Hereditary Graph Classes: When the Complexities of Colouring and Clique Cover Coincide⋆

A graph is (H1,H2)-free for a pair of graphsH1, H2 if it contains no induced subgraph isomorphic toH1 orH2. In 2001, Král’, Kratochvíl, Tuza, and Woeginger initiated a study into the complexity of Colouring for (H1,H2)free graphs. Since then, others have tried to complete their study, but many cases remain open. We focus on those (H1,H2)-free graphs where H2 is H1, the complement of H1. As these classes are closed under complementation, the computational complexities of Colouring and Clique Cover coincide. By combining new and known results, we are able to classify the complexity of Colouring and Clique Cover for (H,H)-free graphs for all cases except when H = sP1 + P3 for s ≥ 3 or H = sP1 + P4 for s ≥ 2. We also classify the complexity of Colouring on graph classes characterized by forbidding a finite number of self-complementary induced subgraphs, and we initiate a study of k-Colouring for (Pr, Pr)-free graphs.