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Pathwidth and ThreeDimensional StraightLine Grid Drawings of Graphs
"... We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for ..."
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Cited by 24 (12 self)
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We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for
ThreeDimensional Grid Drawings with SubQuadratic Volume
, 1999
"... A threedimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight linesegments representing the edges are pairwise noncrossing. A O(n volume bound is proved for threedimensional grid drawings of graphs with bounded ..."
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Cited by 18 (12 self)
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A threedimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight linesegments representing the edges are pairwise noncrossing. A O(n volume bound is proved for threedimensional grid drawings of graphs with bounded degree, graphs with bounded genus, and graphs with no bounded complete graph as a minor. The previous best bound for these graph families was O(n ). These results (partially) solve open problems due to Pach, Thiele, and Toth (1997) and Felsner, Liotta, and Wismath (2001).
The maximum number of edges in a threedimensional griddrawing
 J. Graph Algorithms Appl
, 2003
"... An exact formula is given for the maximum number of edges in a graph that admits a threedimensional griddrawing contained in a given bounding box. A threedimensional (straightline) griddrawing of a graph represents the vertices by distinct points in Z 3, and represents each edge by a linesegme ..."
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Cited by 18 (9 self)
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An exact formula is given for the maximum number of edges in a graph that admits a threedimensional griddrawing contained in a given bounding box. A threedimensional (straightline) griddrawing of a graph represents the vertices by distinct points in Z 3, and represents each edge by a linesegment between its endpoints that does not intersect any other vertex, and does not intersect any other edge except at the endpoints. A folklore result states that every (simple) graph has a threedimensional griddrawing (see [2]). We therefore are interested in griddrawings with small ‘volume’. The bounding box of a threedimensional griddrawing is the axisaligned box of minimum size that contains the drawing. By an X × Y × Z griddrawing we mean a threedimensional griddrawing, such that the edges of the bounding box contain X, Y, and Z gridpoints, respectively. The volume of a threedimensional griddrawing is the number of gridpoints in the bounding box; that is, the volume of an X ×Y ×Z griddrawing is XY Z. (This definition is formulated to ensure that a twodimensional griddrawing has positive volume.) Our main contribution is the following extremal result. Theorem 1. The maximum number of edges in an X × Y × Z griddrawing is exactly (2X − 1)(2Y − 1)(2Z − 1) − XY Z. Proof. Consider an X × Y × Z griddrawing of a graph G with n vertices and m edges. Let P be the set of points (x, y, z) contained in the bounding box such that 2x, 2y, and 2z are all integers. Observe that P  = (2X − 1)(2Y − 1)(2Z − 1). The midpoint of every edge of G is in P, and no two edges share a common midpoint. Hence m ≤ P . In addition, the midpoint of an edge does not intersect a vertex. Thus m ≤ P  − n. (1) A drawing with the maximum number of edges has no edge that passes through a gridpoint. Otherwise, subdivide the edge, and place the new vertex at that gridpoint. Thus n = XY Z, and m ≤ P  − XY Z, as claimed. This bound is attained by the following construction. Associate a vertex with each gridpoint in an X × Y × Z gridbox B. As illustrated in Figure 1, every vertex (x, y, z) is adjacent to each
Queue layouts, treewidth, and threedimensional graph drawing
 Proc. 22nd Foundations of Software Technology and Theoretical Computer Science (FST TCS '02
, 2002
"... Abstract. A threedimensional (straightline grid) drawing of a graph represents the vertices by points in Z 3 and the edges by noncrossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing ’01, Lecture Notes in Comput. Sci., ..."
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Cited by 12 (6 self)
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Abstract. A threedimensional (straightline grid) drawing of a graph represents the vertices by points in Z 3 and the edges by noncrossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing ’01, Lecture Notes in Comput. Sci., 2002]: does every nvertex planar graph have a threedimensional drawing with O(n) volume? We prove that this question is almost equivalent to an existing onedimensional graph layout problem. A queue layout consists of a linear order σ of the vertices of a graph, and a partition of the edges into queues, such that no two edges in the same queue are nested with respect to σ. The minimum number of queues in a queue layout of a graph is its queuenumber. Let G be an nvertex member of a proper minorclosed family of graphs (such as a planar graph). We prove that G has a O(1) × O(1) × O(n) drawing if and only if G has O(1) queuenumber. Thus the above question is almost equivalent to an open problem of Heath, Leighton, and Rosenberg [SIAM J. Discrete Math., 1992], who ask whether every planar graph has O(1) queuenumber? We also present partial solutions to an open problem of Ganley and Heath [Discrete Appl. Math., 2001], who ask whether graphs of bounded treewidth have bounded queuenumber? We prove that graphs with bounded pathwidth, or both bounded treewidth and bounded maximum degree, have bounded queuenumber. As a corollary we obtain threedimensional drawings with optimal O(n) volume, for seriesparallel graphs, and graphs with both bounded treewidth and bounded maximum degree. 1
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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Cited by 1 (1 self)
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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.