Advertisement

Parameterized and Exact Algorithms for Class Domination Coloring

  • R. Krithika
  • Ashutosh Rai
  • Saket Saurabh
  • Prafullkumar TaleEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10139)

Abstract

A class domination coloring (also called as cd-coloring) of a graph is a proper coloring such that for every color class, there is a vertex that dominates it. The minimum number of colors required for a cd-coloring of the graph G, denoted by \(\chi _{cd}(G)\), is called the class domination chromatic number (cd-chromatic number) of G. In this work, we consider two problems associated with the cd-coloring of a graph in the context of exact exponential-time algorithms and parameterized complexity. (1) Given a graph G on n vertices, find its cd-chromatic number. (2) Given a graph G and integers k and q, can we delete at most k vertices such that the cd-chromatic number of the resulting graph is at most q? For the first problem, we give an exact algorithm with running time \(\mathcal {O}(2^n n^4 \log n)\). Also, we show that the problem is \(\mathsf {FPT}\) with respect to the number of colors q as the parameter on chordal graphs. On graphs of girth at least 5, we show that the problem also admits a kernel with \(\mathcal {O}(q^3)\) vertices. For the second (deletion) problem, we show \(\mathsf {NP}\)-hardness for each \(q \ge 2\). Further, on split graphs, we show that the problem is \(\mathsf {NP}\)-hard if q is a part of the input and \(\mathsf {FPT}\) with respect to k and q. As recognizing graphs with cd-chromatic number at most q is \(\mathsf {NP}\)-hard in general for \(q \ge 4\), the deletion problem is unlikely to be \(\mathsf {FPT}\) when parameterized by the size of deletion set on general graphs. We show fixed parameter tractability for \(q \in \{2,3\}\) using the known algorithms for finding a vertex cover and an odd cycle transversal as subroutines.

References

  1. 1.
    Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica 33(4), 461–493 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alon, N., Gutner, S.: Linear time algorithms for finding a dominating set of fixed size in degenerated graphs. Algorithmica 54(4), 544–556 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Comput. 39(2), 546–563 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blum, A., Karger, D.R.: An \(\tilde{\cal{O}}(n^{3/14})\)-coloring algorithm for \(3\)-colorable graphs. Inf. Process. Lett. 61(1), 49–53 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cai, L.: Parameterized complexity of vertex colouring. Discrete Appl. Math. 127(3), 415–429 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chellali, M., Maffray, F.: Dominator Colorings in Some Classes of Graphs. Graph. Comb. 28(1), 97–107 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, J., Kanj, I., Xia, G.: Improved upper bounds for vertex cover. Theor. Comput. Sci. 411(40–42), 3736–3756 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Corneil, D.G., Fonlupt, J.: The complexity of generalized clique covering. Discrete Appl. Math. 22(2), 109–118 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Courcelle, B.: The monadic second-order logic of graphs III: tree-decompositions. Minor Complex. Issues ITA 26, 257–286 (1992)zbMATHGoogle Scholar
  11. 11.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Cham (2015)CrossRefzbMATHGoogle Scholar
  12. 12.
    Diestel, R.: Graph Theory. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  13. 13.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, London (2013)CrossRefzbMATHGoogle Scholar
  14. 14.
    Downey, R.G., Fellows, M.R., McCartin, C., Rosamond, F.A.: Parameterized approximation of dominating set problems. Inf. Process. Lett. 109(1), 68–70 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  16. 16.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  17. 17.
    Gaspers, S., Kratsch, D., Liedloff, M., Todinca, I.: Exponential time algorithms for the minimum dominating set problem on some graph classes. ACM Trans. Algorithms 6(1), 9:1–21 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gaspers, S., Liedloff, M.: A branch-and-reduce algorithm for finding a minimum independent dominating set. Discrete Math. Theor. Comput. Sci. 14(1), 29–42 (2012)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gera, R.: On dominator colorings in graphs. In: Graph Theory Notes of New York LII, pp. 25–30 (2007)Google Scholar
  20. 20.
    Gera, R., Rasmussen, C., Horton, S.: Dominator colorings and safe clique partitions. Congressus Numerantium 181(7–9), 1932 (2006)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Guha, S., Khuller, S.: Improved methods for approximating node weighted steiner trees and connected dominating sets. Inf. Comput. 150(1), 57–74 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kim, D., Zhang, Z., Li, X., Wang, W., Wu, W., Du, D.Z.: A better approximation algorithm for computing connected dominating sets in unit ball graphs. IEEE Trans. Mob. Comput. 9(8), 1108–1118 (2010)CrossRefGoogle Scholar
  23. 23.
    Kratsch, D.: Exact algorithms for dominating set. In: Kao, M.-Y. (ed.) Encyclopedia of Algorithms, pp. 284–286. Springer, New York (2008)CrossRefGoogle Scholar
  24. 24.
    Lawler, E.: A note on the complexity of the chromatic number problem. Inf. Process. Lett. 5(3), 66–67 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lenzen, C., Wattenhofer, R.: Minimum dominating set approximation in graphs of bounded arboricity. In: Lynch, N.A., Shvartsman, A.A. (eds.) DISC 2010. LNCS, vol. 6343, pp. 510–524. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-15763-9_48 CrossRefGoogle Scholar
  26. 26.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20(2), 219–230 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lokshtanov, D., Narayanaswamy, N.S., Raman, V., Ramanujan, M.S., Saurabh, S.: Faster parameterized algorithms using linear programming. ACM Trans. Algorithms 11(2), 15:1–15:31 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lozin, V.V., Kaminski, M.: Coloring edges and vertices of graphs without short or long cycles. Contrib. Discrete Math. 2(1), 61–66 (2007)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  30. 30.
    Panolan, F., Philip, G., Saurabh, S.: B-Chromatic number: beyond NP-hardness. In: 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, pp. 389–401 (2015)Google Scholar
  31. 31.
    Raman, V., Saurabh, S.: Short cycles make w-hard problems hard: FPT algorithms for W-hard problems in graphs with no short cycles. Algorithmica 52(2), 203–225 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    van Rooij, J.M.M., Bodlaender, H.L.: Exact algorithms for dominating set. Discrete Appl. Math. 159(17), 2147–2164 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Schönhage, A., Strassen, V.: Schnelle Multiplikation grosser Zahlen. Computing 7(3–4), 281–292 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Shalu, M.A., Sandhya, T.P.: Personal communication (2016)Google Scholar
  35. 35.
    Shalu, M.A., Sandhya, T.P.: The cd-coloring of graphs. In: Govindarajan, S., Maheshwari, A. (eds.) CALDAM 2016. LNCS, vol. 9602, pp. 337–348. Springer, Heidelberg (2016). doi: 10.1007/978-3-319-29221-2_29 CrossRefGoogle Scholar
  36. 36.
    Venkatakrishnan, Y.B., Swaminathan, V.: Color class domination number of middle graph and center graph of K\(_{1, n}\), C\(_n\), P\(_n\). Adv. Model. Optim. 12, 233–237 (2010)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Yannakakis, M., Gavril, F.: The maximum \(k\)-colorable subgraph problem for chordal graphs. Inf. Process. Lett. 24(2), 133–137 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • R. Krithika
    • 1
  • Ashutosh Rai
    • 1
  • Saket Saurabh
    • 1
    • 2
  • Prafullkumar Tale
    • 1
    Email author
  1. 1.The Institute of Mathematical SciencesHBNIChennaiIndia
  2. 2.University of BergenBergenNorway

Personalised recommendations