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On the Hardness of Point-Set Embeddability

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

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.

Keywords

  • Convex Hull
  • Planar Graph
  • Hamiltonian Cycle
  • Outer Face
  • Outerplanar Graph

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.

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

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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

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