Competition-common enemy graphs of degree-bounded digraphs

TL;DR

This paper characterizes the competition-common enemy graphs of degree-bounded digraphs, specifically for the case where each vertex has indegree and outdegree at most 2, and explores their properties in acyclic cases.

Contribution

It provides a complete characterization of CCE graphs of  2,2 digraphs and investigates their structure in acyclic cases with bounds on components.

Findings

CCE graphs of  2,2 digraphs are fully characterized.

Any CCE graph of an acyclic  2,2 digraph with up to seven components is interval.

The bound of seven components is shown to be sharp.

Abstract

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 $\langle i,j \rangle$ digraph. In this paper, we fully characterize the CCE graphs of $\langle 2,2\rangle$ digraphs. Then we investigate the CCE graphs of acyclic $\langle 2,2 \rangle$ digraphs, and prove that any CCE graph of an acyclic $\langle 2,2 \rangle$ digraph with at most seven components is interval, and the bound is sharp. While characterizing acyclic $\langle 2,2 \rangle$ 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.