Results 1  10
of
41
Applications of the crossing number
 In Proc. 10th Annu. ACM Sympos. Comput. Geom
, 1994
"... Abstract. The crossing number of a graph G is the minimum number of crossings in a drawing of G. The determination of the crossing number is an NPcomplete problem. We present two general lower bounds for the crossing number, and survey their applications and generalizations. 1 ..."
Abstract

Cited by 28 (6 self)
 Add to MetaCart
Abstract. The crossing number of a graph G is the minimum number of crossings in a drawing of G. The determination of the crossing number is an NPcomplete problem. We present two general lower bounds for the crossing number, and survey their applications and generalizations. 1
Decidability of String Graphs
 Proceedings of the 33rd Annual Symposium on the Theory of Computing
, 2003
"... We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a minimal drawing realizing a string graph in the plane. This upper bound confirms a conjecture by Kratochvl and Matousek [KM91] and settles th ..."
Abstract

Cited by 25 (5 self)
 Add to MetaCart
We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a minimal drawing realizing a string graph in the plane. This upper bound confirms a conjecture by Kratochvl and Matousek [KM91] and settles the longstanding open problem of the decidability of string graph recognition (Sinden [Sin66], Graham [Gra76]). Finally we show how to apply the result to solve another old open problem: deciding the existence of Euler diagrams, a fundamental problem of topological inference (Grigni, Papadias, Papadimitriou [GPP95]). The general theory of Euler diagrams turns out to be as hard as secondorder arithmetic.
Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs
, 2006
"... Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e> 4v edges is at least ce3 /v2, where c> 0 is an absolute constant. This result, known as the “Crossing Lemma, ” has found many important applications in discrete and com ..."
Abstract

Cited by 25 (2 self)
 Add to MetaCart
Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e> 4v edges is at least ce3 /v2, where c> 0 is an absolute constant. This result, known as the “Crossing Lemma, ” has found many important applications in discrete and computational geometry. It is tight up to a multiplicative constant. Here we improve the best known value of the constant by showing that the result holds with c> 1024/31827> 0.032. The proof has two new ingredients, interesting in their own right. We show that (1) if a graph can be drawn in the plane so that every edge crosses at most three others, then its number of edges cannot exceed 5.5(v − 2); and (2) the crossing number of any graph is at least 7 25 e − (v − 2). Both bounds are tight upt o
Recognizing string graphs in NP
 J. of Computer and System Sciences
"... A string graph is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. We show that string graphs can be recognized in NP. The recognition problem was not known to be decidable u ..."
Abstract

Cited by 25 (4 self)
 Add to MetaCart
A string graph is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. We show that string graphs can be recognized in NP. The recognition problem was not known to be decidable until very recently, when two independent papers established exponential upper bounds on the number of intersections needed to realize a string graph (Pach and Tóth, 2001; Schaefer and ˇ Stefankovič, 2001). These results implied that the recognition problem lies in NEXP. In the present paper we improve this by showing that the recognition problem for string graphs is in NP, and therefore NPcomplete, since Kratochvíl showed that the recognition problem is NPhard (Kratochvíl, 1991b). The result has consequences for the computational complexity of problems in graph drawing, and topological inference. We also show that the string graph problem is decidable for surfaces of arbitrary genus. Key words: String graphs, NPcompleteness, graph drawing, topological inference, Euler diagrams
Crossing Number is Hard for Cubic Graphs
"... It was proved by [Garey and Johnson, 1983] that computing the crossing number of a graph is an NPhard problem. Their reduction, however, used parallel edges and vertices of very high degrees. We prove here that it is NPhard to determine the crossing number of a simple cubic graph. In particular, t ..."
Abstract

Cited by 19 (0 self)
 Add to MetaCart
It was proved by [Garey and Johnson, 1983] that computing the crossing number of a graph is an NPhard problem. Their reduction, however, used parallel edges and vertices of very high degrees. We prove here that it is NPhard to determine the crossing number of a simple cubic graph. In particular, this implies that the minormonotone version of crossing number is also NPhard, which has been open till now.
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.
Planar decompositions and the crossing number of graphs with an excluded minor
 IN GRAPH DRAWING 2006; LECTURE NOTES IN COMPUTER SCIENCE 4372
, 2007
"... Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded treewidth has linear convex crossing number, and every K3,3minorfree graph with bounded degree has linear rectilinear crossing number.
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 ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
. 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...
On the paircrossing number
 In: Combinatorial and Computational Geometry. Math. Sci. Res. Inst. Publ
, 2005
"... Abstract. By a drawing of a graph G, we mean a drawing in the plane such that vertices are represented by distinct points and edges by arcs. The crossing number cr(G) of a graph G is the minimum possible number of crossings in a drawing of G. The paircrossing number paircr(G) of G is the minimum p ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
Abstract. By a drawing of a graph G, we mean a drawing in the plane such that vertices are represented by distinct points and edges by arcs. The crossing number cr(G) of a graph G is the minimum possible number of crossings in a drawing of G. The paircrossing number paircr(G) of G is the minimum possible number of (unordered) crossing pairs in a drawing of G. Clearly, paircr(G) ≤ cr(G) holds for any graph G. Let f(k) be the maximum cr(G), taken over all graphs G with paircr(G) = k. Obviously, f(k) ≥ k. Pach and Tóth [2000] proved that f(k) ≤ 2k 2. Here we give a slightly better asymptotic upper bound f(k) = O(k 2 / log k). In case of xmonotone drawings (where each vertical line intersects any edge at most once) we get a better upper bound f mon (k) ≤ 4k 4/3. 1.