Two Tricks to Triangulate Chordal Probe Graphs in Polynomial Time

A graph G = (V, E) is chordal probe if its vertices can be partitioned into two sets P (probes) and N (non-probes) where N is a stable set and such that G can be extended to a chordal graph by adding edges between non-probes. The study of chordal probe graphs was originally motivated as a generalization of the interval probe graphs which occur in applications involving physical mapping of DNA. However, chordal probe graphs also have their own computational biology application as a special case of constructing phytogenies, tree structures which model genetic mutations. We give several characterizations of chordal probe graphs, first in the case of a fixed given partition of the vertices into probes and non-probes, and second in the more general case where no partition is given. In both of these cases, our results are obtained by characterizing superclasses, namely, N-triangulatable graphs and cycle-bicolorable graphs, which are first introduced here. We give polynomial time recognition algorithms for each class. The complexity is O(|P||E|), given a partition of the vertices into probes and non-probes, thus also providing a interesting tractible subcase of the chordal graph sandwich problem. If no partition is given in advance, the complexity of our recognition algorithm is O(|V|²|E|).