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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).
Planar separators and the Euclidean norm
 In 1st International Symposium on Algorithms
, 1990
"... In this paper we show that every 2connected embedded planar graph with faces of sizes d!..... df has a simple cycle separator of size 1.58~,/dl 2 +...+d} and we give an almost linear time algorithm for finding these separators, O(m~(n. n)). We show that the new upper bound expressed as a function o ..."
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Cited by 13 (1 self)
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In this paper we show that every 2connected embedded planar graph with faces of sizes d!..... df has a simple cycle separator of size 1.58~,/dl 2 +...+d} and we give an almost linear time algorithm for finding these separators, O(m~(n. n)). We show that the new upper bound expressed as a function of IG { = ~/d ~ +.. + d} is no larger, up to a constant factor than previous bounds that where expressed in terms of ~S. v where d is the maximum face size and v is the number of vertices and is much smaller for many graphs. The algorithms developed are simpler than earlier algorithms in that they work directly with the planar graph and its dual. They need not construct or work with the faceincidence graph as in [Mi186, GM87, GM]. 1
Nested dissection: A survey and comparison of various nested dissection algorithms
, 1992
"... Methods for solving sparse linear systems of equations can be categorized under two broad classes direct and iterative. Direct methods are methods based on gaussian elimination. This report discusses one such direct method namely Nested dissection. Nested Dissection, originally proposed by Alan Geo ..."
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Cited by 8 (1 self)
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Methods for solving sparse linear systems of equations can be categorized under two broad classes direct and iterative. Direct methods are methods based on gaussian elimination. This report discusses one such direct method namely Nested dissection. Nested Dissection, originally proposed by Alan George, is a technique for solving sparse linear systems efficiently. This report is a survey of some of the work in the area of nested dissection and attempts to put it together using a common framework.
Planar crossing numbers of genus g graphs
 PROC. 33RD INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING , LECTURE NOTES IN COMPUTER SCIENCE 4051, PART I
, 2006
"... Pach and Tóth [15] proved that any nvertex graph of genus g and maximum degree d has a planar crossing number at most c g dn, for a constant c>1. We improve on this results by decreasing the bound to O(dgn), if g = o(n), and to O(g²), otherwise, and also prove that our result is tight within a con ..."
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Cited by 7 (0 self)
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Pach and Tóth [15] proved that any nvertex graph of genus g and maximum degree d has a planar crossing number at most c g dn, for a constant c>1. We improve on this results by decreasing the bound to O(dgn), if g = o(n), and to O(g²), otherwise, and also prove that our result is tight within a constant factor.
Computing the Girth of a Planar Graph
 In Proc. 27th International Colloquium on Automata, Languages and Programming ICALP 2000, volume 1853 of LNCS
, 2000
"... The girth of a graph G has been de ned as the length of a shortest cycle of G. We design an O(n 5=4 log n) algorithm for finding the girth of an undirected nvertex planar graph, giving the first o(n 2 ) algorithm for this problem. Our approach combines several techniques such as graph separation, h ..."
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Cited by 6 (0 self)
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The girth of a graph G has been de ned as the length of a shortest cycle of G. We design an O(n 5=4 log n) algorithm for finding the girth of an undirected nvertex planar graph, giving the first o(n 2 ) algorithm for this problem. Our approach combines several techniques such as graph separation, hammock decomposition, covering of a planar graph with graphs of small treewidth, and dynamic shortest path computation. We discuss extensions and generalizations of our result.
Partitioning Graphs with Costs and Weights on Vertices: Algorithms and Applications
 of Lecture Notes in Computer Science
"... We prove separator theorems in which the size of the separator is minimized with respect to nonnegative vertex costs. We show that for any planar graph G there exists a vertex separator of total vertex cost at most c qP v2V (G) (cost(v)) 2 and that this bound is optimal within a constant factor ..."
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Cited by 4 (0 self)
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We prove separator theorems in which the size of the separator is minimized with respect to nonnegative vertex costs. We show that for any planar graph G there exists a vertex separator of total vertex cost at most c qP v2V (G) (cost(v)) 2 and that this bound is optimal within a constant factor. Moreover such a separator can be found in linear time. This theorem implies a variety of other separation results discussed in the paper. We describe application of our separator theorems to graph embedding problems, graph pebbling, and multi commodity flow problems. 1 Introduction Background. A separator is a small set of vertices or edges whose removal divides a graph into two roughly equal parts. The existence of small separators for some important classes of graphs such as planar graphs can be used in the design of efficient divideandconquer algorithms for problems on such graphs. Formally, a separator theorem for a given class of graphs S states that any nvertex graph from S ca...
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.