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On the Hardness of Point-Set Embeddability

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WALCOM: Algorithms and Computation (WALCOM 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7157))

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Abstract

A point-set embedding of a plane graph G with n vertices on a set S of n points is a straight-line drawing of G, where the vertices of G are mapped to distinct points of S. The problem of deciding whether a plane graph admits a point-set embedding on a given set of points is NP-complete for 2-connected planar graphs, but polynomial-time solvable for outerplanar graphs and plane 3-trees. In this paper we prove that the problem remains NP-complete for 3-connected planar graphs, which settles an open question posed by Cabello (Journal of Graph Algorithms and Applications, 10(2), 2000). We then show that the constraint of convexity makes the problem easier for klee graphs, which is a subclass of 3-connected planar graphs. We give a polynomial-time algorithm to decide whether a klee graph with exactly three outer vertices admits a convex point-set embedding on a given set of points and compute such an embedding if one exists.

Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Manitoba, Canada

    Stephane Durocher & Debajyoti Mondal

Authors
  1. Stephane Durocher
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  2. Debajyoti Mondal
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Editor information

Editors and Affiliations

  1. Graph Drawing and Information Visualization Laboratory, Department of Computer Science and Engineering, Bangladesh University of Engineering and Technology, Bangladesh

    Md. Saidur Rahman

  2. , Faculty of Engineering, Department of Computer Science, Gunma University, 1-5-1 Tenjin-Cho, 376-8515, Kiryu-Shi, Japan

    Shin-ichi Nakano

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Durocher, S., Mondal, D. (2012). On the Hardness of Point-Set Embeddability. In: Rahman, M.S., Nakano, Si. (eds) WALCOM: Algorithms and Computation. WALCOM 2012. Lecture Notes in Computer Science, vol 7157. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28076-4_16

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  • DOI: https://doi.org/10.1007/978-3-642-28076-4_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-28075-7

  • Online ISBN: 978-3-642-28076-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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