Abstract
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.
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Acknowledgements
We would like to thank Fedor Fomin and Nicolas Nisse for fruitful discussions on this subject.
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Liedloff, M., Montealegre, P., Todinca, I. (2016). Beyond Classes of Graphs with “Few” Minimal Separators: FPT Results Through Potential Maximal Cliques. In: Mayr, E. (eds) Graph-Theoretic Concepts in Computer Science. WG 2015. Lecture Notes in Computer Science(), vol 9224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53174-7_35
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DOI: https://doi.org/10.1007/978-3-662-53174-7_35
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