On the Pseudolinear Crossing Number
نویسندگان
چکیده
منابع مشابه
On the Pseudolinear Crossing Number
4 A drawing of a graph is pseudolinear if there is a pseudoline arrangement such that 5 each pseudoline contains exactly one edge of the drawing. The pseudolinear crossing 6 number c̃r(G) of a graph G is the minimum number of pairwise crossings of edges in 7 a pseudolinear drawing of G. We establish several facts on the pseudolinear crossing 8 number, including its computational complexity and i...
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Research about crossings is typically about minimization. In this paper, we consider maximizing the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [1] conjectured that any graph has a convex straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructin...
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The degenerate crossing number cr∗(G) of a graph G is the minimum number of crossing points of edges in any drawing of G as a simple topological graph in the plane. This notion was introduced by Pach and Tóth who showed that for a graph G with n vertices and e ≥ 4n edges cr∗(G) = Ω(e/n). In this paper we completely resolve the main open question about degenerate crossing numbers and show that c...
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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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Recently, Aichholzer, Garćıa, Orden, and Ramos derived a remarkably improved lower bound for the number of (≤ k)-edges in an n-point set, and as an immediate corollary an improved lower bound on the rectilinear crossing number of Kn. We use simple allowable sequences to extend all their results to the more general setting of simple generalized configurations of points and slightly improve the l...
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ژورنال
عنوان ژورنال: Journal of Graph Theory
سال: 2016
ISSN: 0364-9024
DOI: 10.1002/jgt.22027