On Separating Path and Tree Systems in Graphs

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.