Forbidden subgraph characterization of (P3-free, K3-free)-colourable cographs

A ($P_3$-free, $K_3$-free)-colouring of a graph $G = (V,E)$ is a partition of $V = A \cup B$ such that $G[A]$ is $P_3$-free and $G[B]$ is $K_3$-free. This problem is known to be NP-complete even when restricted to planar graphs and perfect graphs. In this paper, we provide a finite list of 17 forbidden induced subgraphs for cographs with a ($P_3$-free, $K_3$-free)-colouring. This yields a linear time recognition algorithm.