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 cw and a cw-expression of G, we solve the Minimum Feedback Vertex Set problem in time O(n~22~(O(cw log cw))). 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.
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