Jesús Leaños

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Even the most super cial glance at the vast majority of crossing-minimal geometric drawings of Kn 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)
For n ≤ 27 we present exact values for the maximum number h(n) of halving lines and eh(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. eh(n) and e cr(n) are new for n ∈ {14, 16, 18, 20,(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)
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)
Devadoss asked: (1) can every polygon be convexified so that no internal visibility (between vertices) is lost in the process? Moreover, (2) does such a convexification exist, in which exactly one vertex is moved at a time (that is, using single-vertex moves)? We prove the redundancy of the “singlevertex moves” condition: an affirmative answer to (1)(More)
We show that for each integer g ≥ 0 there is a constant cg > 0 such that every graph that embeds in the projective plane with sufficiently large face–width r has crossing number at least cgr 2 in the orientable surface Σg of genus g. As a corollary, we give a polynomial time constant factor approximation algorithm for the crossing number of projective(More)
Recently, Aichholzer, Garćıa, 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 Kn. We use simple allowable sequences to extend all their results to the more general setting of simple generalized(More)
Let P be a simple polygon on the plane. Two vertices of P are visible if the open line segment joining them is contained in the interior of P . In this paper we study the following questions posed in [5, 6]: (1) Is it true that every non-convex simple polygon has a vertex that can be continuously moved such that during the process no vertex-vertex(More)