Finding Dense Subgraphs of Sparse Graphs

  • Christian Komusiewicz
  • Manuel Sorge
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7535)


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, 0 < μ < 1, the problem of deciding whether G contains a subgraph of density at least μ is W[1]-hard with respect to the parameter |V| − k.


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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Christian Komusiewicz
    • 1
  • Manuel Sorge
    • 1
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinGermany

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