On Separating Path and Tree Systems in Graphs

2023-12-21

arXiv.org

AI Summary

This paper investigates vertex-separating path/tree systems on graphs: families of paths or subtrees such that every pair of vertices is contained in exactly one member—achieving exact separation. We establish the first systematic characterization of the minimum size of such systems for three fundamental graph classes: Θ(log n) for trees, Θ(n) for n×n grid graphs, and Θ(n) for maximal outerplanar graphs. Using combinatorial graph theory, recursive decomposition, extremal constructions, and duality arguments, we derive tight upper and lower bounds and provide matching explicit constructions—achieving both theoretical optimality and constructibility. The core contribution is the introduction and resolution of the “exact vertex separation” model, a novel combinatorial framework that uncovers an intrinsic connection between graph structural complexity and separation efficiency.

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.

Problem

• Vertex-Separator
• Graph Theory
• Path System

Innovation

• Vertex-Separator Path (Tree) Systems
• Graph Theory
• Minimum Path or Tree Quantification