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

TL;DR

This paper investigates the computational complexity of graph colouring in specific classes of graphs defined by forbidden induced subgraphs, providing new hardness results and nearly complete classifications for these classes.

Contribution

It establishes that all connected graphs except two are almost classified regarding colouring complexity and identifies all open cases for $(2P_2,H)$-free and $(P_5,H)$-free graphs.

Findings

Most connected graphs are almost classified for colouring complexity.

New NP-hardness results for $(2P_2,H)$-free graphs.

Open cases for colouring complexity are fully characterized.

Abstract

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.