Beyond Classes of Graphs with “Few” Minimal Separators: FPT Results Through Potential Maximal Cliques

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.