Relaxing the constraints of clustered planarity
نویسندگان
چکیده
منابع مشابه
Relaxing the constraints of clustered planarity
In a drawing of a clustered graph vertices and edges are drawn as points and curves, respectively, while clusters are represented by simple closed regions. A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region, or region-region crossings. Determining the complexity of testing whether a clustered graph admits a c-planar drawing is a long-standing open problem in the Grap...
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According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be drawn in the plane without three pairwise crossing edges, has at most O(n) edges. For straight-line drawing...
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Planarity is an important concept in graph drawing. It is generally accepted that planar drawings are well understandable. Recently, several variations of planarity have been studied for advanced graph concepts such as k-level graphs [6,10–16] and clustered graphs [5,7]. In k-level graphs, the vertices are partitioned into k levels and the vertices of one level are drawn on a horizontal line. I...
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The Hanani–Tutte theorem is a classical result proved for the first time in the 1930s that characterizes planar graphs as graphs that admit a drawing in the plane in which every pair of edges not sharing a vertex cross an even number of times. We generalize this result to clustered graphs with two disjoint clusters, and show that a straightforward extension to flat clustered graphs with three o...
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We study the version of the C-PLANARITY problem in which edges connecting the same pair of clusters must be grouped into pipes, which generalizes the STRIP PLANARITY problem. We give algorithms to decide several families of instances for the two variants in which the order of the pipes around each cluster is given as part of the input or can be chosen by the algorithm.
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ژورنال
عنوان ژورنال: Computational Geometry
سال: 2015
ISSN: 0925-7721
DOI: 10.1016/j.comgeo.2014.08.001