Separating path systems in trees

TL;DR

This paper studies the minimum size of path systems in trees and random graphs that can distinguish between pairs of edges or vertices, providing exact formulas, bounds, and tightness results.

Contribution

It determines the size of minimal edge- and vertex-separating path systems in trees and extends the analysis to a combined separation variant and Erdős–Rényi random graphs.

Findings

Exact formula for edge-separating path systems in trees.

Bounds and tightness results for vertex-separating path systems.

Analysis of minimal vertex-separating path systems in Erdős–Rényi graphs.

Abstract

For a graph $G$, an edge-separating (resp. vertex-separating) path system of $G$ is a family of paths in $G$ such that for any pair of edges $e_1, e_2$ (resp. pair of vertices $v_1, v_2$) of $G$ there is at least one path in the family that contains one of $e_1$ and $e_2$ (resp. $v_1$ and $v_2$) but not the other. We determine the size of a minimum edge-separating path system of an arbitrary tree $T$ as a function of its number of leaves and degree-two vertices. We obtain bounds for the size of a minimal vertex-separating path system for trees, which we show to be tight in many cases. We obtain similar results for a variation of the definition, where we require the path system to separate edges and vertices simultaneously. Finally, we investigate the size of a minimal vertex-separating path system in Erd\H{o}s--R\'enyi random graphs.