Algorithms Parameterized by Vertex Cover and Modular Width, Through Potential Maximal Cliques

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 (\({\text {vc}}\)) and modular width (\({\text {mw}}\)). We prove that for any graph, the number of its minimal separators is \({\mathcal {O}}^*(3^{{\text {vc}}})\) and \({\mathcal {O}}^*(1.6181^{{\text {mw}}})\), and the number of potential maximal cliques is \({\mathcal {O}}^*(4^{{\text {vc}}})\) and \({\mathcal {O}}^*(1.7347^{{\text {mw}}})\), 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 applications of potential maximal cliques, 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 \({\mathcal {O}}^*(4^{{\text {vc}}})\) or \({\mathcal {O}}^*(1.7347^{{\text {mw}}})\). With slightly different techniques, we prove that the Treedepth problem can be also solved in single-exponential time, for both parameters.