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)
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)
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)
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)
For n ≤ 27 we present exact values for the maximum number h(n) of halving lines and e 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(n) and the pseudolinear e cr(n) crossing number of the complete graph Kn. e h(n) and e cr(n) are new for
It is shown that if a simple Euclidean arrangement of n pseudolines has no (≥ 5)–gons, then it has exactly n − 2 triangles and (n − 2)(n − 3)/2 quadrilaterals. We also describe how to construct all such arrangements, and as a consequence we show that they are all stretchable.