منابع مشابه
Maximum Rectilinear Crossing Number
The problem of drawing a graph in the plane with a minimum number of edge crossings—called the crossing number of a graph—is a well-studied problem which dates back to the first half of the twentieth century, as mentioned in [11], and was formulated in full generality in [3]. It was shown that this problem is NP-Complete [4], and that it remains so even when restricted to cubic graphs [5]. Many...
متن کاملApproximating the Rectilinear Crossing Number
A straight-line drawing of a graph G is a mapping which assigns to each vertex a point in the plane and to each edge a straightline segment connecting the corresponding two points. The rectilinear crossing number of a graph G, cr(G), is the minimum number of pairs of crossing edges in any straight-line drawing of G. Determining or estimating cr(G) appears to be a difficult problem, and deciding...
متن کاملNew Lower Bounds for the Number of (<=k)-Edges and the Rectilinear Crossing Number of Kn
We provide a new lower bound on the number of (≤ k)-edges of a set of n points in the plane in general position. We show that for 0 ≤ k ≤ bn−2 2 c the number of (≤ k)-edges is at least Ek(S) ≥ 3 ( k + 2 2 ) + k ∑ j=b3 c (3j − n + 3), which, for b3 c ≤ k ≤ 0.4864n, improves the previous best lower bound in [11]. As a main consequence, we obtain a new lower bound on the rectilinear crossing numbe...
متن کاملApproximating the Maximum Rectilinear Crossing Number
The problem of drawing a graph in the plane with a minimum number of edge crossings—called the crossing number of a graph—is a well-studied problem which dates back to the first half of the twentieth century, as mentioned in [11], and was formulated in full generality in [3]. It was shown that this problem is NP-Complete [4], and that it remains so even when restricted to cubic graphs [5]. Many...
متن کاملk-Sets, Convex Quadrilaterals, and the Rectilinear Crossing Number of Kn
We use circular sequences to give an improved lower bound on the minimum number of (≤ k)– sets in a set of points in general position. We then use this to show that if S is a set of n points in general position, then the number (S) of convex quadrilaterals determined by the points in S is at least 0.37533 ` n 4 ́ + O(n). This in turn implies that the rectilinear crossing number cr(Kn) of the com...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 2017
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2017.04.003