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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
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
Treepartitions of ktrees with applications in graph layout
 Proc. 29th Workshop on Graph Theoretic Concepts in Computer Science (WG’03
, 2002
"... Abstract. A treepartition of a graph is a partition of its vertices into ‘bags ’ such that contracting each bag into a single vertex gives a forest. It is proved that every ktree has a treepartition such that each bag induces a (k − 1)tree, amongst other properties. Applications of this result t ..."
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Cited by 16 (11 self)
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Abstract. A treepartition of a graph is a partition of its vertices into ‘bags ’ such that contracting each bag into a single vertex gives a forest. It is proved that every ktree has a treepartition such that each bag induces a (k − 1)tree, amongst other properties. Applications of this result to two wellstudied models of graph layout are presented. First it is proved that graphs of bounded treewidth have bounded queuenumber, thus resolving an open problem due to Ganley and Heath [2001] and disproving a conjecture of Pemmaraju [1992]. This result provides renewed hope for the positive resolution of a number of open problems regarding queue layouts. In a related result, it is proved that graphs of bounded treewidth have threedimensional straightline grid drawings with linear volume, which represents the largest known class of graphs with such drawings. 1
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
Drawing Kn in Three Dimensions with One Bend per Edge
, 2006
"... We give a drawing of Kn in three dimensions in which vertices are placed at integer grid points and edges are drawn crossingfree with at most one bend per edge in a volume bounded by O(n^2.5). ..."
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Cited by 8 (0 self)
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We give a drawing of Kn in three dimensions in which vertices are placed at integer grid points and edges are drawn crossingfree with at most one bend per edge in a volume bounded by O(n^2.5).
A New Algorithm and Open Problems in ThreeDimensional Orthogonal Graph Drawing
 Curtin University of Technology
, 1999
"... . In this paper we present an algorithm for 3D orthogonal drawing of arbitrary degree nvertex medge multigraphs with O(m 2 = p n) bounding box volume and 6 bends per edge route. This is the smallest known bound on the bounding box volume of 3D orthogonal multigraph drawings. We continue ..."
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Cited by 7 (3 self)
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. In this paper we present an algorithm for 3D orthogonal drawing of arbitrary degree nvertex medge multigraphs with O(m 2 = p n) bounding box volume and 6 bends per edge route. This is the smallest known bound on the bounding box volume of 3D orthogonal multigraph drawings. We continue the study of the tradeoff between bounding box volume and the number of bends in orthogonal graph drawings through a refined algorithm with O(m 2 ) bounding box volume and 5 bends per edge route. Many open problems in 3D orthogonal graph drawing are presented and potential avenues for their solution are discussed. 1 Introduction With applications including VLSI circuit design [4, 18, 20] and software engineering [14, 19, 23], there has been recent interest in 3D graph visualization. Proposed models include straightline drawings [6, 13, 16] and of interest in this paper orthogonal drawings [1, 2, 5, 8, 9, 10, 11, 15, 17, 25, 26, 27, 28]. The 3D orthogonal grid consists of grid po...
ThreeDimensional 1Bend Graph Drawings
 Concordia University
, 2004
"... We consider threedimensional griddrawings of graphs with at most one bend per edge. Under the additional requirement that the vertices be collinear, we prove that the minimum volume of such a drawing is Θ(cn), where n is the number of vertices and c is the cutwidth of the graph. We then prove that ..."
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Cited by 4 (0 self)
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We consider threedimensional griddrawings of graphs with at most one bend per edge. Under the additional requirement that the vertices be collinear, we prove that the minimum volume of such a drawing is Θ(cn), where n is the number of vertices and c is the cutwidth of the graph. We then prove that every graph has a threedimensional griddrawing with O(n 3 / log 2 n) volume and one bend per edge. The best previous bound was O(n 3).
Grid drawings of kcolourable graphs
 Vida Dujmović and Attila Pór
"... www.elsevier.com/locate/comgeo It is proved that every kcolourable graph on n vertices has a grid drawing with O(kn) area, and that this bound is best possible. This result can be viewed as a generalisation of the nothreeinline problem. A further area bound is established that includes the aspec ..."
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Cited by 2 (2 self)
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www.elsevier.com/locate/comgeo It is proved that every kcolourable graph on n vertices has a grid drawing with O(kn) area, and that this bound is best possible. This result can be viewed as a generalisation of the nothreeinline problem. A further area bound is established that includes the aspect ratio as a parameter. © 2004 Elsevier B.V. All rights reserved.
Optimal threedimensional layout of interconnection networks
 THEORETICAL COMPUTER SCIENCE
, 2001
"... The main bene ts of a threedimensional layout of interconnection networks are the savings in material (measured as volume) and the shortening of wires. The result presented in this paper is a general formula for calculating a lower bound on the volume. Moreover, for butter y and Xtree networks we ..."
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Cited by 2 (0 self)
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The main bene ts of a threedimensional layout of interconnection networks are the savings in material (measured as volume) and the shortening of wires. The result presented in this paper is a general formula for calculating a lower bound on the volume. Moreover, for butter y and Xtree networks we show layouts optimizing the maximum wire length and whose upper bounds on the volume are close to the lower bounds.