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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].

Keywords

Planar Graph Connected Graph SIAM Journal Planar Drawing Simple Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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