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Solving Problems on Graphs of High Rank-Width

  • Eduard Eiben
  • Robert Ganian
  • Stefan Szeider
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)

Abstract

A modulator of a graph G to a specified graph class \({\mathcal H}\) is a set of vertices whose deletion puts G into \({\mathcal H}\). The cardinality of a modulator to various graph classes has long been used as a structural parameter which can be exploited to obtain FPT algorithms for a range of hard problems. Here we investigate what happens when a graph contains a modulator which is large but “well-structured” (in the sense of having bounded rank-width). Can such modulators still be exploited to obtain efficient algorithms? And is it even possible to find such modulators efficiently?

We first show that the parameters derived from such well-structured modulators are strictly more general than the cardinality of modulators and rank-width itself. Then, we develop an FPT algorithm for finding such well-structured modulators to any graph class which can be characterized by a finite set of forbidden induced subgraphs. We proceed by showing how well-structured modulators can be used to obtain efficient parameterized algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use the concept of well-structured modulators to develop an algorithmic meta-theorem for efficiently deciding problems expressible in Monadic Second Order (MSO) logic, and prove that this result is tight in the sense that it cannot be generalized to LinEMSO problems.

Keywords

Vertex Cover Maximum Clique Graph Class Split Graph Minimum Vertex Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Algorithms and Complexity Group, Institute of Computer Graphics and AlgorithmsTU WienViennaAustria

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