Feedback Vertex Set on Graphs of Low Cliquewidth

The Feedback Vertex Set problem asks whether a graph contains q vertices meeting all its cycles. This is not a local property, in the sense that we cannot check if q vertices meet all cycles by looking only at their neighbors. Dynamic programming algorithms for problems based on non-local properties are usually more complicated. In this paper, given a graph G of cliquewidth cw and a cw-expression of G, we solve the Minimum Feedback Vertex Set problem in time $O(n^22^{2cw^2 \log cw})$. Our algorithm applies a non-standard dynamic programming on a so-called k-module decomposition of a graph, as defined by Rao [26], which is easily derivable from a k-expression of the graph. The related notion of module-width of a graph is tightly linked to both cliquewidth and nlc-width, and in this paper we give an alternative equivalent characterization of module-width.