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：我们探讨图顶点集的分隔系统概念。集合 $X$ 的分隔系统是 $X$ 的子集族，使得对 $X$ 中任意两个不同元素，分隔系统中存在一个子集恰好包含这两个元素中的一个。若图 $G$ 顶点集的分隔系统的元素均为 $G$ 中的路径（树），则称该分隔系统为 $G$ 的顶点分隔路径（树）系统。本文重点研究不同类型图（包括树、网格图和极大外平面图）的最小顶点分隔路径（树）系统规模。