Solving Problems on Graphs of High Rank-Width

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?

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