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Ordered Level Planarity, Geodesic Planarity and Bi-Monotonicity

  • Boris Klemz
  • Günter Rote
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10692)

Abstract

We introduce and study the problem Ordered Level Planarity which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y-monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Our study of Ordered Level Planarity is motivated by connections to several other graph drawing problems.

Geodesic Planarity asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge e is realized as a polygonal path p composed of line segments with two adjacent directions from a given set S of directions symmetric with respect to the origin. Our results on Ordered Level Planarity imply \(\mathcal {NP}\)-hardness for any S with \(|S|\ge 4\) even if the given graph is a matching. Katz, Krug, Rutter and Wolff claimed that for matchings Manhattan Geodesic Planarity, the case where S contains precisely the horizontal and vertical directions, can be solved in polynomial time [GD 2009]. Our results imply that this is incorrect unless \(\mathcal P=\mathcal NP\). Our reduction extends to settle the complexity of the Bi-Monotonicity problem, which was proposed by Fulek, Pelsmajer, Schaefer and Štefankovič.

Ordered Level Planarity turns out to be a special case of T-Level Planarity, Clustered Level Planarity and Constrained Level Planarity. Thus, our results strengthen previous hardness results. In particular, our reduction to Clustered Level Planarity generates instances with only two non-trivial clusters. This answers a question posed by Angelini, Da Lozzo, Di Battista, Frati and Roselli.

Notes

Acknowledgements

We thank the authors of [16] for providing us with unpublished information regarding their plane sweep approach for Manhattan Geodesic Planarity.

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institute of Computer ScienceFreie Universität BerlinBerlinGermany

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