An Introduction to Virtual Spatial Graph Theory

Two natural generalizations of knot theory are the study of spatial graphs and virtual knots. Our goal is to unify these two approaches into the study of virtual spatial graphs. This paper is a survey, and does not contain any new results. We state the definitions, provide some examples, and survey the known results. We hope that this paper will help lead to rapid development of the area.Comment: 9 pages, 7 figures, presented at the International Workshop on Knot Theory for Scientific Objects at Osaka City University, March 200

Even maps, the Colin de~Verdi\`ere number and representations of graphs

Van der Holst and Pendavingh introduced a graph parameter $\sigma$, which coincides with the more famous Colin de Verdi\`{e}re graph parameter $\mu$ for small values. However, the definition of $\sigma$ is much more geometric/topological directly reflecting embeddability properties of the graph. They proved $\mu(G) \leq \sigma(G) + 2$ and conjectured $\mu(G) \leq \sigma(G)$ for any graph $G$. We confirm this conjecture. As far as we know, this is the first topological upper bound on $\mu(G)$ which is, in general, tight. Equality between $\mu$ and $\sigma$ does not hold in general as van der Holst and Pendavingh showed that there is a graph $G$ with $\mu(G) \leq 18$ and $\sigma(G)\geq 20$. We show that the gap appears on much smaller values, namely, we exhibit a graph $H$ for which $\mu(H)\leq 7$ and $\sigma(H)\geq 8$. We also prove that, in general, the gap can be large: The incidence graphs $H_q$ of finite projective planes of order $q$ satisfy $\mu(H_q) \in O(q^{3/2})$ and $\sigma(H_q) \geq q^2$.Comment: 28 pages, 4 figures. In v2 we slightly changed one of the core definitions (previously "extended representation" now "semivalid representation"). We also use it to introduce a new graph parameter, denoted eta, which did not appear in v1. It allows us to establish an extended version of the main result showing that mu(G) is at most eta(G) which is at most sigma(G) for every graph

Intrinsically S1 3-Linked Graphs and Other Aspects of S1 Embeddings

A graph can be embedded in various spaces. This paper examines S1 embeddings of graphs. Just as links can be defined in spatial embeddings of graphs, links can be defined in S1 embeddings. Because linking properties are preserved under vertex expansion, there exists a finite complete set of minor minimal graphs such that every S1 embedding contains a non-split 3-link. This paper presents a list of minor minimal intrinsically S1 3-linked graphs, along with methods used to find and verify the list, in hopes of obtaining the complete minor minimal set. Other aspects of S1 embeddings are also examined.

A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l such that in both line sets, for both halfplanes delimited by l, there are n^{1/2} lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are (n/3)^{1/2} of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to Graph Drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labelled lines that are universal for all n-vertex labelled planar graphs. As a side note, we prove that every set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar graphs

Approximating branchwidth on parametric extensions of planarity

The \textsl{branchwidth} of a graph has been introduced by Roberson and Seymour as a measure of the tree-decomposability of a graph, alternative to treewidth. Branchwidth is polynomially computable on planar graphs by the celebrated ``Ratcatcher''-algorithm of Seymour and Thomas. We investigate an extension of this algorithm to minor-closed graph classes, further than planar graphs as follows: Let $H_{0}$ be a graph embeddedable in the projective plane and $H_{1}$ be a graph embeddedable in the torus. We prove that every $\{H_{0},H_{1}\}$-minor free graph $G$ contains a subgraph $G'$ where the difference between the branchwidth of $G$ and the branchwidth of $G'$ is bounded by some constant, depending only on $H_{0}$ and $H_{1}$. Moreover, the graph $G'$ admits a tree decomposition where all torsos are planar. This decomposition can be used for deriving an EPTAS for branchwidth: For $\{H_{0},H_{1}\}$-minor free graphs, there is a function $f\colon\mathbb{N}\to\mathbb{N}$ and a $(1+\epsilon)$-approximation algorithm for branchwidth, running in time $\mathcal{O}(n^3+f(\frac{1}{\epsilon})\cdot n),$ for every $\epsilon>0$

Nullspace embeddings for outerplanar graphs

We study relations between geometric embeddings of graphs and the spectrum of associated matrices, focusing on outerplanar embeddings of graphs. For a simple connected graph G=(V,E), we define a "good" G-matrix as a V×V matrix with negative entries corresponding to adjacent nodes, zero entries corresponding to distinct nonadjacent nodes, and exactly one negative eigenvalue. We give an algorithmic proof of the fact that it G is a 2-connected graph, then either the nullspace representation defined by any "good" G-matrix with corank 2 is an outerplanar embedding of G, or else there exists a "good" G-matrix with corank 3

On 22-cycles of graphs

Let $G=(V,E)$ be a finite undirected graph. Orient the edges of $G$ in an arbitrary way. A $2$-cycle on $G$ is a function $d : E^2\to \mathbb{Z}$ such for each edge $e$, $d(e, \cdot)$ and $d(\cdot, e)$ are circulations on $G$, and $d(e, f) = 0$ whenever $e$ and $f$ have a common vertex. We show that each $2$-cycle is a sum of three special types of $2$-cycles: cycle-pair $2$-cycles, Kuratowski $2$-cycles, and quad $2$-cycles. In case that the graph is Kuratowski connected, we show that each $2$-cycle is a sum of cycle-pair $2$-cycles and at most one Kuratowski $2$-cycle. Furthermore, if $G$ is Kuratowski connected, we characterize when every Kuratowski $2$-cycle is a sum of cycle-pair $2$-cycles. A $2$-cycles $d$ on $G$ is skew-symmetric if $d(e,f) = -d(f,e)$ for all edges $e,f\in E$. We show that each $2$-cycle is a sum of two special types of skew-symmetric $2$-cycles: skew-symmetric cycle-pair $2$-cycles and skew-symmetric quad $2$-cycles. In case that the graph is Kuratowski connected, we show that each skew-symmetric $2$-cycle is a sum of skew-symmetric cycle-pair $2$-cycles. Similar results like this had previously been obtained by one of the authors for symmetric $2$-cycles. Symmetric $2$-cycles are $2$-cycles $d$ such that $d(e,f)=d(f,e)$ for all edges $e,f\in E$