Results 1  10
of
22
Which crossing number is it, anyway
 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
, 1998
"... A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the minimum number of crossing points in any drawing of G. We define two new parameters, as follows. The ..."
Abstract

Cited by 45 (8 self)
 Add to MetaCart
A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the minimum number of crossing points in any drawing of G. We define two new parameters, as follows. The pairwise crossing number (resp. the oddcrossing number) of G is the minimum number of pairs of edges that cross (resp. cross an odd number of times) over all drawings of G. We prove that the largest of these numbers (the crossing number) cannot exceed twice the square of the smallest (the oddcrossing number). Our proof is based on the following generalization of an old result of Hanani, which is of independent interest. Let G be a graph and let E0 be a subset of its edges such that there is a drawing of G, in which every edge belonging to E0 crosses any other edge an even number of times. Then G can be redrawn so that the elements of E0 are not involved in any crossing. Finally, we show that the determination of each of these parameters is an NPhard problem and it is NPcomplete in the case of the crossing number and the oddcrossing number. 1
Toward the Rectilinear Crossing Number of K_n : New Embeddings, Upper Bounds, and Asymptotics
, 2000
"... Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by ..."
Abstract

Cited by 22 (1 self)
 Add to MetaCart
Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear. The rectilinear crossing number of Kn is the fewest number of edge crossings attainable over all planar rectilinear embeddings of Kn . For each n we construct a rectilinear embedding of Kn that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of embeddings of Kn with good asymptotics. Finally, we mention some old and new open problems.
Thirteen Problems on Crossing Numbers
, 2000
"... The crossing number of a graph G is the minimum number of crossings in a drawing of G. We introduce several variants of this definition, and present a list of related open problems. The first item is Zarankiewicz's classical conjecture about crossing numbers of complete bipartite graphs, the la ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
The crossing number of a graph G is the minimum number of crossings in a drawing of G. We introduce several variants of this definition, and present a list of related open problems. The first item is Zarankiewicz's classical conjecture about crossing numbers of complete bipartite graphs, the last ones are new and less carefully tested. In Section 5, we state some conjectures about the expected values of various crossing numbers of random graphs, and prove a large deviation result.
Simultaneous Geometric Graph Embeddings
"... Foundation (JU204/101). Abstract. We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straightline drawing is plana ..."
Abstract

Cited by 15 (5 self)
 Add to MetaCart
Foundation (JU204/101). Abstract. We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straightline drawing is planar. We partially settle an open problem of Erten and Kobourov [5] by showing that even for two graphs the problem is NPhard. We also show that the problem of computing the rectilinear crossing number of a graph can be reduced to a simultaneous geometric graph embedding problem; this implies that placing SGE in NP will be hard, since the corresponding question for rectilinear crossing number is a longstanding open problem. However, rather like rectilinear crossing number, SGE can be decided in PSPACE. 1
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 ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
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.
Genetic algorithms for drawing bipartite graphs
 INTERN. J. COMPUTER MATH
, 1994
"... This paper introduces genetic algorithms for the level permutation problem (LPP). The problem is to minimize the number of edge crossings in a bipartite graph when the order of vertices in one of the two vertex subsets is fixed. We show that genetic algorithms outperform the previously known heurist ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
This paper introduces genetic algorithms for the level permutation problem (LPP). The problem is to minimize the number of edge crossings in a bipartite graph when the order of vertices in one of the two vertex subsets is fixed. We show that genetic algorithms outperform the previously known heuristics especially when applied to low density graphs. Values for various parameters of genetic LPP algorithms are tested.
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 ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
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.
A Proposed Algorithm for Calculating the Minimum Crossing Number of a Graph
 Western Michigan University
, 1995
"... In this paper we present a branchandbound algorithm for finding the minimum crossing number of a graph. We begin with the vertex set and add edges by selecting every legal option for creating a crossing or not. After each edge is added we determine if the resulting partial graph is planar. We cont ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
In this paper we present a branchandbound algorithm for finding the minimum crossing number of a graph. We begin with the vertex set and add edges by selecting every legal option for creating a crossing or not. After each edge is added we determine if the resulting partial graph is planar. We continue adding edges until either all edges have been added or we reach a point where the graph cannot be completed as started. At this point we backtrack to see if the graph can be drawn with fewer crossings by selecting other options when adding edges. keywords: Crossing Number, Algorithm 1 Introduction Determining the crossing number of a graph is an important problem with applications in areas such as circuit design and network configuration [17]. It is this importance that has driven our work in finding the minimum crossing number of a graph. Informally, the crossing number of a graph G, denoted (G), is the minimum number of crossings among all good drawings of G in the plane, where a g...
Toward the Rectilinear Crossing Number of K n : New Embeddings, Upper Bounds, and Asymptotics
, 2000
"... Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear. The rectilinear crossing number of Kn is the fewest number of edge crossings attainable over all planar rectilinear embeddings of Kn . For each n we construct a rectilinear embedding of Kn that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of embeddings of Kn with good asymptotics. Finally, we mention some old and new open problems. 1 Introduction and History Given an arbitrary graph G, determining an embedding of G in the plane that produces the fewest number of edge crossings is NPComplete [GJ83]. The same can be said for a...