On the Hardness and Approximability of Planar Biconnectivity Augmentation

  • Carsten Gutwenger
  • Petra Mutzel
  • Bernd Zey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5609)

Abstract

Given a planar graph G = (V,E), the planar biconnectivity augmentation problem (PBA) asks for an edge set E′ ⊆ V×V such that G + E′ is planar and biconnected. This problem is known to be \(\mathcal{NP}\)-hard in general; see [1]. We show that PBA is already \(\mathcal{NP}\)-hard if all cutvertices of G belong to a common biconnected component B*, and even remains \(\mathcal{NP}\)-hard if the SPQR-tree of B* (excluding Q-nodes) has a diameter of at most two. For the latter case, we present a new 5/3-approximation algorithm with runtime \({\mathcal{O}}(|V|^{2.5})\).

Though a 5/3-approximation of PBA has already been presented [12], we give a family of counter-examples showing that this algorithm cannot achieve an approximation ratio better than 2, thus the best known approximation ratio for PBA is 2.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Carsten Gutwenger
    • 1
  • Petra Mutzel
    • 1
  • Bernd Zey
    • 1
  1. 1.Department of Computer ScienceTechnische Universität DortmundGermany

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