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On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
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Cited by 33 (22 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
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 23 (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
Symmetric Tiling of Closed Surfaces: Visualization of Regular Maps
 Proc. Siggraph 2009 Conf
"... A regular map is a tiling of a closed surface into faces, bounded by edges that join pairs of vertices, such that these elements exhibit a maximal symmetry. For genus 0 and 1 (spheres and tori) it is well known how to generate and present regular maps, the Platonic solids are a familiar example. We ..."
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Cited by 7 (0 self)
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A regular map is a tiling of a closed surface into faces, bounded by edges that join pairs of vertices, such that these elements exhibit a maximal symmetry. For genus 0 and 1 (spheres and tori) it is well known how to generate and present regular maps, the Platonic solids are a familiar example. We present a method for the generation of space models of regular maps for genus 2 and higher. The method is based on a generalization of the method for tori. Shapes with the proper genus are derived from regular maps by tubification: edges are replaced by tubes. Tessellations are produced using group theory and hyperbolic geometry. The main results are a generic procedure to produce such tilings, and a collection of intriguing shapes and images. Furthermore, we show how to produce shapes of genus 2 and higher with a highly regular structure. CR Categories: G.2.2 [Discrete Mathematics]: Graph theory—
Crossing Minimization for Symmetries
 Proc. of ISAAC 2002, Lecture Notes in Computer Science
, 2002
"... We consider the problem of drawing a graph with a given symmetry such that the number of edge crossings is minimal. We show that this problem is NPhard, even if the order of orbits around the rotation center or along the reection axis is fixed. Nevertheless, there is a linear time algorithm to test ..."
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Cited by 5 (4 self)
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We consider the problem of drawing a graph with a given symmetry such that the number of edge crossings is minimal. We show that this problem is NPhard, even if the order of orbits around the rotation center or along the reection axis is fixed. Nevertheless, there is a linear time algorithm to test planarity and to construct a planar embedding if possible. Finally, we devise an O(m log m) algorithm for computing a crossing minimal drawing if interorbit edges may not cross orbits, showing in particular that intraorbit edges do not contribute to the NPhardness of the crossing minimization problem for symmetries.
New Results in Graph Layout
 School of Computer Science, Carleton Univ
, 2003
"... A track layout of a graph consists of a vertex colouring, an edge colouring, and a total order of each vertex colour class such that between each pair of vertex colour classes, there is no monochromatic pair of crossing edges. This paper studies track layouts and their applications to other models o ..."
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Cited by 1 (1 self)
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A track layout of a graph consists of a vertex colouring, an edge colouring, and a total order of each vertex colour class such that between each pair of vertex colour classes, there is no monochromatic pair of crossing edges. This paper studies track layouts and their applications to other models of graph layout. In particular, we improve on the results of Enomoto and Miyauchi [SIAM J. Discrete Math., 1999] regarding stack layouts of subdivisions, and give analogous results for queue layouts. We solve open problems due to Felsner, Wismath, and Liotta [Proc. Graph Drawing, 2001] and Pach, Thiele, and Toth [Proc. Graph Drawing, 1997] concerning threedimensional straightline grid drawings. We initiate the study of threedimensional polyline grid drawings and establish volume bounds within a logarithmic factor of optimal.
Visual Analysis of Hierarchical Data Using 2.5D Drawing with Minimum Occlusion
"... In this paper, we consider 2.5D drawing of a pair of trees which are connected by some edges, representing relationships between nodes, as an attempt to develop a tool for analyzing pairwise hierarchical data. We consider two ways of drawing such a graph, called parallel and perpendicular drawings, ..."
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In this paper, we consider 2.5D drawing of a pair of trees which are connected by some edges, representing relationships between nodes, as an attempt to develop a tool for analyzing pairwise hierarchical data. We consider two ways of drawing such a graph, called parallel and perpendicular drawings, where the graph appears as a bipartite graph viewed from two orthogonal angles X and Y. We define the occlusion of a drawing as the sum of the edge crossings that can be seen in the two angles, and propose algorithms to minimize the occlusion based on the fundamental onesided crossing minimization problem. We also give some visualization examples of our method using phylogenetic trees and a mushroom database. 1
High Quality Camera Paths for Navigating Graphs in Three Dimensional Space
"... my mother, Asfia Baig, my wife, Muna Sadary, my daughters, Reem and Tahani, and my siblings Najla, Akram and AmnaAcknowledgements I would like to thank Professor Peter Eades for being a super supervisor; Dr. SeokHee Hong for being an exceptional cosupervisor; Dr. Masahiro Takatsuka for valuable re ..."
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my mother, Asfia Baig, my wife, Muna Sadary, my daughters, Reem and Tahani, and my siblings Najla, Akram and AmnaAcknowledgements I would like to thank Professor Peter Eades for being a super supervisor; Dr. SeokHee Hong for being an exceptional cosupervisor; Dr. Masahiro Takatsuka for valuable research advice on numerous occasions that shaped the outcome of this thesis; Dr. Falk Schreiber and Dr. Kai Xu for helping me in brushing up my bioinformatics skills. Their constant support is the main reason for the existence of this material. I would also like to thank Dr. Joachim Gudmundsson for keeping me on my toes by continuously asking me about my finish date; Dr. Damian Merrick for expert LATEX and 3D advice; Dr. Thomas Wolle for being my main LATEX advisor; Dr. Tim Dwyer for providing the LATEX template; Xiaoyan Fu for I.T. support; Barbara Munday for taking care of all the administrative burden and creating a worryfree environment at National ICT Australia NICTA; Members of the NICTA
Symmetric Graph Drawing
, 2005
"... This chapter gives an overview of the problem of drawing graphs with as much symmetry as possible. It describes a linear time algorithm for constructing maximally symmetric straightline drawings of planar graphs. ..."
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This chapter gives an overview of the problem of drawing graphs with as much symmetry as possible. It describes a linear time algorithm for constructing maximally symmetric straightline drawings of planar graphs.
Stacks, Queues and Tracks: Layouts of Graph Subdivisions
, 2005
"... A kstack layout (respectively, kqueue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of noncrossing (nonnested) edges with respect to the vertex ordering. A ktrack layout of a graph consists of a vertex kcolouring, and a total order of ea ..."
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A kstack layout (respectively, kqueue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of noncrossing (nonnested) edges with respect to the vertex ordering. A ktrack layout of a graph consists of a vertex kcolouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The stacknumber (respectively, queuenumber, tracknumber) of a graph G, denoted by sn(G) (qn(G), tn(G)), is the minimum k such that G has a kstack (kqueue, ktrack) layout. This paper studies stack, queue, and track layouts of graph subdivisions. It is known that every graph has a 3stack subdivision. The best known upper bound on the number of division vertices per edge in a 3stack subdivision of an nvertex graph G is improved from O(log n) to O(log min{sn(G), qn(G)}). This result reduces the question of whether queuenumber is bounded by stacknumber to whether 3stack graphs have bounded queue number. It is proved that every graph has a 2queue subdivision, a 4track subdivision, and a mixed 1stack 1queue subdivision. All these values are optimal for every nonplanar graph. In addition, we characterise those graphs with kstack, kqueue, and ktrack subdivisions, for all values of k. The number of division vertices per edge in the case of 2queue and 4track subdivisions, namely O(log qn(G)), is optimal to within a constant factor, for every graph G. Applications to 3D polyline grid drawings are presented. For example, it is proved that every graph G has a 3D polyline grid drawing with the vertices on a rectangular prism, and with O(log qn(G)) bends per edge. Finally, we