On chordal phylogeny graphs

TL;DR

This paper characterizes when phylogeny graphs of certain acyclic digraphs are chordal, extending previous work and showing conditions under which these graphs are chordal and planar, relevant for Bayesian network evidence propagation.

Contribution

It completely characterizes phylogeny graphs of (1, i) and (i, 1) digraphs and shows that for (2, j) digraphs, chordality of the underlying graph implies chordality of the phylogeny graph.

Findings

Phylogeny graphs of (1, i) and (i, 1) digraphs are characterized.

Phylogeny graphs of (2, j) digraphs are chordal if the underlying graph is chordal.

Chordal underlying graphs lead to chordal and planar phylogeny graphs for (2, 2) digraphs.

Abstract

An acyclic digraph each vertex of which has indegree at most $i$ and outdegree at most $j$ is called an $(i, j)$ digraph for some positive integers $i$ and $j$. Lee {\it et al.} (2017) studied the phylogeny graphs of $(2, 2)$ digraphs and gave sufficient conditions and necessary conditions for $(2, 2)$ digraphs having chordal 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 their moral graphs being chordal (phylogeny graphs are called moral graphs in Bayesian network theory). In this paper, we extend their work. We completely characterize phylogeny graphs of $(1, i)$ digraphs and $(i,1)$ digraphs, respectively, for a positive integer $i$. Then, we study phylogeny graphs of a $(2,j)$ digraphs, which is worthwhile in the context that a child has two biological parents in most species, to show that the phylogeny graph of a $(2,j)$ digraph $D$ is chordal if the underlying graph of $D$ is chordal for any positive integer $j$. Especially, we show that as long as the underlying graph of a $(2,2)$ digraph is chordal, its phylogeny graph is not only chordal but also planar.