Approximating 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 straight-line segment connecting the corresponding two points. The rectilinear crossing number of a graph \(G, \text {cr}(G)\), is the minimum number of pairs of crossing edges in any straight-line drawing of G. Determining or estimating \(\overline{\mathrm{cr}}(G)\) appears to be a difficult problem, and deciding if \(\overline{\mathrm{cr}}(G)\le k\) is known to be NP-hard. In fact, the asymptotic behavior of \(\overline{\mathrm{cr}}(K_n)\) is still unknown.

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