Triangulating vertex colored graphs

This paper examines the class of vertex-colored graphs that can be triangulated without the introduction of edges between vertices of the same color. This is related to a fundamental and long-standing problem for numerical taxonomists, called the Perfect Phylogeny Problem. These problems are known to be polynomially equivalent and NP-complete. This paper presents a dynamic programming algorithm that can be used to determine whether a given vertex-colored graph can be so triangulated and that runs in $O((n+m(k-2))^{k+1})$ time, where the graph has $n$ vertices, $m$ edges, and $k$ colors. The corresponding algorithm for the Perfect Phylogeny Problem runs in $O(r^{k+1} k^{k+1} + sk^2 )$ time, where $s$ species are defined by $k$ $r$-state characters.