Algorithms Parameterized by Vertex Cover and Modular Width, through Potential Maximal Cliques

  • Fedor V. Fomin
  • Mathieu Liedloff
  • Pedro Montealegre
  • Ioan Todinca
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8503)


In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover (vc) and modular width (mw). We prove that for any graph, the number of minimal separators is \(\mathcal{O}^*(3^{\operatorname{vc}})\) and \(\mathcal{O}^*(1.6181^{\operatorname{mw}})\), the number of potential maximal cliques is \(\mathcal{O}^*(4^{\operatorname{vc}})\) and \(\mathcal{O}^*(1.7347^{\operatorname{mw}})\), and these objects can be listed within the same running times. (The \(\mathcal{O}^*\) notation suppresses polynomial factors in the size of the input.) Combined with known results [3,12], we deduce that a large family of problems, e.g., Treewidth, Minimum Fill-in, Longest Induced Path, Feedback vertex set and many others, can be solved in time \(\mathcal{O}^*(4^{\operatorname{vc}})\) or \(\mathcal{O}^*(1.7347^{\operatorname{mw}})\).


Vertex Cover Maximal Clique Input Graph Minimal Separator Minimum Vertex Cover 
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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Fedor V. Fomin
    • 1
  • Mathieu Liedloff
    • 2
  • Pedro Montealegre
    • 2
  • Ioan Todinca
    • 2
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.INSA Centre Val de Loire, LIFO EA 4022Univ. OrléansOrléans Cedex 2France

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