Hereditary Graph Classes: When the Complexities of Colouring and Clique Cover Coincide
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A graph is (H_1,H_2)-free for a pair of graphs H_1,H_2 if it contains no induced subgraph isomorphic to H_1 or H_2. In 2001, Král', Kratochvíl, Tuza, and Woeginger initiated a study into the complexity of Colouring for (H_1,H_2)-free graphs. Since then, others have tried to complete their study, but many cases remain open. We focus on those (H_1,H_2)-free graphs where H_2 is H_1, the complement of H_1. 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=sP_1+ P_3 for s≥ 3 or H=sP_1+P_4 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 (P_r,P_r)-free graphs.
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