Recognizing Chordal Probe Graphs and Cycle-Bicolorable Graphs

A graph $G=(V,E)$ is a chordal probe graph 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. 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 introducing new classes, namely, $N$-triangulatable graphs and cycle-bicolorable graphs. We give polynomial time recognition algorithms for each class. $N$-triangulatable graphs have properties similar to chordal graphs, and we characterize them using graph separators and using a vertex elimination ordering. For cycle-bicolorable graphs, which are shown to be perfect, we prove that any cycle-bicoloring of a graph renders it $N$-triangulatable. The corresponding recognition complexity for chordal probe graphs, given a partition of the vertices into probes and non-probes, is $O(|P||E|)$, thus also providing an interesting tractable subcase of the chordal graph sandwich problem. If no partition is given in advance, the complexity of our recognition algorithm is $O(|E|^2)$.