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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 32 (20 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 26 (15 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
Drawing planar graphs symmetrically, III: Oneconnected planar graphs
 ALGORITHMICA
, 2006
"... Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. This paper discusses symmetric drawings of oneconnected planar graphs. More specifically, we discuss planar (geometric) automorphisms, that is, automorphisms of a graph G that can b ..."
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Cited by 8 (5 self)
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Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. This paper discusses symmetric drawings of oneconnected planar graphs. More specifically, we discuss planar (geometric) automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a planar drawing of G. Finding planar automorphisms is the first and most difficult step in constructing planar symmetric drawings of graphs. The problem of determining whether a given graph has a nontrivial geometric automorphism is NPcomplete for general graphs. The two previous papers in this series have discussed the problem of drawing planar graphs with a maximum number of symmetries, for the restricted cases where the graph is triconnected and biconnected. This paper extends the previous results to cover planar graphs that are oneconnected. We present a linear time algorithm for drawing oneconnected planar graphs with a maximum number of 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.
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.
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
Geometric Automorphism Groups of Graphs ⋆
"... Abstract. Constructing symmetric drawings of graphs is NPhard. In this paper, we present a new method for drawing graphs symmetrically based on group theory. More formally, we define an ngeometric automorphism group as a subgroup of the automorphism group of a graph that can be displayed as symmet ..."
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Abstract. Constructing symmetric drawings of graphs is NPhard. In this paper, we present a new method for drawing graphs symmetrically based on group theory. More formally, we define an ngeometric automorphism group as a subgroup of the automorphism group of a graph that can be displayed as symmetries of a drawing of the graph in n dimensions. Then we present an algorithm to find all 2 and 3geometric automorphism groups of a given graph. We implement the algorithm using Magma [29] and the experimental results show that our approach is very efficient in practice. We also present a drawing algorithm to display 2 and 3geometric automorphism groups.
1.1 Stack and Queue Layouts................................ 3
, 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 ..."
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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