Local structures in polyhedral maps on surfaces, and path transferability of graphs

高阶Whitehead映射之间的关系

Relations among higher Whitehead maps

本文定义了聚合同态积之间的广义高阶Whitehead映射，通过研究聚合同态积的同伦性质与单纯复形的组合性质之间的关系，描述了新的映射关系家族，并恢复和推广了已知的Whitehead积恒等式。

Connectivity and $W_v$-Paths in Polyhedral Maps on Surfaces

The $W_v$-Path Conjecture due to Klee and Wolfe states that any two vertices of a simple polytope can be joined by a path that does not revisit any facet. This is equivalent to the well-known Hirsch Conjecture. Klee proved that the $W_v$-Path Conjecture is true for all 3-polytopes (3-connected plane graphs), and conjectured even more, namely that the $W_v$-Path Conjecture is true for all general cell complexes. This general $W_v$-Path Conjecture was verified for polyhedral maps on the projective plane and the torus by Barnette, and on the Klein bottle by Pulapaka and Vince. Let $G$ be a graph …

The homotopy theory of polyhedral products associated with flag complexes

If $K$ is a simplicial complex on $m$ vertices the flagification of $K$ is the minimal flag complex $K^f$ on the same vertex set that contains $K$. Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\to K\to K^f$. This induces a sequence of maps of polyhedral products $(\underline X,\underline A)^L\stackrel g\longrightarrow(\underline X,\underline A)^K\stackrel f\longrightarrow (\underline X,\underline A)^{K^f}$. We show that $Ωf$ and $Ωf\circΩg$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\unde…

Cycle Traversability for Claw-free Graphs and Polyhedral Maps

Let $G$ be a graph, and $v\in V(G)$ and $S\subseteq V(G)\backslash v$ of size at least $k$. An important result on graph connectivity due to Perfect states that, if $v$ and $S$ are $k$-linked, then a $(k-1)$-link between a vertex $v$ and $S$ can be extended to a $k$-link between $v$ and $S$ such that the endvertices of the $(k-1)$-link are also the endvertices of the $k$-link. We begin by proving a generalization of Perfect's result by showing that, if two disjoint sets $S_1$ and $S_2$ are $k$-linked, then a $t$-link ($t< k$) between two disjoint sets $S_1$ and $S_2$ can be extended to a $k$-l…

平面多项式映射的多面体类型

The polyhedral type of a polynomial map on the plane

本文利用Newton多面体对描述了平面一般复多项式映射的拓扑，提出构造给定次数拓扑不等价映射的方法，并给出了d=3和4时拓扑类型数量T(d)的下界。

Generating maps on surfaces

We describe procedures for generating all 2-cell embedded simple graphs with up to a fixed number of vertices on a given surface. We also modify these procedures to generate closed 2-cell embeddings and polyhedral embeddings. We give results of computer implementations of these procedures for seven surfaces: the sphere, the torus, the double torus, the projective plane, the Klein bottle, the triple cross surface, and the quadruple cross surface.

Shortest Paths and Convex Hulls in 2D Complexes with Non-Positive Curvature

Globally non-positively curved, or CAT(0), polyhedral complexes arise in a number of applications, including evolutionary biology and robotics. These spaces have unique shortest paths and are composed of Euclidean polyhedra, yet many algorithms and properties of shortest paths and convex hulls in Euclidean space fail to transfer over. We give an algorithm, using linear programming, to compute the convex hull of a set of points in a 2-dimensional CAT(0) polyhedral complex with a single vertex. We explore the use of shortest path maps to answer single-source shortest path queries in 2-dimensiona…