Beyond Classes of Graphs with “Few” Minimal Separators: FPT Results Through Potential Maximal Cliques

  • Mathieu Liedloff
  • Pedro Montealegre
  • Ioan TodincaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)


In many graph problems, like Longest Induced Path, Maximum Induced Forest, etc., we are given as input a graph G and the goal is to compute a largest induced subgraph G[F], of treewidth at most a constant t, and satisfying some property \(\mathcal {P}\). Fomin et al. [12] proved that this generic problem is polynomial on the class of graphs \({\mathcal {G}}_{{\text {poly}}}\), i.e., the graphs having at most \({\text {poly}}(n)\) minimal separators for some polynomial \({\text {poly}}\), when property \(\mathcal {P}\) is expressible in counting monadic second order logic (CMSO).

Here we consider the class \({\mathcal {G}}_{{\text {poly}}}+ kv\), formed by graphs of \({\mathcal {G}}_{{\text {poly}}}\) to which we may add a set of at most k vertices with arbitrary adjacencies, called modulator. We prove that the generic optimization problem is fixed parameter tractable on \({\mathcal {G}}_{{\text {poly}}}+ kv\), with parameter k, if the modulator is also part of the input. The running time is of type \(\mathcal {O}\left( f(k+t, \mathcal {P})\cdot n^{t+5} \cdot ({\text {poly}}(n)^2)\right) \), for some function f.



We would like to thank Fedor Fomin and Nicolas Nisse for fruitful discussions on this subject.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Mathieu Liedloff
    • 1
  • Pedro Montealegre
    • 1
  • Ioan Todinca
    • 1
    Email author
  1. 1.Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022OrléansFrance

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