On the Complexity of Planar Covering of Small Graphs

  • Ondřej Bílka
  • Jozef Jirásek
  • Pavel Klavík
  • Martin Tancer
  • Jan Volec
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6986)

Abstract

The problem Cover(H) asks whether an input graph G covers a fixed graph H (i.e., whether there exists a homomorphism G → H which locally preserves the structure of the graphs). Complexity of this problem has been intensively studied. In this paper, we consider the problem PlanarCover(H) which restricts the input graph G to be planar.

PlanarCover(H) is polynomially solvable if Cover(H) belongs to P, and it is even trivially solvable if H has no planar cover. Thus the interesting cases are when H admits a planar cover, but Cover(H) is NP-complete. This also relates the problem to the long-standing Negami Conjecture which aims to describe all graphs having a planar cover. Kratochvíl asked whether there are non-trivial graphs for which Cover(H) is NP-complete but Planarcover(H) belongs to P.

We examine the first nontrivial cases of graphs H for which Cover(H) is NP-complete and which admit a planar cover. We prove NP-completeness of Planarcover(H) in these cases.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ondřej Bílka
    • 1
  • Jozef Jirásek
    • 1
  • Pavel Klavík
    • 1
  • Martin Tancer
    • 1
  • Jan Volec
    • 1
  1. 1.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic

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