We explore the concept of separating systems of vertex sets of graphs. A separating system of a set $X$ is a collection of subsets of $X$ such that for any pair of distinct elements in $X$, there exists a set in the separating system that contains exactly one of the two elements. A separating system of the vertex set of a graph $G$ is called a vertex-separating path (tree) system of $G$ if the elements of the separating system are paths (trees) in the graph $G$. In this paper, we focus on the size of the smallest vertex-separating path (tree) system for different types of graphs, including trees, grids, and maximal outerplanar graphs.

翻译：本文探讨图的顶点集分离系统概念。集合$X$的分离系统指$X$子集的集合，满足对$X$中任意两个相异元素，总存在分离系统中的某个集合恰好包含其中一个元素。若图$G$顶点集的分离系统中所有元素均为$G$中的路径（树），则称该系统为$G$的顶点分离路径（树）系统。本文重点研究不同类型图（包括树、网格图及极大外平面图）的最小顶点分离路径（树）系统的规模问题。