Results 1  10
of
18
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 25 (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
Counting patternfree set partitions. II: Noncrossing and other hypergraphs
 J. Combin
, 2000
"... A (multi)hypergraph H with vertices in N contains a permutation p = a 1 a 2: : : a k of 1; 2; : : : ; k if one can reduce H by omitting vertices from the edges so that the resulting hypergraph is isomorphic, via an increasing mapping, to H p = (fi; k + a i g: i = 1; : : : ; k). We formulate six co ..."
Abstract

Cited by 20 (9 self)
 Add to MetaCart
A (multi)hypergraph H with vertices in N contains a permutation p = a 1 a 2: : : a k of 1; 2; : : : ; k if one can reduce H by omitting vertices from the edges so that the resulting hypergraph is isomorphic, via an increasing mapping, to H p = (fi; k + a i g: i = 1; : : : ; k). We formulate six conjectures stating that if H has n vertices and does not contain p then the size of H is O(n) and the number of such Hs is O(c n). The latter part generalizes the StanleyWilf conjecture on permutations. Using generalized DavenportSchinzel sequences, we prove the conjectures with weaker bounds O(nfi(n)) and O(fi(n) n), where fi(n) ! 1 very slowly. We prove the conjectures fully if p first increases and then decreases or if p
Generalized DavenportSchinzel sequences: results, problems, and applications
, 1994
"... We survey in detail... ..."
Geometric Graph Theory
, 1999
"... A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications. ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications.
Geometric Graphs with Few Disjoint Edges
, 1998
"... A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the corresponding points. Improving a result of Pach and Töröcsik, we show that a geometric graph on n vertices with no k ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the corresponding points. Improving a result of Pach and Töröcsik, we show that a geometric graph on n vertices with no k + 1 pairwise disjoint edges has at most k³(n + 1) edges. On the other hand, we construct geometric graphs with n vertices and approximately 3/2 (k  1)n edges, containing no k + 1 pairwise disjoint edges. We also improve both the lower and upper bounds of Goddard, Katchalski and Kleitman on the maximum number of edges in a geometric graph with no four pairwise disjoint edges.
Geometric Graphs With No SelfIntersecting Path Of Length Three
"... Let G be a geometric graph with n vertices, i.e., a graph drawn in the plane with straightline edges. It is shown that if G has no selfintersecting path of length 3, then its number of edges is O(n log n). This result is asymptotically tight. Analogous questions for curvilinear drawings and for ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
Let G be a geometric graph with n vertices, i.e., a graph drawn in the plane with straightline edges. It is shown that if G has no selfintersecting path of length 3, then its number of edges is O(n log n). This result is asymptotically tight. Analogous questions for curvilinear drawings and for longer paths are also considered.
Relaxing planarity for topological graphs
 Discrete and Computational Geometry, Lecture Notes in Comput. Sci., 2866
, 2003
"... Abstract. According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Abstract. According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be drawn in the plane without three pairwise crossing edges, has at most O(n) edges. For straightline drawings, this statement has been established by Agarwal et al., using a more complicated argument, but for the general case previously no bound better than O(n 3/2) was known. 1
On Crossing Sets, Disjoint Sets and the Pagenumber
, 1998
"... Let G = (V; E) be a tpartite graph with n vertices and m edges, where the partite sets are given. We present an O(n 2 m 1:5 ) time algorithm to construct drawings of G in the plane so that the size of the largest set of pairwise crossing edges, and at the same time, the size of the largest set ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Let G = (V; E) be a tpartite graph with n vertices and m edges, where the partite sets are given. We present an O(n 2 m 1:5 ) time algorithm to construct drawings of G in the plane so that the size of the largest set of pairwise crossing edges, and at the same time, the size of the largest set of disjoint (pairwise noncrossing) edges are O( p t \Delta m). As an application we embed G in a book of O( p t \Delta m) pages, in O(n 2 m 1:5 ) time, resolving an open question for the pagenumber problem. A similar result is obtained for the dual of the pagenumber or the queuenumber. Our algorithms are obtained by derandomizing a probabilistic proof. 1 Introduction and Summary 1.1 Preliminaries Throughout this paper G = (V; E) is an undirected graph with jV j = n and jEj = m. A linear ordering of a set S is a bijection from S to f1; 2; : : : ; jSjg. Let h be a linear ordering of V . Consider a drawing of G that is obtained by placing the vertices along a straight line in the pl...
Notes on Large Angle Crossing Graphs
, 2010
"... A graph G is an α angle crossing (αAC) graph if every pair of crossing edges in G intersect at an angle of at least α. The concept of right angle crossing (RAC) graphs (α = π/2) was recently introduced by Didimo et al. [6]. It was shown that any RAC graph with n vertices has at most 4n −10 edges and ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
A graph G is an α angle crossing (αAC) graph if every pair of crossing edges in G intersect at an angle of at least α. The concept of right angle crossing (RAC) graphs (α = π/2) was recently introduced by Didimo et al. [6]. It was shown that any RAC graph with n vertices has at most 4n −10 edges and that there are infinitely many values of n for which there exists a RAC graph with n vertices and 4n − 10 edges. In this paper, we give upper and lower bounds for the number of edges in αAC graphs for all 0 < α < π/2. 1