Manhattan-Geodesic Embedding of Planar Graphs

  • Bastian Katz
  • Marcus Krug
  • Ignaz Rutter
  • Alexander Wolff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5849)

Abstract

In this paper, we explore a new convention for drawing graphs, the (Manhattan-) geodesic drawing convention. It requires that edges are drawn as interior-disjoint monotone chains of axis-parallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1-bend embeddability on the grid. For the latter question an efficient algorithm has been proposed. Second, we consider geodesic point-set embeddability where the task is to decide whether a given graph can be embedded on a given point set. We show that this problem is \(\mathcal{NP}\)-hard. In contrast, we efficiently solve geodesic polygonization—the special case where the graph is a cycle. Third, we consider geodesic point-set embeddability where the vertex–point correspondence is given. We show that on the grid, this problem is \(\mathcal{NP}\)-hard even for perfect matchings, but without the grid restriction, we solve the matching problem efficiently.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Bastian Katz
    • 1
  • Marcus Krug
    • 1
  • Ignaz Rutter
    • 1
  • Alexander Wolff
    • 2
  1. 1.Faculty of InformaticsUniversität Karlsruhe (TH), KITGermany
  2. 2.Institut für InformatikUniversität WürzburgGermany

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