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On Computational Aspects of Greedy Partitioning of Graphs

  • Piotr BorowieckiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10336)

Abstract

In this paper we consider a problem of graph \({\mathcal P}\)-coloring consisting in partitioning the vertex set of a graph such that each of the resulting sets induces a graph in a given additive, hereditary class of graphs \({\mathcal P}\). We focus on partitions generated by the greedy algorithm. In particular, we show that given a graph G and an integer k deciding if the greedy algorithm outputs a \({\mathcal P}\)-coloring with a least k colors is \(\mathbb {NP}\)-complete for an infinite number of classes \({\mathcal P}\). On the other hand we get a polynomial-time certifying algorithm if k is fixed and the family of minimal forbidden graphs defining the class \({\mathcal P}\) is finite. We also prove \(\mathrm{co}\mathbb {NP}\)-completeness of the problem of deciding whether for a given graph G the difference between the largest number of colors used by the greedy algorithm and the minimum number of colors required in any \({\mathcal P}\)-coloring of G is bounded by a given constant. A new Brooks-type bound on the largest number of colors used by the greedy \({\mathcal P}\)-coloring algorithm is given.

Keywords

Graph partitioning Computational complexity Graph coloring Greedy algorithm Grundy number Minimal graphs 

Notes

Acknowledgements

Special thanks to D. Dereniowski, E. Drgas-Burchardt and anonymous referees for their remarks on preliminary version of this paper, and to S. Vishwanathan for sending the manuscript [10].

References

  1. 1.
    Araujo, J., Linhares-Sales, C.: On the Grundy number of graphs with few \(P_4\)’s. Discret. Appl. Math. 160(18), 2514–2522 (2012). doi: 10.1016/j.dam.2011.08.016 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Borowiecki, P., Rautenbach, D.: New potential functions for greedy independence and coloring. Discret. Appl. Math. 182, 61–72 (2015). doi: 10.1016/j.dam.2013.12.011 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borowiecki, P., Sidorowicz, E.: Dynamic coloring of graphs. Fund. Inform. 114(2), 105–128 (2012). doi: 10.3233/FI-2012-620 MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brown, J.I.: The complexity of generalized graph colorings. Discret. Appl. Math. 69(3), 257–270 (1996). doi: 10.1016/0166-218X(96)00096-0 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Choudum, S.A., Karthick, T.: First-Fit coloring of \(\{P_5, K_4-e\}\)-free graphs. Discret. Appl. Math. 158(6), 620–626 (2010). doi: 10.1016/j.dam.2009.12.009 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Christen, C.A., Selkow, S.M.: Some perfect coloring properties of graphs. J. Comb. Theory Ser. B 27(1), 49–59 (1979). doi: 10.1016/0095-8956(79)90067-4 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cockayne, E.J., Miller, G.G., Prins, G.: An interpolation theorem for partitions which are complete with respect to hereditary properties. J. Comb. Theory Ser. B 13, 290–297 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. Theoret. Comput. Sci. 1(3), 237–267 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Golovach, P.A., Johnson, M., Paulusma, D., Song, J.: A survey on the complexity of coloring graphs with forbidden subgraphs. J. Graph Theory 84, 331–363 (2017)CrossRefzbMATHGoogle Scholar
  10. 10.
    Goyal, N., Vishwanathan, S.: NP-completeness of undirected Grundy numberings and related problems (1998, unpublished manuscript)Google Scholar
  11. 11.
    Gyárfás, A., Király, Z., Lehel, J.: On-line 3-chromatic graphs II. Critical graphs. Discret. Math. 177(1–3), 99–122 (1997). doi: 10.1016/S0012-365X(96)00359-7 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gyárfás, A., Lehel, J.: First-Fit and on-line chromatic number of families of graphs. Ars Comb. 29C, 168–176 (1990)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hedetniemi, S.M., Hedetniemi, S.T., Beyer, T.: A linear algorithm for the Grundy (coloring) number of a tree. Congr. Numer. 36, 351–363 (1982)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kierstead, H.A.: Coloring graphs on-line. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms. LNCS, vol. 1442, pp. 281–305. Springer, Heidelberg (1998). doi: 10.1007/BFb0029574 CrossRefGoogle Scholar
  15. 15.
    Kortsarz, G.: A lower bound for approximating Grundy numbering. Discret. Math. Theor. Comput. Sci. 9, 7–22 (2007)zbMATHGoogle Scholar
  16. 16.
    Narayanaswamy, N.S., Babu, R.S.: A note on first-fit coloring of interval graphs. Order 25(1), 49–53 (2008). doi: 10.1007/s11083-008-9076-6 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zaker, M.: Results on the Grundy chromatic number of graphs. Discret. Math. 306(23), 3166–3173 (2006). doi: 10.1016/j.disc.2005.06.044 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zaker, M.: New bounds for the chromatic number of graphs. J. Graph Theory 58(2), 110–122 (2008). doi: 10.1002/jgt.20298 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Faculty of Electronics, Telecommunications and InformaticsGdańsk University of TechnologyGdańskPoland

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