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24
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.
CrossingNumber Critical Graphs have Bounded Pathwidth
, 2000
"... . The crossing number of a graph G, denoted by cr(G), is dened as the smallest possible number of edgecrossings in a drawing of G in the plane. A graph G is crossingcritical if cr(G e) < cr(G) for all edges e of G. We prove that crossingcritical graphs have \bounded pathwidth" (by a function ..."
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Cited by 13 (1 self)
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. The crossing number of a graph G, denoted by cr(G), is dened as the smallest possible number of edgecrossings in a drawing of G in the plane. A graph G is crossingcritical if cr(G e) < cr(G) for all edges e of G. We prove that crossingcritical graphs have \bounded pathwidth" (by a function of the crossing number), which roughly means that such graphs are made up of small pieces joined in a linear way on small cutsets. Equivalently, a crossingcritical graph cannot contain a subdivision of a \large" binary tree. This assertion was conjectured earlier by Salazar in [J. Geelen, B. Richter, G. Salazar, Embedding grids on surfaces, manuscript, 2000]. 1 Introduction We begin with the most important denitions here. Additional denitions and comments will be presented in the subsequent section. If % : [0; 1] ! IR 2 is a simple continuous function, then %([0; 1]) is a simple curve, and %((0; 1)) is a simple open curve. Denition. A graph G is drawn in the plane if the ver...
The Rectilinear Crossing Number of K_10 is 62
 Electron. J. Combin., 8(1):Research Paper
, 2000
"... The rectilinear crossing number of a graph G is the minimum number of edge crossings that can occur in any drawing of G in which the edges are straight line segments and no three vertices are collinear. This number has been known for G = K n if n # 9. Using a combinatorial argument we show that fo ..."
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Cited by 11 (0 self)
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The rectilinear crossing number of a graph G is the minimum number of edge crossings that can occur in any drawing of G in which the edges are straight line segments and no three vertices are collinear. This number has been known for G = K n if n # 9. Using a combinatorial argument we show that for n =10the number is 62.
On the crossing numbers of Cartesian products with paths
"... Zip product was recently used in a note establishing the crossing number of the Cartesian product K1,n✷Pm. In this paper, we further investigate the relations of this graph operation with the crossing numbers of graphs. First, we use a refining of the embedding method bound for crossing numbers to w ..."
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Cited by 11 (5 self)
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Zip product was recently used in a note establishing the crossing number of the Cartesian product K1,n✷Pm. In this paper, we further investigate the relations of this graph operation with the crossing numbers of graphs. First, we use a refining of the embedding method bound for crossing numbers to weaken the connectivity condition under which the crossing number is additive for the zip product. Next, we deduce a general theorem for bounding the crossing numbers of (capped) Cartesian products of graphs with trees, which yields exact results under certain symmetry conditions. We apply this theorem to obtain exact and approximate results on crossing numbers of various graphs with trees.
Crossing numbers of Sierpińskilike graphs
 J. Graph Theory
, 2005
"... The crossing number of Sierpiński graphs S(n, k) and their regularizations S + (n, k) and S ++ (n, k) is studied. Explicit drawings of these graphs are presented and proved to be optimal for S + (n, k) and S ++ (n, k) for every n ≥ 1 and k ≥ 1. These are the first nontrivial families of graphs of “f ..."
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Cited by 10 (0 self)
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The crossing number of Sierpiński graphs S(n, k) and their regularizations S + (n, k) and S ++ (n, k) is studied. Explicit drawings of these graphs are presented and proved to be optimal for S + (n, k) and S ++ (n, k) for every n ≥ 1 and k ≥ 1. These are the first nontrivial families of graphs of “fractal ” type whose crossing number is known.
Bounding the Crossing Number of a Graph in terms of the Crossing Number of a Minor with Small Maximum Degree
, 2000
"... We show that if G has a minor M with maximum degree at most 4, then the crossing number of G in a surface is at least one fourth the crossing number of M in . We use this result to show that every graph embedded on the torus with representativity r 6 has Klein bottle crossing number at least b ..."
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Cited by 6 (1 self)
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We show that if G has a minor M with maximum degree at most 4, then the crossing number of G in a surface is at least one fourth the crossing number of M in . We use this result to show that every graph embedded on the torus with representativity r 6 has Klein bottle crossing number at least b2r=3c =64.
Unavoidable Configurations in Complete Topological Graphs
 Proc. Graph Drawing 2000., LNCS
, 1984
"... A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological ..."
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Cited by 5 (3 self)
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A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topological complete graph with n vertices has a canonical subgraph of size at least c log 1/8 n, which belongs to one of these classes. We also show that every complete topological graph with n vertices has a noncrossing subgraph isomorphic to any fixed tree with at most c log 1/6 n vertices.
The Splitting Number of the 4Cube
, 1998
"... The splitting number of a graph G consists in the smallest positive integer k 0, such that a planar graph can be obtained from G by k splitting operations, such operation replaces v by two nonadjacent vertices v1 and v2 , and attaches the neighbors of v either to v1 or to v2 . One of the most usef ..."
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Cited by 3 (3 self)
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The splitting number of a graph G consists in the smallest positive integer k 0, such that a planar graph can be obtained from G by k splitting operations, such operation replaces v by two nonadjacent vertices v1 and v2 , and attaches the neighbors of v either to v1 or to v2 . One of the most useful graphs in computer science is the ncube. Dean and Richter devoted an article to proving that the minimum number of crossings in an optimum drawing of the 4cube is 8, but no results about splitting number of nonplanar ncubes are known. In this note we give a proof that the splitting number of the 4cube is 4. In addition, we give the lower bound 2 n\Gamma2 for the splitting number of the ncube. In particular, because it is known that the splitting number of the ncube is O(2 n ), our result implies that the splitting number of the ncube is \Theta(2 n ).
HananiTutte and Related Results
, 2011
"... We are taking the view that crossings of adjacent edges are trivial, and easily got rid of. ..."
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Cited by 2 (2 self)
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We are taking the view that crossings of adjacent edges are trivial, and easily got rid of.