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On Distance-d Independent Set and Other Problems in Graphs with “few” Minimal Separators

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Graph-Theoretic Concepts in Computer Science (WG 2016)

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Abstract

Fomin and Villanger ([14], STACS 2010) proved that Maximum Independent Set, Feedback Vertex Set, and more generally the problem of finding a maximum induced subgraph of treewith at most a constant t, can be solved in polynomial time on graph classes with polynomially many minimal separators. We extend these results in two directions. Let \(\mathcal {G}_{{\text {poly}}}\) be the class of graphs with at most \({\text {poly}}(n)\) minimal separators, for some polynomial \({\text {poly}}\).

We show that the odd powers of a graph G have at most as many minimal separators as G. Consequently, Distance-d Independent Set, which consists in finding maximum set of vertices at pairwise distance at least d, is polynomial on \(\mathcal {G}_{{\text {poly}}}\), for any even d. The problem is NP-hard on chordal graphs for any odd \(d \ge 3\) [12].

We also provide polynomial algorithms for Connected Vertex Cover and Connected Feedback Vertex Set on subclasses of \(\mathcal {G}_{{\text {poly}}}\) including chordal and circular-arc graphs, and we discuss variants of independent domination problems.

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Notes

  1. 1.

    To be precise, the \(\gamma \) function is not required at this stage, if we only compute largest induced subgraphs of treewidth at most t. However it becomes necessary when we request the solution to satisfy additional properties, as it will happen in Sect. 4.

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Acknowledgements

We thank Iyad Kanj for fruitful discussions on the subject.

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Authors and Affiliations

  1. Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, Orléans, France

    Pedro Montealegre & Ioan Todinca

Authors
  1. Pedro Montealegre
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  2. Ioan Todinca
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Correspondence to Pedro Montealegre .

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Editors and Affiliations

  1. University of Bergen, Bergen, Norway

    Pinar Heggernes

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Montealegre, P., Todinca, I. (2016). On Distance-d Independent Set and Other Problems in Graphs with “few” Minimal Separators. In: Heggernes, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2016. Lecture Notes in Computer Science(), vol 9941. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53536-3_16

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  • DOI: https://doi.org/10.1007/978-3-662-53536-3_16

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