Results 1 
8 of
8
Pfaffian graphs, tjoins, and crossing numbers
"... Abstract. We prove a technical theorem about the numbers of crossings in Tjoins in different drawings of a fixed graph. As a corollary we characterize Pfaffian graphs in terms of their drawings in the plane and give a new proof of a theorem of Kleitman on the parity of crossings in drawings of K2j+ ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Abstract. We prove a technical theorem about the numbers of crossings in Tjoins in different drawings of a fixed graph. As a corollary we characterize Pfaffian graphs in terms of their drawings in the plane and give a new proof of a theorem of Kleitman on the parity of crossings in drawings of K2j+1 and K2j+1,2k+1. This gives a new proof of the HananiTutte theorem. 1.
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
Removing even crossings on surfaces
 European Journal of Combinatorics, 30(7):1704
"... We give a new, topological proof that the weak HananiTutte theorem is true on orientable surfaces and extend the result to nonorientable surfaces. That is, we show that if a graph G cannot be embedded on a surface S, then any drawing of G on S must contain two edges that cross an odd number of time ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
We give a new, topological proof that the weak HananiTutte theorem is true on orientable surfaces and extend the result to nonorientable surfaces. That is, we show that if a graph G cannot be embedded on a surface S, then any drawing of G on S must contain two edges that cross an odd number of times. We apply the result and proof techniques to obtain new and old results about generalized thrackles, including that every bipartite generalized thrackle in a surface S can be embedded in S. We also extend to arbitrary surfaces a result of Pach and Tóth that allows the redrawing of a graph so as to remove all crossings with even edges (an edge is even if it crosses every other edge an even number of times). From this result we can conclude that crS(G), the crossing number of a graph G on surface S, is bounded by 2 ocrS(G) 2,whereocrS(G) istheoddcrossing number of G on surface S. Finally, we show that ocrS(G) =crS(G) whenever ocrS(G) ≤ 2, for any surface S. Keywords: number.
HananiTutte and Related Results
, 2011
"... We are taking the view that crossings of adjacent edges are trivial, and easily got rid of. ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
We are taking the view that crossings of adjacent edges are trivial, and easily got rid of.
On the Boundary of the Union of Planar Convex Sets
 Discrete Comput. Geom
"... We give two alternative proofs leading to different generalizations of the following theorem of [1]. Given n convex sets in the plane, such that the boundaries of each pair of sets cross at most twice, then the boundary of their union consists of at most 6n \Gamma 12 arcs. (An arc is a connected ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We give two alternative proofs leading to different generalizations of the following theorem of [1]. Given n convex sets in the plane, such that the boundaries of each pair of sets cross at most twice, then the boundary of their union consists of at most 6n \Gamma 12 arcs. (An arc is a connected piece of the boundary of one of the sets.) In the generalizations we allow pairs of boundaries to cross more than twice.
On the Complexity of the Union of Fat Objects in
 Proc. 13th ACM Symposium on Computational Geometry
, 1997
"... We prove a nearlinear bound on the combinatorial complexity of the union of n fat convex objects in the plane, each pair of whose boundaries cross at most a constant number of times. ..."
Abstract
 Add to MetaCart
We prove a nearlinear bound on the combinatorial complexity of the union of n fat convex objects in the plane, each pair of whose boundaries cross at most a constant number of times.
Removing even crossings on surfaces
"... a b s t r a c t In this paper we investigate how certain results related to the Hanani–Tutte theorem can be extended from the plane to surfaces. We give a simple topological proof that the weak Hanani–Tutte theorem is true on arbitrary surfaces, both orientable and nonorientable. We apply these resu ..."
Abstract
 Add to MetaCart
a b s t r a c t In this paper we investigate how certain results related to the Hanani–Tutte theorem can be extended from the plane to surfaces. We give a simple topological proof that the weak Hanani–Tutte theorem is true on arbitrary surfaces, both orientable and nonorientable. We apply these results and the proof techniques to obtain new and old results about generalized thrackles, including that every bipartite generalized thrackle on a surface S can be embedded on S. We also extend to arbitrary surfaces a result of Pach and Tóth that allows the redrawing of a graph so as to remove all crossings with even edges. From this we can conclude that crS(G), the crossing number of a graph G on surface S, is bounded by 2 ocrS(G) 2, where ocrS(G) is the odd crossing number of G on surface S. Finally, we prove that ocrS(G) = crS(G) whenever ocrS(G) ≤ 2, for any surface S. 1.