The Elimination Procedure for the Phylogeny Number

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. Appeared in Ars Combinatoria, 75 (2005), 297–311. Supported by a National Defense Science and Engineering Graduate Fellowship.