arXiv subjects

David Orden

Publications and source records attributed to David Orden.

43 records · Page 3Linked to original sources

arXiv2006

New lower bounds for the number of $(\leq k)$-edges and the rectilinear crossing number of $K_n$

We provide a new lower bound on the number of $(\leq k)$-edges of a set of $n$ points in the plane in general position. We show that for $0 \leq k \leq\lfloor\frac{n-2}{2}\rfloor$ the number of $(\leq k)$-edges is at least $$ E_k(S) \geq 3\binom{k+2}{2} + \sum_{j=\lfloor\frac{n}{3}\rfloor}^k (3j-n+3), $$ which, for $k\geq \lfloor\tfrac{n}{3}\rfloor$, improves the previous best lower bound in [J. Balogh, G. Salazar, Improved bounds for the number of ($\leq k$)-sets, convex quadrilaterals, and the rectilinear crossing number of $K_n$]. As a main consequence, we obtain a new lower bound on the rectilinear crossing number of the complete graph or, in other words, on the minimum number of convex quadrilaterals determined by $n$ points in the plane in general position. We show that the crossing number is at least $$ \Bigl({41/108}+ε\Bigr) \binom{n}{4} + O(n^3) \geq 0.379631 \binom{n}{4} + O(n^3), $$ which improves the previous bound of $0.37533 \binom{n}{4} + O(n^3)$ in [J. Balogh, G. Salazar, Improved bounds for the number of ($\leq k$)-sets, convex quadrilaterals, and the rectilinear crossing number of $K_n$] and approaches the best known upper bound $0.38058\binom{n}{4}$ in [O. Aichholzer, H. Krasser, Abstract order type extension and new results on the rectilinear crossing number].

math.CO↗

arXiv2006

From graphs to tensegrity structures: Geometric and symbolic approaches

A form-finding problem for tensegrity structures is studied; given an abstract graph, we show an algorithm to provide a necessary condition for it to be the underlying graph of a tensegrity in $\mathbb{R}^d$ (typically $d=2,3$) with vertices in general position. Furthermore, for a certain class of graphs our algorithm allows to obtain necessary and sufficient conditions on the relative position of the vertices in order to underlie a tensegrity, for what we propose both a geometric and a symbolic approach.

math.CO↗

arXiv2005

Combinatorial pseudo-Triangulations

We prove that a planar graph is generically rigid in the plane if and only if it can be embedded as a pseudo-triangulation. This generalizes the main result of math.CO/0307347 which treats the minimally generically rigid case. The proof uses the concept of combinatorial pseudo-triangulation, CPT, in the plane and has two main steps: showing that a certain ``generalized Laman property'' is a necessary and sufficient condition for a CPT to be ``stretchable'', and showing that all generically rigid plane graphs admit a CPT assignment with that property. Additionally, we propose the study of combinatorial pseudo-triangulations on closed surfaces.

math.CO↗

arXiv2003

Non-crossing frameworks with non-crossing reciprocals

We study non-crossing frameworks in the plane for which the classical reciprocal on the dual graph is also non-crossing. We give a complete description of the self-stresses on non-crossing frameworks whose reciprocals are non-crossing, in terms of: the types of faces (only pseudo-triangles and pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a geometric condition on the stress vectors at some of the vertices. As in other recent papers where the interplay of non-crossingness and rigidity of straight-line plane graphs is studied, pseudo-triangulations show up as objects of special interest. For example, it is known that all planar Laman circuits can be embedded as a pseudo-triangulation with one non-pointed vertex. We show that if such an embedding is sufficiently generic, then the reciprocal is non-crossing and again a pseudo-triangulation embedding of a planar Laman circuit. For a singular (i.e., non-generic) pseudo-triangulation embedding of a planar Laman circuit, the reciprocal is still non-crossing and a pseudo-triangulation, but its underlying graph may not be a Laman circuit. Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal arise as the reciprocals of such, possibly singular, stresses on pseudo-triangulation embeddings of Laman circuits. All self-stresses on a planar graph correspond to liftings to piece-wise linear surfaces in 3-space. We prove characteristic geometric properties of the lifts of such non-crossing reciprocal pairs.

math.MG↗

arXiv2003

Planar Minimally Rigid Graphs and Pseudo-Triangulations

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (adjacent to an angle larger than 180 degrees. In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. We provide two proofs, which both yield efficient embedding algorithms. One based on Henneberg inductive constructions from combinatorial rigidity theory, the other on a generalization of Tutte's barycentric embeddings to directed graphs.

math.CO↗

arXiv2003

The polytope of non-crossing graphs on a planar point set

For any finite set $\A$ of $n$ points in $\R^2$, we define a $(3n-3)$-dimensional simple polyhedron whose face poset is isomorphic to the poset of ``non-crossing marked graphs'' with vertex set $\A$, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of non-crossing graphs on $\A$ appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension $2n_i +n -3$ where $n_i$ is the number of points of $\A$ in the interior of $\conv(\A)$. The vertices of this polytope are all the pseudo-triangulations of $\A$, and the edges are flips of two types: the traditional diagonal flips (in pseudo-triangulations) and the removal or insertion of a single edge. As a by-product of our construction we prove that all pseudo-triangulations are infinitesimally rigid graphs.

math.CO↗

arXiv2003

Asymptotically efficient triangulations of the d-cube

Let $P$ and $Q$ be polytopes, the first of "low" dimension and the second of "high" dimension. We show how to triangulate the product $P \times Q$ efficiently (i.e., with few simplices) starting with a given triangulation of $Q$. Our method has a computational part, where we need to compute an efficient triangulation of $P \times Δ^m$, for a (small) natural number $m$ of our choice. $Δ^m$ denotes the $m$-simplex. Our procedure can be applied to obtain (asymptotically) efficient triangulations of the cube $I^n$: We decompose $I^n = I^k \times I^{n-k}$, for a small $k$. Then we recursively assume we have obtained an efficient triangulation of the second factor and use our method to triangulate the product. The outcome is that using $k=3$ and $m=2$, we can triangulate $I^n$ with $O(0.816^{n} n!)$ simplices, instead of the $O(0.840^{n} n!)$ achievable before.

math.CO↗