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Colouring Diamond-free Graphs

Authors: Konrad K. Dabrowski, François Dross, and Daniël Paulusma

Published in: LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

The Colouring problem is that of deciding, given a graph G and an integer k, whether G admits a (proper) k-colouring. For all graphs H up to five vertices, we classify the computational complexity of Colouring for (diamond,H)-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for (diamond,P_1+2P_2)-free graphs. Our technique for handling this case is to reduce the graph under consideration to a k-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of (H_1,H_2)-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of (H_1,H_2)-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8.

Cite as

Konrad K. Dabrowski, François Dross, and Daniël Paulusma. Colouring Diamond-free Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 16:1-16:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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  author =	{Dabrowski, Konrad K. and Dross, Fran\c{c}ois and Paulusma, Dani\"{e}l},
  title =	{{Colouring Diamond-free Graphs}},
  booktitle =	{15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)},
  pages =	{16:1--16:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-011-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{53},
  editor =	{Pagh, Rasmus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.16},
  URN =		{urn:nbn:de:0030-drops-60380},
  doi =		{10.4230/LIPIcs.SWAT.2016.16},
  annote =	{Keywords: colouring, clique-width, diamond-free, graph class, hereditary graph class}
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