Mim-Width II. The Feedback Vertex Set Problem

Abstract

We give a first polynomial-time algorithm for (Weighted) Feedback Vertex Set on graphs of bounded maximum induced matching width (mim-width). Explicitly, given a branch decomposition of mim-width w, we give an $$n^{\mathcal {O}(w)}$$-time algorithm that solves Feedback Vertex Set. This provides a unified polynomial-time algorithm for many well-known classes, such as Interval graphs, Permutation graphs, and Leaf power graphs (given a leaf root), and furthermore, it gives the first polynomial-time algorithms for other classes of bounded mim-width, such as Circular Permutation and Circular k-Trapezoid graphs (given a circular k-trapezoid model) for fixed k. We complement our result by showing that Feedback Vertex Set is $$\textsf {W}[1]$$-hard when parameterized by w and the hardness holds even when a linear branch decomposition of mim-width w is given.