components is interval, and the bound is sharp. While characterizing acyclic $⟨2,2⟩$ digraphs that have interval graphs as their competition graphs, Hefner~{\it et al}. (1991) initiated the study of competition graphs of degree-bounded digraphs. Recently, Lee~{\em et al}. (2017) and Eoh and Kim (2021) studied phylogeny graphs of degree-bounded digraphs to extend their work.