Extensions of results on phylogeny graphs of degree bounded digraphs

Abstract

An acyclic digraph in which every vertex has indegree at most $i$ and outdegree at most $j$ is called an $(i,j)$ digraph for some positive integers $i$ and $j$. The phylogeny graph of a digraph $D$ has $V(D)$ as the vertex set and an edge $uv$ if and only if one of the following is true: $(u,v) \in A(D)$; $(v,u) \in A(D)$; $(u,w) \in A(D)$ and $(v,w) \in A(D)$ for some $w \in V(D)$. A graph $G$ is a phylogeny graph (resp.\ an $(i,j)$ phylogeny graph) if there is an acyclic digraph $D$ (resp.\ an $(i,j)$ digraph $D$) such that the phylogeny graph of $D$ is isomorphic to $G$. Lee~{\em et al.} (2017) and Eoh and Kim (2021) studied the $(2,2)$ phylogeny graphs, $(1,j)$ phylogeny graphs, $(i,1)$ phylogeny graphs, and $(2,j)$ phylogeny graphs. Their work was motivated by problems related to evidence propagation in a Bayesian network for which it is useful to know which acyclic digraphs have chordal moral graphs (phylogeny graphs are called moral graphs in Bayesian network theory). In this paper, we extend their work by giving necessary conditions of chordal $(i,2)$ phylogeny graphs. We go further to give necessary conditions of $(i,j)$ phylogeny graphs by listing forbidden induced subgraphs.