Feedback vertex set on graphs of low clique-width

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 clique-width c w and a c w -expression of G , we solve the Minimum Feedback Vertex Set problem in time O ( n 2 2 O ( c w log c w ) ) . Our algorithm applies dynamic programming on a so-called k -module decomposition of a graph, as defined by Rao (2008) [29], 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 clique-width and NLC-width, and in this paper we give an alternative equivalent characterization of module-width.