Characterisations and Linear-Time Recognition of Probe Cographs

Cographs are those graphs without induced path on four vertices. A graph G is a probe cograph if its vertex set can be partitioned into two sets, N (non-probes) and P (probes) where N is independent and G can be extended to a cograph by adding edges between certain non-probes. A partitioned probe cograph is a probe cograph with a given partition in N and P . We characterise probe cographs in several ways. Moreover, we characterise partitioned probe cographs in terms of five forbidden induced subgraphs. Using the forbidden induced subgraph characterisation, we give a linear-time recognition algorithm for probe cographs, improving the recent quadratic-time recognition algorithm by Chandler et al. Our algorithm is a modification of the linear-time recognition algorithm for cographs by Corneil et al.