Kernelization Hardness of Connectivity Problems in d-Degenerate Graphs

  • Marek Cygan
  • Marcin Pilipczuk
  • Michał Pilipczuk
  • Jakub Onufry Wojtaszczyk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)


A graph is d-degenerate if its every subgraph contains a vertex of degree at most d. For instance, planar graphs are 5-degenerate. Inspired by recent work by Philip, Raman and Sikdar, who have shown the existence of a polynomial kernel for Dominating Set in d-degenerate graphs, we investigate kernelization hardness of problems that include connectivity requirement in this class of graphs.

Our main contribution is the proof that Connected Dominating Set does not admit a polynomial kernel in d-degenerate graphs for d ≥ 2 unless the polynomial hierarchy collapses up to the third level. We prove this using a problem originated from bioinformatics – Colourful Graph Motif – analyzed and proved to be NP-hard by Fellows et al. This problem nicely encapsulates the hardness of the connectivity requirement in kernelization. Our technique yields also an alternative proof that, under the same complexity assumption, Steiner Tree does not admit a polynomial kernel. The original proof, via reduction from Set Cover, is due to Dom, Lokshtanov and Saurabh.

We extend our analysis by showing that, unless \(PH = \Sigma_p^3\), there do not exist polynomial kernels for Steiner Tree, Connected Feedback Vertex Set and Connected Odd Cycle Transversal in d-degenerate graphs for d ≥ 2. On the other hand, we show a polynomial kernel for Connected Vertex Cover in graphs that do not contain the biclique K i,j as a subgraph.


Planar Graph STEINER Tree Vertex Cover Polynomial Kernel Free Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Marek Cygan
    • 1
  • Marcin Pilipczuk
    • 1
  • Michał Pilipczuk
    • 1
  • Jakub Onufry Wojtaszczyk
    • 1
  1. 1.Faculty of Mathematics, Computer Science and MechanicsUniversity of WarsawPoland

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