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- Bernardo M. Ábrego, Mario Cetina, Silvia Fernández-Merchant, Jesús Leaños, Gelasio Salazar
- Discrete Applied Mathematics
- 2010

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)

- Bernardo M. Ábrego, Silvia Fernández-Merchant, Jesús Leaños, Gelasio Salazar
- Electronic Notes in Discrete Mathematics
- 2008

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)

- Bernardo M. Ábrego, Silvia Fernández-Merchant, Jesús Leaños, Gelasio Salazar
- Electronic Notes in Discrete Mathematics
- 2008

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)

- B. Ábrego, J. Balogh, S. Fernández–Merchant, J. Leaños, G. Salazar
- 2006

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)

We describe the relationship between the crossing number of a graph G with a 2-edge-cut C and the crossing numbers of the components of G− C. Let G be a connected graph with a 2-edge-cut C := [V1, V2]. Let u1u2, v1v2 be the edges of C, so that ui, vi ∈ Vi for i = 1, 2, and let Gi := G[Vi] and Gi := Gi +uivi. We show that if either G1 or G2 is not connected,… (More)

- Bernardo M. Ábrego, Mario Cetina, Jesús Leaños, Gelasio Salazar
- Inf. Process. Lett.
- 2012

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)

- Isidoro Gitler, Petr Hlinený, Jesús Leaños, Gelasio Salazar
- Electronic Notes in Discrete Mathematics
- 2007

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)

- Bernardo M. Ábrego, József Balogh, Silvia Fernández-Merchant, Jesús Leaños, Gelasio Salazar
- J. Comb. Theory, Ser. A
- 2008

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)

- Jesús Leaños, Mario Lomelí, Criel Merino, Gelasio Salazar, Jorge Urrutia
- Discrete & Computational Geometry
- 2007

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.