Tight complexity bounds for FPT subgraph problems parameterized by clique-width

  • Authors:

  • Hajo Broersma;Petr A. Golovach;Viresh Patel

  • Affiliations:

  • School of Engineering and Computing Sciences, Durham University, Science Laboratories, Durham, UK;School of Engineering and Computing Sciences, Durham University, Science Laboratories, Durham, UK;School of Engineering and Computing Sciences, Durham University, Science Laboratories, Durham, UK

  • Venue:
  • IPEC'11 Proceedings of the 6th international conference on Parameterized and Exact Computation
  • Year:
  • 2011

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Abstract

We give tight algorithmic lower and upper bounds for some double-parameterized subgraph problems when the clique-width of the input graph is one of the parameters. Let G be an arbitrary input graph on n vertices with clique-width at most w. We prove the following results. — The Dense (Sparse)k-Subgraph problem, which asks whether there exists an induced subgraph of G with k vertices and at least q edges (at most q edges, respectively), can be solved in time kO(w)·n, but it cannot be solved in time 2o(wlogk)·nO(1) unless the Exponential Time Hypothesis (ETH) fails. — The d-Regular Induced Subgraph problem, which asks whether there exists a d-regular induced subgraph of G, and the Minimum Subgraph of Minimum Degree at leastd problem, which asks whether there exists a subgraph of G with k vertices and minimum degree at least d, can be solved in time dO(w)·n, but they cannot be solved in time 2o(wlogd)·nO(1) unless ETH fails.