On colouring (2P2,H)-free and (P5,H)-free graphs
Konrad K. Dabrowski, Daniël Paulusma · Information Processing Letters · 2018
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 ( 2 P 2 , H ) -free graphs. This enables us to list all graphs H for which the complexity of Colouring is open on ( 2 P 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 ( 2 P 2 , H ) -free graphs and for ( P 5 , H ) -free graphs are the same for all known cases.