On Colouring $(2P_2,H)$-Free and $(P_5,H)$-Free Graphs

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.