منابع مشابه
Removing even crossings
An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a the...
متن کاملRemoving Independently Even Crossings
We show that cr(G) ≤ (2 iocr(G) 2 ) settling an open problem of Pach and Tóth [5,1]. Moreover, iocr(G) = cr(G) if iocr(G) ≤ 2.
متن کاملRemoving Even Crossings, Continued
In this paper we investigate how certain results related to the HananiTutte theorem can be lifted to orientable surfaces of higher genus. We give a new simple, geometric proof that the weak Hanani-Tutte theorem is true for higher-genus surfaces. We extend the proof to prove that bipartite generalized thrackles in a surface S can be embedded in S. We also show that a result of Pach and Tóth that...
متن کاملRemoving Even Crossings on Surfaces
We give a new, topological proof that the weak Hanani-Tutte 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 thrac...
متن کاملALCOMFT - TR - 01 - 148 Removing Cycles for Minimizing Crossings
We consider the one-sided crossing minimization problem (CP): given a bipartite graph G and a permutation x 0 of the vertices on a layer, nd a permutation x 1 of the vertices on the other layer which minimizes the number of edge crossings in any straightline drawing of G where vertices are placed on two parallel lines and sorted according to x 0 and x 1. Solving CP represents a fundamental step...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series B
سال: 2007
ISSN: 0095-8956
DOI: 10.1016/j.jctb.2006.08.001