Drawing Planar Graphs with Many Collinear Vertices

Da Lozzo, Giordano and Dujmović, Vida and Frati, Fabrizio and Mchedlidze, Tamara and Roselli, Vincenzo (2016) Drawing Planar Graphs with Many Collinear Vertices. In: Graph Drawing and Network Visualization. GD 2016, September, 19. - 21., 2016 , pp. 152-165(Official URL: http://dx.doi.org/10.1007/978-3-319-50106-2_13).

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Abstract

Given a planar graph G, what is the maximum number of collinear vertices in a planar straight-line drawing of G? This problem resides at the core of several graph drawing problems, including universal point subsets, untangling, and column planarity. The following results are known: Every n-vertex planar graph has a planar straight-line drawing with Ω(√n) collinear vertices; for every n, there is an n-vertex planar graph whose every planar straight-line drawinghas O(n^0.986) collinear vertices; every n-vertex planar graph of treewidth at most two has a planar straight-line drawingwith Θ(n) collinear vertices. We extend the linear bound to planar graphs of treewidth at most three and to triconnected cubic planar graphs, partially answering two problems posed by Ravsky and Verbitsky. Similar results are not possible for all bounded treewidth or bounded degree planar graphs. For planar graphs of treewidth at most three, our results also imply asymptotically tight bounds for all of the other above mentioned graph drawing problems.

Item Type: Conference Paper
Classifications: P Styles > P.540 Planar
P Styles > P.720 Straight-line
Z Theory > Z.250 Geometry
URI: http://gdea.informatik.uni-koeln.de/id/eprint/1540

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