Planar Biconnectivity Augmentation with Fixed Embedding

  • Carsten Gutwenger
  • Petra Mutzel
  • Bernd Zey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5874)

Abstract

A combinatorial embedding\({\it \Pi}\) of a planar graph G = (V,E) is defined by the cyclic order of incident edges around each vertex in a planar drawing of G. The planar biconnectivity augmentation problem with fixed embedding (PBA-Fix) asks for a minimum edge set E′ ⊆ V×V that augments \({\it \Pi}\) to a combinatorial embedding \({\it \Pi}'\) of G + E′ such that G + E′ is biconnected and \({\it \Pi}\) is preserved, i.e., \({\it \Pi}'\) restricted to G yields again \({\it \Pi}\).

In this paper, we show that PBA-Fix is NP-hard in general, i.e., for not necessarily connected graphs, by giving a reduction from 3-PARTITION. For connected graphs, we present an \(\mathcal{O}(|V|(1+\alpha(|V|)))\) time algorithm solving PBA-Fix optimally. Moreover, we show that—considering each face of \({\it \Pi}\) separately—this algorithm meets the lower bound for the general biconnectivity augmentation problem proven by Eswaran and Tarjan [1].

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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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