arXiv subjects

Alan Arroyo

Publications and source records attributed to Alan Arroyo.

13 recordsLinked to original sources

arXiv2022

On Compatible Matchings

A matching is compatible to two or more labeled point sets of size $n$ with labels $\{1,\dots,n\}$ if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of $n$ points there exists a compatible matching with $\lfloor \sqrt {2n}\rfloor$ edges. More generally, for any $\ell$ labeled point sets we construct compatible matchings of size $Ω(n^{1/\ell})$. As a corresponding upper bound, we use probabilistic arguments to show that for any $\ell$ given sets of $n$ points there exists a labeling of each set such that the largest compatible matching has ${\mathcal{O}}(n^{2/({\ell}+1)})$ edges. Finally, we show that $Θ(\log n)$ copies of any set of $n$ points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.

cs.CG↗

arXiv2022

Inserting one edge into a simple drawing is hard

A {\em simple drawing} $D(G)$ of a graph $G$ is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge $e$ in the complement of $G$ can be {\em inserted} into $D(G)$ if there exists a simple drawing of $G+e$ extending $D(G)$. As a result of Levi's Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of $G$ can be inserted. In contrast, we show that it is NP -complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles $\mathcal{A}$ and a pseudosegment $σ$, it can be decided in polynomial time whether there exists a pseudocircle $Φ_σ$ extending $σ$ for which $\mathcal{A}\cup\{Φ_σ\}$ is again an arrangement of pseudocircles.

cs.CG↗

arXiv2021

Approximating the Bundled Crossing Number

Bundling crossings is a strategy which can enhance the readability of drawings. In this paper we consider good drawings, i.e., we require that any two edges have at most one common point which can be a common vertex or a crossing. Our main result is that there is a polynomial time algorithm to compute an 8-approximation of the bundled crossing number of a good drawing (up to adding a term depending on the facial structure of the drawing). In the special case of circular drawings the approximation factor is 8 (no extra term), this improves upon the 10-approximation of Fink et al. (Bundled crossings in embedded graphs, Proc. Latin'16). Our approach also works with the same approximation factor for families of pseudosegments, i.e., curves intersecting at most once. We also show how to compute a 9/2-approximation when the intersection graph of the pseudosegments is bipartite.

cs.CG↗

arXiv2021

Extending drawings of complete graphs into arrangements of pseudocircles

Motivated by the successful application of geometry to proving the Harary-Hill Conjecture for "pseudolinear" drawings of $K_n$, we introduce "pseudospherical" drawings of graphs. A spherical drawing of a graph $G$ is a drawing in the unit sphere $\mathbb{S}^2$ in which the vertices of $G$ are represented as points -- no three on a great circle -- and the edges of $G$ are shortest-arcs in $\mathbb{S}^2$ connecting pairs of vertices. Such a drawing has three properties: (1) every edge $e$ is contained in a simple closed curve $γ_e$ such that the only vertices in $γ_e$ are the ends of $e$; (2) if $e\ne f$, then $γ_e\capγ_f$ has precisely two crossings; and (3) if $e\ne f$, then $e$ intersects $γ_f$ at most once, either in a crossing or an end of $e$. We use Properties (1)--(3) to define a pseudospherical drawing of $G$. Our main result is that, for the complete graph, Properties (1)--(3) are equivalent to the same three properties but with "precisely two crossings" in (2) replaced by "at most two crossings". The proof requires a result in the geometric transversal theory of arrangements of pseudocircles. This is proved using the surprising result that the absence of special arcs ( coherent spirals) in an arrangement of simple closed curves characterizes the fact that any two curves in the arrangement have at most two crossings. Our studies provide the necessary ideas for exhibiting a drawing of $K_{10}$ that has no extension to an arrangement of pseudocircles and a drawing of $K_9$ that does extend to an arrangement of pseudocircles, but no such extension has all pairs of pseudocircles crossing twice.

math.CO↗

arXiv2021

Drawings of complete graphs in the projective plane

Hill's Conjecture states that the crossing number $\text{cr}(K_n)$ of the complete graph $K_n$ in the plane (equivalently, the sphere) is $\frac{1}{4}\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor\lfloor\frac{n-2}{2}\rfloor\lfloor\frac{n-3}{2}\rfloor=n^4/64 + O(n^3)$. Moon proved that the expected number of crossings in a spherical drawing in which the points are randomly distributed and joined by geodesics is precisely $n^4/64+O(n^3)$, thus matching asymptotically the conjectured value of $\text{cr}(K_n)$. Let $\text{cr}_P(G)$ denote the crossing number of a graph $G$ in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of $K_n$ is $(n^4/8π^2)+O(n^3)$. In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if $\lim_{n\to\infty} \text{cr}_P(K_n)/n^4=1/8π^2$. We construct drawings of $K_n$ in the projective plane that disprove this.

math.CO↗

arXiv2019

The unavoidable rotation systems

For each positive integer $m$, Pach, Solymosi, and Tóth identified two canonical complete topological subgraphs $C_m$ and $T_m$, and proved that every sufficiently large topological complete graph contains $C_m$ or $T_m$ as a subgraph. We generalize this result in the setting of abstract rotation systems.

math.CO↗

arXiv2019

Extending Simple Drawings

Simple drawings of graphs are those in which each pair of edges share at most one point, either a common endpoint or a proper crossing. In this paper we study the problem of extending a simple drawing $D(G)$ of a graph $G$ by inserting a set of edges from the complement of $G$ into $D(G)$ such that the result is a simple drawing. In the context of rectilinear drawings, the problem is trivial. For pseudolinear drawings, the existence of such an extension follows from Levi's enlargement lemma. In contrast, we prove that deciding if a given set of edges can be inserted into a simple drawing is NP-complete. Moreover, we show that the maximization version of the problem is APX-hard. We also present a polynomial-time algorithm for deciding whether one edge $uv$ can be inserted into $D(G)$ when $\{u,v\}$ is a dominating set for the graph $G$.

cs.CG↗

arXiv2019

Graphs with at most one crossing

The crossing number of a graph $G$ is the least number of crossings over all possible drawings of $G$. We present a structural characterization of graphs with crossing number one.

math.CO↗

arXiv2018

Extending Drawings of Graphs to Arrangements of Pseudolines

A pseudoline is a homeomorphic image of the real line in the plane so that its complement is disconnected. An arrangement of pseudolines is a set of pseudolines in which every two cross exactly once. A drawing of a graph is pseudolinear if the edges can be extended to an arrangement of pseudolines. In the recent study of crossing numbers, pseudolinear drawings have played an important role as they are a natural combinatorial extension of rectilinear drawings. A characterization of the pseudolinear drawings of $K_n$ was found recently. We extend this characterization to all graphs, by describing the set of minimal forbidden subdrawings for pseudolinear drawings. Our characterization also leads to a polynomial-time algorithm to recognize pseudolinear drawings and construct the pseudolines when it is possible.

math.CO↗

arXiv2017

Convex drawings of the complete graph: topology meets geometry

In this work, we introduce and develop a theory of convex drawings of the complete graph $K_n$ in the sphere. A drawing $D$ of $K_n$ is convex if, for every 3-cycle $T$ of $K_n$, there is a closed disc $Δ_T$ bounded by $D[T]$ such that, for any two vertices $u,v$ with $D[u]$ and $D[v]$ both in $Δ_T$, the entire edge $D[uv]$ is also contained in $Δ_T$. As one application of this perspective, we consider drawings containing a non-convex $K_5$ that has restrictions on its extensions to drawings of $K_7$. For each such drawing, we use convexity to produce a new drawing with fewer crossings. This is the first example of local considerations providing sufficient conditions for suboptimality. In particular, we do not compare the number of crossings {with the number of crossings in} any known drawings. This result sheds light on Aichholzer's computer proof (personal communication) showing that, for $n\le 12$, every optimal drawing of $K_n$ is convex. Convex drawings are characterized by excluding two of the five drawings of $K_5$. Two refinements of convex drawings are h-convex and f-convex drawings. The latter have been shown by Aichholzer et al (Deciding monotonicity of good drawings of the complete graph, Proc.~XVI Spanish Meeting on Computational Geometry (EGC 2015), 2015) and, independently, the authors of the current article (Levi's Lemma, pseudolinear drawings of $K_n$, and empty triangles, \rbr{J. Graph Theory DOI: 10.1002/jgt.22167)}, to be equivalent to pseudolinear drawings. Also, h-convex drawings are equivalent to pseudospherical drawings as demonstrated recently by Arroyo et al (Extending drawings of complete graphs into arrangements of pseudocircles, submitted).

math.CO↗

arXiv2016

The crossing number of the cone of a graph

Motivated by a problem asked by Richter and by the long standing Harary-Hill conjecture, we study the relation between the crossing number of a graph $G$ and the crossing number of its cone $CG$, the graph obtained from $G$ by adding a new vertex adjacent to all the vertices in $G$. Simple examples show that the difference $cr(CG)-cr(G)$ can be arbitrarily large for any fixed $k=cr(G)$. In this work, we are interested in finding the smallest possible difference, that is, for each non-negative integer $k$, find the smallest $f(k)$ for which there exists a graph with crossing number at least $k$ and cone with crossing number $f(k)$. For small values of $k$, we give exact values of $f(k)$ when the problem is restricted to simple graphs, and show that $f(k)=k+Θ(\sqrt {k})$ when multiple edges are allowed.

math.CO↗

arXiv2016

Drawings of Kn with the same rotation scheme are the same up to Reidemeister moves. Gioan's Theorem

A {\em good drawing\/} of $K_n$ is a drawing of the complete graph with $n$ vertices in the sphere such that: no two edges with a common end cross; no two edges cross more than once; and no three edges all cross at the same point. Gioan's Theorem asserts that any two good drawings of $K_n$ that have the same rotations of incident edges at every vertex are equivalent up to Reidemeister moves. At the time of preparation, 10 years had passed between the statement in the WG 2005 conference proceedings and our interest in the proposition. Shortly after we completed our preprint, Gioan independently completed a preprint.

math.CO↗

arXiv2015

Levi's Lemma, pseudolinear drawings of $K_n$, and empty triangles

There are three main thrusts to this article: a new proof of Levi's Enlargement Lemma for pseudoline arrangements in the real projective plane; a new characterization of pseudolinear drawings of the complete graph; and proofs that pseudolinear and convex drawings of $K_n$ have $n^2+{}$O$(n\log n)$ and O$(n^2)$, respectively, empty triangles. All the arguments are elementary, algorithmic, and self-contained.

math.CO↗