Unexpected behaviour of crossing sequences

Let $K_{n}$ ($n>1$) be an $n$--punctured non--orientable surface of genus $2$ with 1 boundary component. We give formulae for calculating the geometric intersection number of an arbitrary multicurve with a relaxed multicurve on $K_{n}$ given their generalized Dynnikov coordinates.

On the one-sided crossing minimization in a bipartite graph with large degrees

<abstract><p>The skewness of a graph $ G $, $ sk(G) $, is the smallest number of edges that need to be removed from $ G $ to make it planar. The crossing number of a graph $ G $, $ cr(G) $, is the minimum number of crossings over all possible drawings of $ G $. There is minimal work concerning the relationship between skewness and crossing numbers. In this work, we first introduce an inequality relation for these two parameters, and then we construct infinitely many near-planar graphs such that the inequality is equal. In addition, we give a necessary and sufficient condition for a graph to have its skewness equal to the crossing number and characterize some special graphs with $ sk(G) = cr(G) $.</p></abstract>

The crossing number of a graph [Formula: see text], denoted by [Formula: see text], is defined to be the minimum number of crossings that arise among all its drawings in the plane. This concept has been of interest to many researchers who have studied it for many families of graphs. In this paper, we introduce the crossing number of Cartesian product of sunlet graph [Formula: see text] with path [Formula: see text]. Further, we prove that the crossing number of [Formula: see text] is equal to [Formula: see text], along with giving a conjecture for the general case. In addition, we utilize the vertex’s rotation concept in order to prove some necessary conditions for the complete bipartite graph [Formula: see text] to be optimal when it is drawn in the plane, by presenting an upper bound for the crossing number of any subgraph in it together with determining the exact number of crossings in case the vertices of subgraph have the same rotation.

Connecting colored point sets

A book with $k$ pages consists of a straight line (the spine) and $k$ half-planes (the pages), such that the boundary of each page is the spine. If a graph is drawn on a book with $k$ pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a $k$-page drawing). The $k$-page crossing number $\nu_k(G)$ of a graph $G$ is the minimum number of crossings in a $k$-page drawing of $G$. In this paper we investigate the $k$-page crossing numbers of complete graphs. We use semidefinite programming techniques to give improved lower bounds on $\nu_k(K_n)$ for various values of $k$. We also use a maximum satisfiability reformulation to obtain a computer-aided calculation of the exact value of $\nu_k(K_n)$ for several values of $k$ and $n$. Finally, we investigate the best construction known for drawing $K_n$ in $k$ pages, calculate the resulting number of crossings, and discuss this upper bound in light of the new results reported in this paper.

In phylogenetics, tree-based networks are used to model and visualize the evolutionary history of species where reticulate events such as horizontal gene transfer have occurred. Formally, a tree-based network N consists of a phylogenetic tree T (a rooted, binary, leaf-labeled tree) and so-called reticulation edges that span between edges of T. The network N is typically visualized by drawing T downward and planar and reticulation edges with one of several different styles. One aesthetic criteria is to minimize the number of crossings between tree edges and reticulation edges. This optimization problem has not yet been researched. We show that, if reticulation edges are drawn x-monotone, the problem is NP-complete, but fixed-parameter tractable in the number of reticulation edges. If, on the other hand, reticulation edges are drawn like “ears”, the crossing minimization problem can be solved in quadratic time.

The delunification process and minimal diagrams

We study a boundary labeling problem, where multiple points may connect to the same label. In this new many-to-one model, a horizontal backbone reaches out of each label into the feature-enclosing rectangle. Feature points that need to be connected to this label are linked via vertical line segments to the backbone. We present dynamic programming algorithms for minimizing the total number of label occurrences and for minimizing the total leader length of crossing-free backbone labelings. When crossings are allowed, we aim at obtaining solutions with the minimum number of crossings. This can be achieved efficiently in the case of fixed label order; however, in the case of flexible label order we show that minimizing the number of leader crossings is NP-hard.

Let cr(Kn) be the minimum number of crossings over all rectilinear drawings of the complete graph on n vertices in the plane. In this paper we prove that cr(Kn) < 0:380473 n + ( n 3 ); improving thus on the previous best known upper bound. This is done by obtaining new rectilinear drawings of Kn for small values of n, and then using known constructions to obtain arbitrarily large good drawings from smaller ones. The \small sets were found using a simple heuristic detailed in this paper.

On the classification of rational tangles

We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$, for $i\in\{0,\ldots, n-1\}$. The problem of realizing $A$ by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an angle graph. In 2D, we characterize sequences $A$ for which every generic polygon $P\subset \mathbb{R}^2$ realizing $A$ has at least $c$ crossings, for every $c\in \mathbb{N}$, and describe an efficient algorithm that constructs, for a given sequence $A$, a generic polygon $P\subset \mathbb{R}^2$ that realizes $A$ with the minimum number of crossings. In 3D, we describe an efficient algorithm that tests whether a given sequence $A$ can be realized by a (not necessarily generic) polygon $P\subset \mathbb{R}^3$, and for every realizable sequence the algorithm finds a realization.

The classical theorem of Fary states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fary’s theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces. We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property. Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil–Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.

A map is a \(2\) -cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. An automorphism of a map can be thought of as a permutation of the vertices, which preserves the vertex-edge-face incidences in the embedding. Every automorphism of a map determines an angle-preserving homeomorphism of the surface. While it is conjectured that there is no “truly subquadratic” algorithm for testing map isomorphism for unconstrained genus, we present a linear-time algorithm for computing the generators of the automorphism group of a map on an orientable surface of genus \(g\neq 0\) , parametrized by the genus \(g\) . A map on an orientable surface is uniform if the cyclic vector of sizes of faces incident to a vertex \(v\) does not depend on the choice of \(v\) . The algorithm applies a sequence of local reductions and produces a uniform map while preserving the automorphism group. The automorphism group of the original map can be reconstructed from the automorphism group of the associated uniform map in linear time. We also extend the algorithm to non-orientable surfaces by making use of the antipodal double-cover. The algorithm can be used to solve the map isomorphism problem between maps (orientable or non-orientable) of bounded negative Euler characteristic.

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various sub- groups of the mapping class group—the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration—and additionally to the outer automorphism group of a free group and to certain linear groups. There are many analogies and interconnections between the theories of right- angled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest (13), the work of Koberda (27), and the work of the first two authors with Mangahas (12), there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved (30) that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups (4, Theorem 1.2). We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index sub- groups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph , we can associate a right-angled Artin grou p: this is the group with one generator for each vertex of , and one defin ing relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the set of marked points. As discussed in Koberda's paper (27, Theorem 1.5), no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g,n) 6 (2,0); see