Solving Problems on Graphs of High Rank-Width

Eduard Eiben; Robert Ganian; Stefan Szeider

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arXiv · 1507.05463

Solving Problems on Graphs of High Rank-Width

Eduard Eiben·Robert Ganian·Stefan Szeider

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Abstract

A modulator of a graph G to a specified graph class H is a set of vertices whose deletion puts G into H. The cardinality of a modulator to various tractable 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 well-structured modulators to develop an algorithmic meta-theorem for 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.

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Eduard Eiben, Robert Ganian, Stefan Szeider. 2015-07-20. Solving Problems on Graphs of High Rank-Width. https://arxiv.org/abs/1507.05463

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