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11
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 ..."
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Cited by 6 (3 self)
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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
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+ ..."
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Cited by 6 (0 self)
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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.
Removing even crossings on surfaces
 EUROPEAN JOURNAL OF COMBINATORICS, 30(7):1704
, 2009
"... 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 ..."
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Cited by 5 (4 self)
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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, where ocr S(G) is the odd crossing number of G on surface S. Finally, we show that ocrS(G) =crS(G) whenever ocrS(G) ≤ 2, for any surface S.
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.
Gauss Codes and Thrackles  On Characterizations of Closed Curves in the Plane with an Application to the Thrackle Conjecture
, 2006
"... ..."
A Reduction of Conway’s Thrackle Conjecture
"... Abstract. A thrackle is a drawing of a simple graph on the plane, where each edge is drawn as a smooth arc with distinct endpoints, and every two arcs have exactly one common point, at which they have distinct tangents. Conway, who coined the term thrackle, conjectured that there is no thrackle wit ..."
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Abstract. A thrackle is a drawing of a simple graph on the plane, where each edge is drawn as a smooth arc with distinct endpoints, and every two arcs have exactly one common point, at which they have distinct tangents. Conway, who coined the term thrackle, conjectured that there is no thrackle with more edges than vertices – a question which is still unsolved. A full thrackle is one with n vertices and n edges, and it is called nonextensible if it cannot be a subthrackle of a counterexample to Conway’s conjecture on n vertices. We define the notion of incidence type for a thrackle, which is the sequence of degrees of all vertices in increasing order. We introduce three reduction operations that can be applied to full subthrackles of thrackles. These reductions enable us to rule out the extensibility of many infinite series of incidence types of full thrackles. After defining the 123 set, we reduce Conway’s conjecture to the problem of proving that thrackles from the 123 set are not extensible. Our result proves the hypothesis of Wehner, who predicted that a potential counterexample to Conway’s conjecture would have certain graphtheoretic properties. 1
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. ..."
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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 A Study of Conway’s Thrackle Conjecture
"... A thrackle is a drawing of a simple graph on the plane, where each edge is drawn as a smooth arc with distinct endpoints, and every two arcs have exactly one common point, at which they have distinct tangents. Conway, who coined the term thrackle, conjectured that there is no thrackle with more edg ..."
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A thrackle is a drawing of a simple graph on the plane, where each edge is drawn as a smooth arc with distinct endpoints, and every two arcs have exactly one common point, at which they have distinct tangents. Conway, who coined the term thrackle, conjectured that there is no thrackle with more edges than vertices – a question which is still unsolved. A full thrackle is one with n vertices and n edges, and it is called nonextensible, if it cannot be a subthrackle of a counterexample to Conway’s conjecture on n vertices. We define the notion of incidence type for a thrackle, which is the sequence of degrees of all vertices in increasing order. We introduce three reduction operations that can be applied to full subthrackles of thrackles. These reductions enable us to rule out the extensibility of many infinite series of incidence types of full thrackles. After defining the 123 group, we reduce Conway’s conjecture to the problem of proving that thrackles from the 123 group are not extensible. Our result proves the hypothesis of Wehner, who predicted that a potential counterexample to Conway’s conjecture would have certain graphtheoretic properties, which he described in [4]. 1