License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2018.35
URN: urn:nbn:de:0030-drops-84922
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8492/
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Gaspers, Serge ; Huang, Shenwei ; Paulusma, Daniel

Colouring Square-Free Graphs without Long Induced Paths

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LIPIcs-STACS-2018-35.pdf (0.5 MB)

Abstract

The Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a given integer k such that no two adjacent vertices are coloured alike. The complexity of Colouring is fully understood for graph classes characterized by one forbidden induced subgraph H. Despite a huge body of existing work, there are still major complexity gaps if two induced subgraphs H_1 and H_2 are forbidden. We let H_1 be the s-vertex cycle C_s and H_2 be the t-vertex path P_t. We show that Colouring is polynomial-time solvable for s=4 and t<=6, which unifies several known results for Colouring on (H_1,H_2)-free graphs. Our algorithm is based on a novel decomposition theorem for (C_4,P_6)-free graphs without clique cutsets into homogeneous pairs of sets and a new framework for bounding the clique-width of a graph by the clique-width of its subgraphs induced by homogeneous pairs of sets. To apply this framework, we also need to use divide-and-conquer to bound the clique-width of subgraphs induced by homogeneous pairs of sets. To complement our positive result we also prove that Colouring is NP-complete for s=4 and t>=9, which is the first hardness result on Colouring for (C_4,P_t)-free graphs.

BibTeX - Entry

@InProceedings{gaspers_et_al:LIPIcs:2018:8492,
  author =	{Serge Gaspers and Shenwei Huang and Daniel Paulusma},
  title =	{{Colouring Square-Free Graphs without Long Induced Paths}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{35:1--35:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Rolf Niedermeier and Brigitte Vall{\'e}e},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8492},
  URN =		{urn:nbn:de:0030-drops-84922},
  doi =		{10.4230/LIPIcs.STACS.2018.35},
  annote =	{Keywords: graph colouring, hereditary graph class, clique-width, cycle, path}
}

Keywords: graph colouring, hereditary graph class, clique-width, cycle, path
Collection: 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)
Issue Date: 2018
Date of publication: 27.02.2018

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