Document Open Access Logo

Colouring Diamond-free Graphs

Authors Konrad K. Dabrowski, François Dross, Daniël Paulusma

PDF

File

Thumbnail PDF
PDF
LIPIcs.SWAT.2016.16.pdf
  • Filesize: 483 kB
  • 14 pages

Document Identifiers

Subject Classification

Keywords
  • colouring
  • clique-width
  • diamond-free
  • graph class
  • hereditary graph class

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

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 Get BibTex

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) https://doi.org/10.4230/LIPIcs.SWAT.2016.16

Author Details

Konrad K. Dabrowski
François Dross
Daniël Paulusma

References

  1. Claudio Arbib and Raffaele Mosca. On (P₅,diamond)-free graphs. Discrete Mathematics, 250(1-3):1-22, 2002. Google Scholar
  2. Rodica Boliac and Vadim V. Lozin. On the clique-width of graphs in hereditary classes. Proc. ISAAC 2002, LNCS, 2518:44-54, 2002. Google Scholar
  3. Flavia Bonomo, Maria Chudnovsky, Peter Maceli, Oliver Schaudt, Maya Stein, and Mingxian Zhong. Three-coloring and list three-coloring of graphs without induced paths on seven vertices. preprint, 2015. Google Scholar
  4. Flavia Bonomo, Luciano N. Grippo, Martin Milanič, and Martín D. Safe. Graph classes with and without powers of bounded clique-width. Discrete Applied Mathematics, 199:3-15, 2016. Google Scholar
  5. Andreas Brandstädt, Konrad K. Dabrowski, Shenwei Huang, and Daniël Paulusma. Bounding the clique-width of H-free chordal graphs. Proc. MFCS 2015 Part II, LNCS, 9235:139-150, 2015. Google Scholar
  6. Andreas Brandstädt, Konrad K. Dabrowski, Shenwei Huang, and Daniël Paulusma. Bounding the clique-width of H-free split graphs. Discrete Applied Mathematics, (to appear). Google Scholar
  7. Andreas Brandstädt, Joost Engelfriet, Hoàng-Oanh Le, and Vadim V. Lozin. Clique-width for 4-vertex forbidden subgraphs. Theory of Computing Systems, 39(4):561-590, 2006. Google Scholar
  8. Andreas Brandstädt, Vassilis Giakoumakis, and Frédéric Maffray. Clique separator decomposition of hole-free and diamond-free graphs and algorithmic consequences. Discrete Applied Mathematics, 160(4-5):471-478, 2012. Google Scholar
  9. Andreas Brandstädt, Tilo Klembt, and Suhail Mahfud. P₆- and triangle-free graphs revisited: structure and bounded clique-width. Discrete Mathematics and Theoretical Computer Science, 8(1):173-188, 2006. Google Scholar
  10. Andreas Brandstädt, Hoàng-Oanh Le, and Raffaele Mosca. Gem- and co-gem-free graphs have bounded clique-width. International Journal of Foundations of Computer Science, 15(1):163-185, 2004. Google Scholar
  11. Andreas Brandstädt, Hoàng-Oanh Le, and Raffaele Mosca. Chordal co-gem-free and (P₅,gem)-free graphs have bounded clique-width. Discrete Applied Mathematics, 145(2):232-241, 2005. Google Scholar
  12. Andreas Brandstädt and Suhail Mahfud. Maximum weight stable set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time. Information Processing Letters, 84(5):251-259, 2002. Google Scholar
  13. Hajo Broersma, Petr A. Golovach, Daniël Paulusma, and Jian Song. Determining the chromatic number of triangle-free 2P₃-free graphs in polynomial time. Theoretical Computer Science, 423:1-10, 2012. Google Scholar
  14. Maria Chudnovsky. Coloring graphs with forbidden induced subgraphs. Proc. ICM 2014, IV:291-302, 2014. Google Scholar
  15. Maria Chudnovsky, Jan Goedgebeur, Oliver Schaudt, and Mingxian Zhong. Obstructions for three-coloring graphs with one forbidden induced subgraph. Proc. SODA 2016, pages 1774-1783, 2016. Google Scholar
  16. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33(2):125-150, 2000. Google Scholar
  17. Konrad K. Dabrowski, François Dross, and Daniël Paulusma. Colouring diamond-free graphs. arXiv, 1512.07849, 2015. Google Scholar
  18. Konrad K. Dabrowski, Petr A. Golovach, and Daniël Paulusma. Colouring of graphs with Ramsey-type forbidden subgraphs. Theoretical Computer Science, 522:34-43, 2014. Google Scholar
  19. Konrad K. Dabrowski, Shenwei Huang, and Daniël Paulusma. Bounding clique-width via perfect graphs. Proc. LATA 2015, LNCS, 8977:676-688, 2015. Google Scholar
  20. Konrad K. Dabrowski, Vadim V. Lozin, Rajiv Raman, and Bernard Ries. Colouring vertices of triangle-free graphs without forests. Discrete Mathematics, 312(7):1372-1385, 2012. Google Scholar
  21. Konrad K. Dabrowski and Daniël Paulusma. Classifying the clique-width of H-free bipartite graphs. Discrete Applied Mathematics, 200:43-51, 2016. Google Scholar
  22. Konrad K. Dabrowski and Daniël Paulusma. Clique-width of graph classes defined by two forbidden induced subgraphs. The Computer Journal, (in press). Google Scholar
  23. Wolfgang Espelage, Frank Gurski, and Egon Wanke. How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time. Proc. WG 2001, LNCS, 2204:117-128, 2001. Google Scholar
  24. Jean-Luc Fouquet, Vassilis Giakoumakis, and Jean-Marie Vanherpe. Bipartite graphs totally decomposable by canonical decomposition. International Journal of Foundations of Computer Science, 10(04):513-533, 1999. Google Scholar
  25. Petr A. Golovach, Matthew Johnson, Daniël Paulusma, and Jian Song. A survey on the computational complexity of colouring graphs with forbidden subgraphs. Journal of Graph Thoery, (in press). Google Scholar
  26. Martin Grötschel, László Lovász, and Alexander Schrijver. Polynomial algorithms for perfect graphs. Annals of Discrete Mathematics, 21:325-356, 1984. Google Scholar
  27. Frank Gurski. Graph operations on clique-width bounded graphs. CoRR, abs/cs/0701185, 2007. Google Scholar
  28. Pinar Heggernes, Daniel Meister, and Charis Papadopoulos. Characterising the linear clique-width of a class of graphs by forbidden induced subgraphs. Discrete Applied Mathematics, 160(6):888-901, 2012. Google Scholar
  29. Chính T. Hoàng and D. Adam Lazzarato. Polynomial-time algorithms for minimum weighted colorings of (P₅, ̅P₅)-free graphs and similar graph classes. Discrete Applied Mathematics, 186:106-111, 2015. Google Scholar
  30. Shenwei Huang, Matthew Johnson, and Daniël Paulusma. Narrowing the complexity gap for colouring (C_s,P_t)-free graphs. The Computer Journal, 58(11):3074-3088, 2015. Google Scholar
  31. Marcin Kamiński, Vadim V. Lozin, and Martin Milanič. Recent developments on graphs of bounded clique-width. Discrete Applied Mathematics, 157(12):2747-2761, 2009. Google Scholar
  32. Ton Kloks, Haiko Müller, and Kristina Vušković. Even-hole-free graphs that do not contain diamonds: A structure theorem and its consequences. Journal of Combinatorial Theory, Series B, 99(5):733-800, 2009. Google Scholar
  33. Daniel Kobler and Udi Rotics. Edge dominating set and colorings on graphs with fixed clique-width. Discrete Applied Mathematics, 126(2-3):197-221, 2003. Google Scholar
  34. Daniel Král', Jan Kratochvíl, Zsolt Tuza, and Gerhard J. Woeginger. Complexity of coloring graphs without forbidden induced subgraphs. Proc. WG 2001, LNCS, 2204:254-262, 2001. Google Scholar
  35. László Lovász. Coverings and coloring of hypergraphs. Congressus Numerantium, VIII:3-12, 1973. Google Scholar
  36. Vadim V. Lozin and Dmitriy S. Malyshev. Vertex coloring of graphs with few obstructions. Discrete Applied Mathematics, (in press). Google Scholar
  37. Vadim V. Lozin and Dieter Rautenbach. On the band-, tree-, and clique-width of graphs with bounded vertex degree. SIAM Journal on Discrete Mathematics, 18(1):195-206, 2004. Google Scholar
  38. Johann A. Makowsky and Udi Rotics. On the clique-width of graphs with few P₄’s. International Journal of Foundations of Computer Science, 10(03):329-348, 1999. Google Scholar
  39. Dmitriy S. Malyshev. The coloring problem for classes with two small obstructions. Optimization Letters, 8(8):2261-2270, 2014. Google Scholar
  40. Dmitriy S. Malyshev. Two cases of polynomial-time solvability for the coloring problem. Journal of Combinatorial Optimization, 31(2):833-845, 2016. Google Scholar
  41. Sang-Il Oum. Approximating rank-width and clique-width quickly. ACM Transactions on Algorithms, 5(1):10, 2008. Google Scholar
  42. Bert Randerath and Ingo Schiermeyer. Vertex colouring and forbidden subgraphs - a survey. Graphs and Combinatorics, 20(1):1-40, 2004. Google Scholar
  43. Michaël Rao. MSOL partitioning problems on graphs of bounded treewidth and clique-width. Theoretical Computer Science, 377(1-3):260-267, 2007. Google Scholar
  44. David Schindl. Some new hereditary classes where graph coloring remains NP-hard. Discrete Mathematics, 295(1-3):197-202, 2005. Google Scholar
  45. Alan Tucker. Coloring perfect (K₄-e)-free graphs. Journal of Combinatorial Theory, Series B, 42(3):313-318, 1987. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact