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A Polyhedral Approach to the MultiLayer Crossing Minimization Problem
 PROCEEDINGS OF THE INTERNATIONAL SYMPOSIUM ON GRAPH DRAWING, LECTURE NOTES IN COMPUTER SCIENCE 1353
, 1997
"... We study the multilayer crossing minimization problem from a polyhedral point of view. After the introduction of an integer programming formulation of the multilayer crossing minimization problem, we examine the 2layer case and derive several classes of facets of the associated polytope. Prelimin ..."
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Cited by 20 (2 self)
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We study the multilayer crossing minimization problem from a polyhedral point of view. After the introduction of an integer programming formulation of the multilayer crossing minimization problem, we examine the 2layer case and derive several classes of facets of the associated polytope. Preliminary computational results for 2 and 3layer instances indicate, that the usage of the corresponding facetdefining inequalities in a branchandcut approach may only lead to a practically useful algorithm, if deeper polyhedral studies are conducted.
Inserting an Edge Into a Planar Graph
 Algorithmica
, 2000
"... Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e in which all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a planar combinatorial embedding of a planar graph G in which ..."
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Cited by 18 (9 self)
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Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e in which all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a planar combinatorial embedding of a planar graph G in which the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NPhard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQRtrees, which is able to find a crossing minimum solution.
Graph Drawing
 Lecture Notes in Computer Science
, 1997
"... INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as software engineering, database systems, visual interfaces, and computeraideddesign. Research on graph drawing has been conducte ..."
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Cited by 14 (3 self)
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INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as software engineering, database systems, visual interfaces, and computeraideddesign. Research on graph drawing has been conducted within several diverse areas, including discrete mathematics (topological graph theory, geometric graph theory, order theory), algorithmics (graph algorithms, data structures, computational geometry, vlsi), and humancomputer interaction (visual languages, graphical user interfaces, software visualization). This chapter overviews aspects of graph drawing that are especially relevant to computational geometry. Basic definitions on drawings and their properties are given in Section 1.1. Bounds on geometric and topological properties of drawings (e.g., area and crossings) are presented in Section 1.2. Section 1.3 deals with the time complexity of fundamental graph drawin
Crossing Numbers: Bounds and Applications
 I. B'AR'ANY AND K. BOROCZKY, BOLYAI SOCIETY MATHEMATICAL STUDIES 6
, 1997
"... We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the autho ..."
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Cited by 13 (5 self)
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We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the authors. We also show applications of crossing numbers to other areas of discrete mathematics, like discrete geometry.
Bisection Widths of Transposition Graphs
 Discrete Applied Mathematics
, 1998
"... Introduction Several interconnection networks of parallel computers based on permutations have appeared recently, e.g. bubblesort and star graph, alternating group graphs ..., see a survey paper of Lakshmivarahan et al. [8]. Leighton [10] introduced the n\Gammadimensional complete transposition grap ..."
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Cited by 1 (0 self)
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Introduction Several interconnection networks of parallel computers based on permutations have appeared recently, e.g. bubblesort and star graph, alternating group graphs ..., see a survey paper of Lakshmivarahan et al. [8]. Leighton [10] introduced the n\Gammadimensional complete transposition graph CT n : It consists of n! vertices, each of which corresponds to a permutation on n numbers. Two vertices are adjacent iff their corresponding permutations differ by a single transposition. Leighton suggested to study a computational power of a parallel computer with the complete transposition graph interconnection network. Especially, he asked what the bisection width of the complete transposition graph is (problem (R) 3.356). The bisection width is the size of the smallest edge cut of a graph which divides it into two equal parts. This graph invariant is a fundamental concept in the theory of
Bipartite Crossing Numbers of Meshes and Hypercubes
 Proc. 4th Intl. Symposium on Graph Drawing, Lecture Notes in Computer Science 1027
, 1997
"... . Let G = (V0 ; V1 ; E) be a connected bipartite graph, where V0 ; V1 is the bipartition of the vertex set V (G) into independent sets. A bipartite drawing of G consists of placing the vertices of V0 and V1 into distinct points on two parallel lines x0 ; x1 , respectively, and then drawing each ed ..."
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Cited by 1 (1 self)
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. Let G = (V0 ; V1 ; E) be a connected bipartite graph, where V0 ; V1 is the bipartition of the vertex set V (G) into independent sets. A bipartite drawing of G consists of placing the vertices of V0 and V1 into distinct points on two parallel lines x0 ; x1 , respectively, and then drawing each edge with one straight line segment which connects the points of x0 and x1 where the endvertices of the edge were placed. The bipartite crossing number of G, denoted by bcr(G) is the minimum number of crossings of edges over all bipartite drawings of G. We develop a new lower bound method for estimating bcr(G). It relates bipartite crossing numbers to edge isoperimetric inequalities and Laplacian eigenvalues of graphs. We apply the method, which is suitable for "well structured" graphs, to hypercubes and 2dimensional meshes. E.g. for the n\Gammadimensional hypercube graph we get n4 n\Gamma2 \Gamma O(4 n ) bcr(Qn) n4 n\Gamma1 : We also consider a more general setting of the method whi...
Bisection Widths of Transposition Graphs and Their Applications
, 1998
"... We prove lower and upper bounds on bisection widths of the transposition graphs. This class of graphs contains several frequently studied interconnection networks including star graphs and hypercubes. In particular, we prove that the bisection width of the complete transposition graph is of order \T ..."
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We prove lower and upper bounds on bisection widths of the transposition graphs. This class of graphs contains several frequently studied interconnection networks including star graphs and hypercubes. In particular, we prove that the bisection width of the complete transposition graph is of order \Theta(n:n!) which solves the open problem (R) 3.356 of the Leighton's book [11] and determine nearly exact value of bisection width of the star graph. The results have applications to VLSI layouts, cutwidths and crossing numbers of transposition graphs. We also study bandwidths of these graphs. 1 Introduction Several interconnection networks of parallel computers based on permutations have appeared recently, e.g. bubblesort and star graph, alternating group graphs . . ., see a survey paper of Lakshmivarahan et al. [9]. Leighton [11] introduced the n\Gammadimensional complete transposition graph CT n : It consists of n! vertices, each of which corresponds to a permutation on n numbers. Two ve...
Intersection of Curves and Crossing Number of C_m × C_n on Surfaces
, 1998
"... Let (K 1 , 2 ) be two families of closed curves on a surface S, such that = m,K 2 =n,m #m#n, each curve in 1 intersects each curve in 2 , and no point of is covered three times. When is the plane, the projective plane, or the Klein bottle, we prove that the total number of int ..."
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Let (K 1 , 2 ) be two families of closed curves on a surface S, such that = m,K 2 =n,m #m#n, each curve in 1 intersects each curve in 2 , and no point of is covered three times. When is the plane, the projective plane, or the Klein bottle, we prove that the total number of intersections in 2 is at least 10mn/9, 12mn/11, and 10 13 m , respectively. Moreover, when m is close to n, the constants are improved. For instance, the constant for the plane, 10/9, is improved to 8/5, for n 1)/4. Consequently, we prove lower bounds on the crossing number of the Cartesian product of two cycles, in the plane, projective plane, and the Klein bottle. All lower bounds are within small multiplicative factors from easily derived upper bounds. No general lower bound has been previously known, even on the plane. 1.