Phylogeny numbers Original Research Article

Motivated by problems of phylogenetic tree reconstruction, we introduce notions of phylogeny graph and phylogeny number. These notions are analogous to and can be considered as natural generalizations of notions of competition graph and competition number that arise from problems of ecology. Given an acyclic digraph D = (V,A), define its phylogeny graph G = P(D) by taking the same vertex set as D and, for x ≠ y, letting xy ϵ E(G) if and only if (x, y)ϵ A or (y,x) ϵ A or (x,a),(y,a) ϵ A for some vertex a ϵ V. Given a graph G = (V,E), we shall call the acyclic digraph D a phylogeny digraph for G if G is an induced subgraph of P(D) and D has no arcs from vertices outside of G to vertices in G. The phylogeny number p(G) is defined to be the smallest r such that G has a phylogeny digraph D with ¦V(D)¦ − ¦V(G)¦ = r. In this paper, we obtain results about phylogeny number analogous to a number of well-known results about competition number. In particular, we show that the computation of p(G) is NP-complete and we calculate the phylogeny number for triangulated graphs, line graphs, and graphs with zero or one triangle.