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Tree Spanners in Planar Graphs

(Extended Abstract)
  • Sándor P. Fekete
  • Jana Kremer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1517)

Abstract

A tree t-spanner of a graph G is a spanning subtree T of G in which the distance between every pair of vertices is at most t times their distance in G. Spanner problems have received some attention, mostly in the context of communication networks. It is known that for general unweighted graphs, the problem of deciding the existence of a tree t-spanner can be solved in polynomial time for t=2, while it is NP-hard for any t≥ 4; the case t=3 is open, but has been conjectured to be hard.

In this paper, we consider tree spanners in planar graphs. We show that even for planar unweighted graphs, it is NP-hard to determine the minimum t for which a tree t-spanner exists. On the other hand, we give a polynomial algorithm for any fixed t that decides for planar unweighted graphs with bounded face length whether there is a tree t-spanner. Furthermore, we prove that it can be decided in polynomial time whether a planar unweighted graph has a tree t-spanner for t=3.

Keywords

Polynomial Time Tree Spanner Planar Graph Polynomial Algorithm 3SAT Instance 
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 1998

Authors and Affiliations

  • Sándor P. Fekete
    • 1
  • Jana Kremer
    • 2
  1. 1.Center for Parallel ComputingUniversität zu KölnKölnGermany
  2. 2.Lehrstuhl für VolkswirtschaftslehreOtto-Friedrich Universität BambergBambergGermany

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