Algorithms parameterized by vertex cover and modular width, through potential maximal cliques

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 ($\operatorname{vc}$) and modular width ($\operatorname{mw}$). We prove that for any graph, the number of minimal separators is $\mathcal{O}^*(3^{\operatorname{vc}})$ and $\mathcal{O}^*(1.6181^{\operatorname{mw}})$, and 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, 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 $\mathcal{O}^*(1.7347^{\operatorname{mw}})$.