相关论文: Projective Planar Cartesian Products of Graphs
A graph is apex if it becomes planar after the deletion of one vertex. The family of apex graphs is closed under taking minors, so it is characterized by a finite set of forbidden minors. Determining the finite set of forbidden minors for…
In the area of beyond-planar graphs, i.e. graphs that can be drawn with some local restrictions on the edge crossings, the recognition problem is prominent next to the density question for the different graph classes. For 1-planar graphs,…
A pseudo-Cartan inclusion is a regular inclusion having a Cartan envelope. Unital pseudo-Cartan inclusions were classified by Pitts; we extend this classification to include the non-unital case. The class of pseudo-Cartan inclusions…
Let $G$ be a subgroup of the three dimensional projective group $\mathrm{PGL}(3,q)$ defined over a finite field $\mathbb{F}_q$ of order $q$, viewed as a subgroup of $\mathrm{PGL}(3,K)$ where $K$ is an algebraic closure of $\mathbb{F}_q$.…
Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median…
Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded…
In this paper, we give new, tight subexponential lower bounds for a number of graph embedding problems. We introduce two related combinatorial problems, which we call String Crafting and Orthogonal Vector crafting, and show that these…
Given a function $g=g(n)$ we let ${\mathcal E}^g$ be the class of all graphs $G$ such that if $G$ has order $n$ (that is, has $n$ vertices) then it is embeddable in some surface of Euler genus at most $g(n)$, and let ${\widetilde{\mathcal…
We generalize Jones' planar algebras by internalising the notion to a pivotal braided tensor category $\mathcal{C}$. To formulate the notion, the planar tangles are now equipped with additional `anchor lines' which connect the inner circles…
A minimal separating set in a connected topological space $X$ is a subset $L \subset X$ with the property that $X \setminus L$ is disconnected, but if $L^{\prime}$ is a proper subset of $L$, then $X \setminus L^{\prime}$ is connected. Such…
A fake projective plane is a complex surface with the same Betti numbers as $\mathbb{C} P^2$ but not biholomorphic to it. We study the fake projective plane $\mathbb{P}_{\operatorname{fake}}^2 = (a = 7, p = 2, \emptyset, D_3 2_7)$ in the…
Given a smooth complex variety $X$, an algebraically skew embedding of $X$ is an embedding of $X$ into a complex projective space $\mathbb{P}^N$ such that for any two points $x,y\in X$, their embedded tangent spaces in $\mathbb{P}^N$ do not…
An Euclidean greedy embedding of a graph is a straight-line embedding in the plane, such that for every pair of vertices $s$ and $t$, the vertex $s$ has a neighbor $v$ with smaller distance to $t$ than $s$. This drawing style is motivated…
We introduce the labelling map and the quasi-free action of a locally compact abelian group on a graph $C^*$-algebra of a row-finite directed graph. Some necessary conditions for embedding the crossed product to an $AF$ algebra are…
We call a (not necessarily planar) embedding of a graph $G$ in the plane \emph{sequential} if its vertices lie in $\mathbb Z^2$ and the line segments between adjacent vertices contain no interior integer points. In this note, we prove (i) a…
The embeddability of graphs into surfaces has been studied for nearly a century. While the complete set of topological obstructions is known for the sphere and the real projective plane, there are only partial results for the torus. Here we…
The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising frommaximal (120,8)-arcs in the known projective planes of order 16 are computed. It is shown that each of these designs is embeddable in a unique…
We introduce and study, for a process P delivering edges on the Cartesian product of the vertex sets of a given set of graphs, the P-product of these graphs, thereby generalizing many types of product graph. Analogous to the notion of a…
Sparse graphs and their associated matroids play an important role in rigidity theory, where they capture the combinatorics of generically rigid structures. We define a new family called {\bf graded sparse graphs}, arising from generically…
It is shown that a (curved) projective structure on a smooth manifold determines on the Poisson algebra of smooth, fiberwise-polynomial functions on the cotangent bundle a one-parameter family of graded star products. For a particular value…