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We investigate the computational complexity of the Densest-k-Subgraph (DkS) problem, where the input is an undirected graph G=(V,E) and one wants to find a subgraph on exactly k vertices with a maximum number of edges. We extend previous work on DkS by studying its parameterized complexity. On the positive side, we show that, when fixing some constant minimum density μ of the sought subgraph, DkS becomes fixed-parameter tractable with respect to either of the parameters maximum degree and h-index of G. Furthermore, we obtain a fixed-parameter algorithm for DkS with respect to the combined parameter "degeneracy of G and |V|−k". On the negative side, we find that DkS is W[1]-hard with respect to the combined parameter "solution size k and degeneracy of G". We furthermore strengthen a previous hardness result for DkS [Cai, Comput. J., 2008] by showing that for every fixed μ, 0μG contains a subgraph of density at least μ is W[1]-hard with respect to the parameter |V|−k.