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)


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.


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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

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