We initiate the first systematic study of the NP-hard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixed-parameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. The parameter is the number of vertex deletions. We present efficient fixed-parameter algorithms for CVD. Our iterative compression algorithm for CVD seems to be the first nontrivial application of

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... fairly new technique to a problem that is not a feedback set problem. Moreover, we study the variant of CVD where the number of cliques to be generated is specified. Here, we detect connections to fixed-parameter algorithms for (weighted) Vertex Cover.