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An acyclic digraph in which every vertex has indegree at most $i$ andoutdegree 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 setand 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 thereis an acyclic digraph $D$ (resp.\ an $(i,j)$ digraph $D$) such that thephylogeny graph of $D$ is isomorphic to $G$. Lee~{\em et al.} (2017) and Eohand Kim (2021) studied the $(2,2)$ phylogeny graphs, $(1,j)$ phylogeny graphs,$(i,1)$ phylogeny graphs, and $(2,j)$ phylogeny graphs. Their work wasmotivated by problems related to evidence propagation in a Bayesian network forwhich it is useful to know which acyclic digraphs have chordal moral graphs(phylogeny graphs are called moral graphs in Bayesian network theory). In thispaper, 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.

Extensions of results on phylogeny graphs of degree bounded digraphs