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2006

Upward Three-Dimensional Grid Drawings of Graphs

8 years 9 months ago
Upward Three-Dimensional Grid Drawings of Graphs
A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. Our first main result is that every n-vertex graph with bounded degeneracy has a three-dimensional grid drawing with O(n3/2 ) volume. This is the largest known class of graphs that have such drawings. A three-dimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the z-direction. We prove that every dag has an upward three-dimensional grid drawing with O(n3 ) volume, which is tight for the complete dag. The previous best upper bound was O(n4 ). Our main result concerning upward drawings is that every c-colourable dag (c constant) has an upward three-dimensional grid drawing with O(n2 ) volume. This result matches the bound in the undirected case, and improves the best kno...
Vida Dujmovic, David R. Wood
Added 14 Dec 2010
Updated 14 Dec 2010
Type Journal
Year 2006
Where ORDER
Authors Vida Dujmovic, David R. Wood
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