On Separating Path and Tree Systems in Graphs

TL;DR

This paper investigates the minimal size of vertex-separating systems composed of paths or trees in various graph classes, advancing understanding of graph separation properties.

Contribution

It introduces the concept of separating systems of paths and trees in graphs and analyzes their minimal sizes across different graph types.

Findings

Characterized minimal vertex-separating path systems in trees.

Analyzed size bounds for grid graphs.

Studied vertex-separating systems in maximal outerplanar graphs.

Abstract

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.