Given a clique-width $k$-expression of a graph $G$, we provide $2^{O(k)}cdot n$ time algorithms for connectivity constraints on locally checkable properties such as Node-Weighted Steiner Tree, Connected Dominating Set, or Connected Vertex Cover. We also propose a $2^{O(k)}cdot n$ time algorithm for Feedback Vertex Set. The best running times for all the considered cases were either $2^{O(kcdot log(k))}cdot n^{O(1)}$ or worse.
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