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24
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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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 19 (11 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
Drawing SeriesParallel Graphs on a Box
 The University of Lethbridge
, 1997
"... A box is a restricted portion of the threedimensional integer grid consisting of four parallel lines of in nite length placed one grid unit apart. A boxdrawing of a graph is a straightline crossingfree drawing where vertices are located at integer grid points along the four lines. It is known t ..."
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Cited by 13 (0 self)
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A box is a restricted portion of the threedimensional integer grid consisting of four parallel lines of in nite length placed one grid unit apart. A boxdrawing of a graph is a straightline crossingfree drawing where vertices are located at integer grid points along the four lines. It is known that some planar graphs with triconnected components do not admit a boxdrawing. This paper shows that even structurally simpler planar graphs, namely seriesparallel graphs, are not boxdrawable in general. On the positive side, it is proved that every seriesparallel graph whose vertices have maximum degree at most three is boxdrawable. A drawing algorithm is presented that computes a box drawing of a 3planar seriesparallel graph in optimal time and with optimal volume.
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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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 10 (7 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
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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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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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.
It’s in the Bag: Plane Decompositions as Tools for Approximation
, 2005
"... 2. Background............................... 8 3. Tree Decompositions.......................... 13 ..."
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2. Background............................... 8 3. Tree Decompositions.......................... 13
3D straightline drawings of ktrees
"... This paper studies the problem of computing 3D crossingfree straightline grid drawings of graphs such that the overall volume is small. We show that every 2tree (and therefore every seriesparallel graph) can be drawn on an integer 3D grid consisting of 15 parallel lines and having linear volume. ..."
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This paper studies the problem of computing 3D crossingfree straightline grid drawings of graphs such that the overall volume is small. We show that every 2tree (and therefore every seriesparallel graph) can be drawn on an integer 3D grid consisting of 15 parallel lines and having linear volume. We extend the study to the problem of drawing general ktrees on a set of parallel grid lines. Lower bounds and upper bounds on the number of such grid lines are presented. The results in this paper extend and improve similar ones already described in the literature.