On Colouring (2P_2,H)-Free and (P_5,H)-Free Graphs
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The Colouring problem asks whether the vertices of a graph can be coloured with at most k colours for a given integer k in such a way that no two adjacent vertices receive the same colour. A graph is (H_1,H_2)-free if it has no induced subgraph isomorphic to H_1 or H_2. A connected graph H_1 is almost classified if Colouring on (H_1,H_2)-free graphs is known to be polynomial-time solvable or NP-complete for all but finitely many connected graphs H_2. We show that every connected graph H_1 apart from the claw K_1,3 and the 5-vertex path P_5 is almost classified. We also prove a number of new hardness results for Colouring on (2P_2,H)-free graphs. This enables us to list all graphs H for which the complexity of Colouring is open on (2P_2,H)-free graphs and all graphs H for which the complexity of Colouring is open on (P_5,H)-free graphs. In fact we show that these two lists coincide. Moreover, we show that the complexities of Colouring for (2P_2,H)-free graphs and for (P_5,H)-free graphs are the same for all known cases.
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