Publication detailsDabrowski, K.K. & Paulusma, D. (2018). On colouring (2P2,H)-free and (P5,H)-free graphs. Information Processing Letters 134: 35-41.
- Publication type: Journal Article
- ISSN/ISBN: 0020-0190
- DOI: 10.1016/j.ipl.2018.02.003
- Further publication details on publisher web site
- Durham Research Online (DRO) - may include full text
Author(s) from Durham
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 (H1,H2)-free if it has no induced subgraph isomorphic to H1 or H2. A connected graph H1 is almost classified if Colouring on (H1,H2)-free graphs is known to be polynomial-time solvable or NP-complete for all but finitely many connected graphs H2. We show that every connected graph H1 apart from the claw K1,3 and the 5-vertex path P5 is almost classified. We also prove a number of new hardness results for Colouring on (2P2,H)-free graphs. This enables us to list all graphs H for which the complexity of Colouring is open on (2P2,H)-free graphs and all graphs H for which the complexity of Colouring is open on (P5,H)-free graphs. In fact we show that these two lists coincide. Moreover, we show that the complexities of Colouring for (2P2,H)-free graphs and for (P5,H)-free graphs are the same for all known cases.