Triangulating three-colored graphs

We are interested here in the problem of determining whether we can triangulate a vertex-colored graph without introducing edges between vertices ofthe same color. This problem ismotivated byalongstanding and fundamental problem in numerical taxonomy called the Perfect Phylogeny Problem, which is concerned with the inference of evolutionary history. This problem is also related to the problem of recognizing partial k-trees, a class of graphs that has received a lot of attention recently. In this paper we present an almost linear time algorithm for this problem in the case that the graph is three colored.