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On the Exact Complexity of Hamiltonian Cycle and q-Colouring in Disk Graphs

  • Sándor Kisfaludi-Bak
  • Tom C. van der Zanden
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10236)

Abstract

We study the exact complexity of the Hamiltonian Cycle and the q-Colouring problem in disk graphs. We show that the Hamiltonian Cycle problem can be solved in \(2^{O(\sqrt{n})}\) on n-vertex disk graphs where the ratio of the largest and smallest disk radius is O(1). We also show that this is optimal: assuming the Exponential Time Hypothesis, there is no \(2^{o(\sqrt{n})}\)-time algorithm for Hamiltonian Cycle, even on unit disk graphs. We give analogous results for graph colouring: under the Exponential Time Hypothesis, for any fixed q, q-Colouring does not admit a \(2^{o(\sqrt{n})}\)-time algorithm, even when restricted to unit disk graphs, and it is solvable in \(2^{O(\sqrt{n})}\)-time on disk graphs.

Notes

Acknowledgements

This work was initiated at the Lorentz Center workshop ‘Fixed-Parameter Computational Geometry’. We are grateful to Hans L. Bodlaender and Mark de Berg for discussions and their help with improving this paper.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Sándor Kisfaludi-Bak
    • 1
  • Tom C. van der Zanden
    • 2
  1. 1.Department of Mathematics and Computer ScienceTU EindhovenEindhovenThe Netherlands
  2. 2.Department of Computer ScienceUtrecht UniversityUtrechtThe Netherlands

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