Advertisement

Maximum Colorful Cliques in Vertex-Colored Graphs

  • Giuseppe F. Italiano
  • Yannis Manoussakis
  • Nguyen Kim Thang
  • Hong Phong PhamEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10976)

Abstract

In this paper we study the problem of finding a maximum colorful clique in vertex-colored graphs. Specifically, given a graph with colored vertices, we wish to find a clique containing the maximum number of colors. Note that this problem is harder than the maximum clique problem, which can be obtained as a special case when each vertex has a different color. In this paper we aim to give a dichotomy overview on the complexity of the maximum colorful clique problem. We first show that the problem is NP-hard even for several cases where the maximum clique problem is easy, such as complement graphs of bipartite permutation graphs, complement graphs of bipartite convex graphs, and unit disk graphs, and also for properly vertex-colored graphs. Next, we provide a XP parameterized algorithm and polynomial-time algorithms for classes of complement graphs of bipartite chain graphs, complete multipartite graphs and complement graphs of cycle graphs, which are our main contributions.

References

  1. 1.
    Akbari, S., Liaghat, V., Nikzad, A.: Colorful paths in vertex coloring of graphs. Electron. J. Comb. 18(1), P17 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bruckner, S., Hüffner, F., Komusiewicz, C., Niedermeier, R.: Evaluation of ILP-based approaches for partitioning into colorful components. In: Bonifaci, V., Demetrescu, C., Marchetti-Spaccamela, A. (eds.) SEA 2013. LNCS, vol. 7933, pp. 176–187. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38527-8_17CrossRefGoogle Scholar
  3. 3.
    Cohen, J., Manoussakis, Y., Pham, H., Tuza, Z.: Tropical matchings in vertex-colored graphs. In: Latin and American Algorithms, Graphs and Optimization Symposium (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cohen, J., Italiano, G.F., Manoussakis, Y., Nguyen, K.T., Pham, H.P.: Tropical paths in vertex-colored graphs. In: Gao, X., Du, H., Han, M. (eds.) COCOA 2017. LNCS, vol. 10628, pp. 291–305. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-71147-8_20CrossRefGoogle Scholar
  5. 5.
    Corel, E., Pitschi, F., Morgenstern, B.: A min-cut algorithm for the consistency problem in multiple sequence alignment. Bioinformatics 26(8), 1015–1021 (2010)CrossRefGoogle Scholar
  6. 6.
    Fellows, M.R., Fertin, G., Hermelin, D., Vialette, S.: Upper and lower bounds for finding connected motifs in vertex-colored graphs. J. Comput. Syst. Sci. 77(4), 799–811 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Foucaud, F., Harutyunyan, A., Hell, P., Legay, S., Manoussakis, Y., Naserasr, R.: Tropical homomorphisms in vertex-coloured graphs. Discrete Appl. Math. 229, 1–168 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Italiano, G.F., Manoussakis, Y., Kim Thang, N., Pham, H.P.: Maximum colorful cycles in vertex-colored graphs. In: Fomin, F., Podolskii, V. (eds.) CSR 2018. LNCS, vol. 10846, pp. 106–117. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-90530-3_10CrossRefGoogle Scholar
  9. 9.
    Li, H.: A generalization of the Gallai–Roy theorem. Graphs and Combinatorics 17(4), 681–685 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lin, C.: Simple proofs of results on paths representing all colors in proper vertex-colorings. Graphs and Combinatorics 23(2), 201–203 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Marx, D.: Graph colouring problems and their applications in scheduling. Periodica Polytech. Electr. Eng. 48(1–2), 11–16 (2004)Google Scholar
  12. 12.
    Micali, S., Vazirani, V.V.: An \({O}(\sqrt{|V|} |{E}|)\) algorithm for finding maximum matching in general graphs. In: Proceedings of 21st Symposium on Foundations of Computer Science, pp. 17–27 (1980)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Giuseppe F. Italiano
    • 2
  • Yannis Manoussakis
    • 1
  • Nguyen Kim Thang
    • 3
  • Hong Phong Pham
    • 1
    Email author
  1. 1.LRIUniversity Paris-SaclayOrsayFrance
  2. 2.University of Rome Tor VergataRomeItaly
  3. 3.IBISCUniversity Paris-SaclayEvryFrance

Personalised recommendations