论文标题
DIGRAPHS的离散莫尔斯理论
A Discrete Morse Theory for Digraphs
论文作者
论文摘要
图形是图形的概括，其中每个边缘都有一个或两个方向分配。在本文中，我们定义了离散的摩尔斯在挖掘上的功能，并证明莫尔斯复合物和路径同源性的同源性对于及其及其传递的挖掘是同构。我们还研究由离散梯度矢量场定义的崩溃。令$ g $为digraph和$ f $ a离散的摩尔斯功能。假设$ g $上$ f $的任何零点的超级和内度都是1。我们证明原始的digraph $ g $及其$ \ mathcal {m} $ - 倒塌$ \ tilde {g} $具有相同的路径同源性组。
Digraphs are generalizations of graphs in which each edge is assigned with a direction or two directions. In this paper, we define discrete Morse functions on digraphs, and prove that the homology of the Morse complex and the path homology are isomorphic for a transitive digraph. We also study the collapses defined by discrete gradient vector fields. Let $G$ be a digraph and $f$ a discrete Morse function. Assume the out-degree and in-degree of any zero-point of $f$ on $G$ are both 1. We prove that the original digraph $G$ and its $\mathcal{M}$-collapse $\tilde{G}$ have the same path homology groups.
