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The Target Set Selection problem takes as an input a graph $G$ and anon-negative integer threshold $ \mbox{thr}(v) $ for every vertex $v$. A vertex$v$ can get active as soon as at least $ \mbox{thr}(v) $ of its neighbors havebeen activated. The objective is to select a smallest possible initial set ofvertices, the target set, whose activation eventually leads to the activationof all vertices in the graph.  We show that Target Set Selection is in FPT when parameterized with thecombined parameters clique-width of the graph and the maximum threshold value.This generalizes all previous FPT-membership results for the parameterizationby maximum threshold, and thereby solves an open question from the literature.We stress that the time complexity of our algorithm is surprisinglywell-behaved and grows only single-exponentially in the parameters.

Target Set Selection Parameterized by Clique-Width and Maximum Threshold