Colouring on Hereditary Graph Classes Closed under Complementation

The Colouring problem is that of deciding whether a given graph G admits a (proper) k-colouring for a given integer k. A graph is (H1,H2)-free for a pair of graphs H1,H2 if it contains no induced subgraph isomorphic to H1 or H2. We continue a study into the complexity of Colouring for (H1,H2)-free graphs. The complement H of a graph H has the same vertices as H and an edge between two distinct vertices if and only if these vertices are not adjacent in H . By combining new and known results we classify the computational complexity of Colouring for (H,H)-free graphs except when H = sP1 + P3 or H = sP1 + P4 for s ≥ 2. We also show that these are the only open cases when considering all bigenic graph classes closed under complementation.

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