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It is known that every generalized configuration with n points has at least 3 ` k+2 2 ´ (≤ k)–pseudoedges, and that this bound is tight for k ≤ n/3 − 1. Here we show that this bound is no longer tight for (any) k > n/3 − 1. As a corollary, we prove that the usual and the pseudolinear (and hence the rectilinear) crossing numbers of the complete graph Kn are(More)
For n ≤ 27 we present exact values for the maximum number h(n) of halving lines and h(n) of halving pseudolines, determined by n points in the plane. For this range of values of n we also present exact values of the rectilinear cr(K n) and the pseudolinear cr(K n) crossing numbers of the complete graph K n. h(n) and cr(K n) are new for
We show that for each integer g ≥ 0 there is a constant c g > 0 such that every graph that embeds in the projective plane with sufficiently large face–width r has crossing number at least c g r 2 in the orientable surface Σ g of genus g. As a corollary, we give a polynomial time constant factor approximation algorithm for the crossing number of projective(More)
Recently, Aichholzer, García, 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 K n. We use simple allowable sequences to extend all their results to the more general setting of simple generalized(More)
A generalized configuration is a set of n points and n 2 pseudolines such that each pseudoline passes through exactly two points, two pseudolines intersect exactly once, and no three pseudolines are concurrent. Following the approach of allowable sequences we prove a recursive inequality for the number of (≤ k)-sets for generalized configurations. As a(More)
Even the most supercial glance at the vast majority of crossing-minimal geometric drawings of K n reveals two hard-to-miss features. First, all such drawings appear to be 3-fold symmetric (or simply 3-symmetric). And second, they all are 3-decomposable, that is, there is a triangle T enclosing the drawing, and a balanced partition A, B, C of the underlying(More)
Let P be a set of points in general position in the plane. Join all pairs of points in P with straight line segments. The number of segment-crossings in such a drawing, denoted by cr(P), is the rectilinear crossing number of P. A halving line of P is a line passing though two points of P that divides the rest of the points of P in (almost) half. The number(More)
A pseudolinear (respectively, rectilinear) drawing of a graph G is optimal if it has the smallest number of crossings among all pseudolinear (respectively, rectilinear) drawings of G. We show that the convex hull of every optimal pseudolinear drawing of the complete graph Kn is a triangle. This is closely related to the recently announced result that the(More)
1 Let P be a simple polygon on the plane. Two vertices of P are visible if 2 the open line segment joining them is contained in the interior of P. In this 3 paper we study the following questions posed in [5, 6]: (1) Is it true that 4 every non-convex simple polygon has a vertex that can be continuously 5 moved such that during the process no vertex-vertex(More)