There is a unique crossing-minimal rectilinear drawing of \(K_{18}\) (Q6545086)

The rectilinear crossing number \(\overline{\operatorname{cr}}(G)\) of a graph \(G\) is the minimum number of edge crossings in a rectilinear drawing of \(G\) in the plane, i.e., a drawing of \(G\) in the plane where the vertices are points in general position and the edges are straight line segments. A drawing of \(G\) with exactly \(\overline{\operatorname{cr}}(G)\) crossings is called crossing-minimal. In this paper, the authors show that, up to order type isomorphism, there is a unique crossing-minimal rectilinear drawing of \(K_{18}\), namely: Up to order-type isomorphism, there is a unique 18-point set whose induced rectilinear drawing of \(K_{18}\) has \(\overline{\operatorname{cr}}(K_{18})\) crossings. It is easily verified that this drawing does not contain any crossing-minimal drawing of \(K_{17}\). Therefore this settles, in the negative, the following question from \textit{O. Aichholzer} and \textit{H. Krasser} [Comput. Geom. 36, No. 1, 2--15 (2007; Zbl 1110.65019)]: is it true that, for every integer \(n \geq 4\), there exists a crossing-minimal drawing of \(K_n\) that contains a crossing-minimal drawing of \(K_{n-1}\)? A conjecture and an open question conclude the paper.