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A Linear-Time Algorithm for Star-Shaped Drawings of Planar Graphs with the Minimum Number of Concave Corners

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Abstract

A star-shaped drawing of a graph is a straight-line drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. In this paper, we consider the problem of finding a star-shaped drawing of a biconnected planar graph with the minimum number of concave corners. We first show new structural properties of planar graphs to derive a lower bound on the number of concave corners. Based on the lower bound, we prove that the problem can be solved in linear time by presenting a linear-time algorithm for finding a best plane embedding of a biconnected planar graph with the minimum number of concave corners. This is in spite of the fact that a biconnected planar graph may have an exponential number of different plane embeddings.

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Authors and Affiliations

  1. School of Information Technologies, University of Sydney, Sydney, Australia

    Seok-Hee Hong

  2. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, Japan

    Hiroshi Nagamochi

Authors
  1. Seok-Hee Hong
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  2. Hiroshi Nagamochi
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Correspondence to Seok-Hee Hong.

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Hong, SH., Nagamochi, H. A Linear-Time Algorithm for Star-Shaped Drawings of Planar Graphs with the Minimum Number of Concave Corners. Algorithmica 62, 1122–1158 (2012). https://doi.org/10.1007/s00453-011-9513-6

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  • DOI: https://doi.org/10.1007/s00453-011-9513-6

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