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Upward threedimensional grid drawings of graphs. arXiv.org math.CO/0510051
, 2005
"... Abstract. A threedimensional 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 threedimensional grid drawings with small bounding box volume. Our first ..."
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Abstract. A threedimensional 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 threedimensional grid drawings with small bounding box volume. Our first main result is that every nvertex graph with bounded degeneracy has a threedimensional grid drawing with O(n 3/2) volume. This is the largest known class of graphs that have such drawings. A threedimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the zdirection. We prove that every dag has an upward threedimensional grid drawing with O(n 3) volume, which is tight for the complete dag. The previous best upper bound was O(n 4). Our main result concerning upward drawings is that every ccolourable dag (c constant) has an upward threedimensional grid drawing with O(n 2) volume. This result matches the bound in the undirected case, and improves the best known bound from O(n 3) for many classes of dags, including planar, series parallel, and outerplanar. Improved bounds are also obtained for tree dags. We prove a strong relationship between upward threedimensional grid drawings, upward track layouts, and upward queue layouts. Finally, we study upward threedimensional grid drawings with bends in the edges. 1.
Linkless symmetric drawings of series parallel digraphs
, 2004
"... In this paper, we present a linear time algorithm for constructing linkless drawings of series parallel digraphs with maximum number of symmetries. Linkless drawing in three dimensions is a natural extension to planar drawing in two dimensions. Symmetry is one of the most important aesthetic criteri ..."
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In this paper, we present a linear time algorithm for constructing linkless drawings of series parallel digraphs with maximum number of symmetries. Linkless drawing in three dimensions is a natural extension to planar drawing in two dimensions. Symmetry is one of the most important aesthetic criteria in graph drawing. More specifically, we present two algorithms: a symmetry finding algorithm which finds maximum number of three dimensional symmetries, and a drawing algorithm which constructs linkless symmetric drawings of series parallel digraphs in three dimensions.