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On colouring (2P2,H)-free and (P5,H)-free graphs.

Dabrowski, K.K. and Paulusma, D. (2018) 'On colouring (2P2,H)-free and (P5,H)-free graphs.', Information processing letters., 134 . pp. 35-41.

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 (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.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1016/j.ipl.2018.02.003
Publisher statement:© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Date accepted:02 February 2018
Date deposited:06 February 2018
Date of first online publication:05 February 2018
Date first made open access:No date available

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