Note on the Pair-crossing Number and the Odd-crossing Number
نویسندگان
چکیده
منابع مشابه
Note on the Pair-Crossing Number and the Odd-Crossing Number
The crossing number cr(G) of a graph G is the minimum possible number of edge-crossings in a drawing of G, the pair-crossing number pcr(G) is the minimum possible number of crossing pairs of edges in a drawing of G, and the odd-crossing number ocr(G) is the minimum number of pairs of edges that cross an odd number of times. Clearly, ocr(G) ≤ pcr(G) ≤ cr(G). We construct graphs with 0.855 · pcr(...
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The crossing number of a graph is the minimum number of edge intersections in a plane drawing of a graph, where each intersection is counted separately. If instead we count the number of pairs of edges that intersect an odd number of times, we obtain the odd crossing number. We show that there is a graph for which these two concepts differ, answering a well-known open question on crossing numbe...
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The crossing number crðGÞ of a graph G is the minimum possible number of edge crossings in a drawing of G in the plane, while the pair-crossing number pcrðGÞ is the smallest number of pairs of edges that cross in a drawing of G in the plane. While crðGÞXpcrðGÞ holds trivially, it is not known whether a strict inequality can ever occur (this question was raised by Mohar and Pach and Tóth). We ai...
متن کاملOdd Crossing Number and Crossing Number Are Not the Same
The crossing number of a graph is the minimum number of edge intersections in a plane drawing of a graph, where each intersection is counted separately. If instead we count the number of pairs of edges that intersect an odd number of times, we obtain the odd crossing number. We show that there is a graph for which these two concepts differ, answering a well-known open question on crossing numbe...
متن کاملOn the Pair-Crossing Number
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 pair-crossing number pair-cr(G) of G is the minimum possible number of (unordered) crossing pairs in a drawing of G. Clearly, pair-cr(G) ≤ cr(G) holds for any g...
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ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2007
ISSN: 0179-5376,1432-0444
DOI: 10.1007/s00454-007-9024-z