Ordered Level Planarity, Geodesic Planarity and Bi-Monotonicity

Klemz, Boris and Rote, Günter (2017) Ordered Level Planarity, Geodesic Planarity and Bi-Monotonicity. In: Graph Drawing and Network Visualization. GD 2017, September 25-27 , pp. 440-454(Official URL: https://doi.org/10.1007/978-3-319-73915-1_34).

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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  -hardness for any S with |S|≥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 =P . 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.

Item Type: Conference Paper
Classifications: G Algorithms and Complexity > G.770 Planarity Testing
P Styles > P.300 Curved
P Styles > P.480 Layered
URI: http://gdea.informatik.uni-koeln.de/id/eprint/1622

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