The Elimination Procedure for the Phylogeny Number

Abstract

Given an acyclic digraph \(D\), the phylogeny graph \(P(D)\) is defined to be the undirected graph with \(V(D)\) as its vertex set and with adjacencies as follows: two vertices \(x\) and \(y\) are adjacent if one of the arcs \((x,y)\) or \((y,x)\) is present in \(D\), or if there exists another vertex \(z\) such that the arcs \((x,z)\) and \((y,z)\) are both present in \(D\). Phylogeny graphs were introduced by Roberts and Sheng [6] from an idealized model for reconstructing phylogenetic trees in molecular biology, and are closely related to the widely studied competition graphs. The phylogeny number \(p(G)\) for an undirected graph \(G\) is the least number \(r\) such that there exists an acyclic digraph \(D\) on \(|V(G)| + r\) vertices where \(G\) is an induced subgraph of \(P(D)\). We present an elimination procedure for the phylogeny number analogous to the elimination procedure of Kim and Roberts [2] for the competition number arising in the study of competition graphs. We show that our elimination procedure computes the phylogeny number exactly for so-called “kite-free” graphs. The methods employed also provide a simpler proof of Kim and Roberts’ theorem on the exactness of their elimination procedure for the competition number on kite-free graphs.