Abstract
In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover (vc) and modular width (mw). We prove that for any graph, the number of minimal separators is \(\mathcal{O}^*(3^{\operatorname{vc}})\) and \(\mathcal{O}^*(1.6181^{\operatorname{mw}})\), the number of potential maximal cliques is \(\mathcal{O}^*(4^{\operatorname{vc}})\) and \(\mathcal{O}^*(1.7347^{\operatorname{mw}})\), and these objects can be listed within the same running times. (The \(\mathcal{O}^*\) notation suppresses polynomial factors in the size of the input.) Combined with known results [3,12], we deduce that a large family of problems, e.g., Treewidth, Minimum Fill-in, Longest Induced Path, Feedback vertex set and many others, can be solved in time \(\mathcal{O}^*(4^{\operatorname{vc}})\) or \(\mathcal{O}^*(1.7347^{\operatorname{mw}})\).
Supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 267959.
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Fomin, F.V., Liedloff, M., Montealegre, P., Todinca, I. (2014). Algorithms Parameterized by Vertex Cover and Modular Width, through Potential Maximal Cliques. In: Ravi, R., Gørtz, I.L. (eds) Algorithm Theory – SWAT 2014. SWAT 2014. Lecture Notes in Computer Science, vol 8503. Springer, Cham. https://doi.org/10.1007/978-3-319-08404-6_16
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