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Crossing-Free Spanning Trees in Visibility Graphs of Points between Monotone Polygonal Obstacles

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Computer Science - Theory and Applications (CSR 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8476))

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Abstract

We consider the problem of deciding whether or not a geometric graph has a crossing-free spanning tree. This problem is known to be NP-hard even for very restricted types of geometric graphs. In this paper, we present an O(n 5) time algorithm to solve this problem for the special case of geometric graphs that arise as visibility graphs of a finite set of n points between two monotone polygonal obstacles. In addition, we give a combinatorial characterization of those visibility graphs induced by such obstacles that have a crossing-free spanning tree. As a byproduct, we obtain a family of counterexamples to the following conjecture by Rivera-Campo: A geometric graph has a crossing-free spanning tree if every subgraph obtained by removing a single vertex has a crossing-free spanning tree.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Greifswald, Germany

    Julia Schüler & Andreas Spillner

Authors
  1. Julia Schüler
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  2. Andreas Spillner
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Editor information

Editors and Affiliations

  1. Steklov Institute of Mathematics at St. Petersburg, Russian Academy of Sciences, 27 Fontanka,, 191023, St. Petersburg, Russia

    Edward A. Hirsch

  2. Higher School of Economics, National Research University, 109028, Moscow, Russia

    Sergei O. Kuznetsov

  3. CNRS and University Paris Diderot, Paris, France

    Jean-Éric Pin

  4. Moscow State University, 119992, Moscow, Russia

    Nikolay K. Vereshchagin

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Schüler, J., Spillner, A. (2014). Crossing-Free Spanning Trees in Visibility Graphs of Points between Monotone Polygonal Obstacles. In: Hirsch, E.A., Kuznetsov, S.O., Pin, JÉ., Vereshchagin, N.K. (eds) Computer Science - Theory and Applications. CSR 2014. Lecture Notes in Computer Science, vol 8476. Springer, Cham. https://doi.org/10.1007/978-3-319-06686-8_26

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  • DOI: https://doi.org/10.1007/978-3-319-06686-8_26

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06685-1

  • Online ISBN: 978-3-319-06686-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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