Separating path systems

TL;DR

This paper investigates the existence and bounds of separating systems of paths in graphs, proving the conjecture for specific graph classes like random graphs and those with linear minimum degree, and analyzing trees.

Contribution

It introduces a conjecture that every n-vertex graph has a separating path system of size O(n) and proves it for certain graph classes, providing tight bounds for trees.

Findings

Proved the conjecture for random graphs.

Established the conjecture for graphs with linear minimum degree.

Derived tight bounds for separating path systems in trees.

Abstract

We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every $n$-vertex graph admits a separating path system of size $O(n)$ and prove this in certain interesting special cases. In particular, we establish this conjecture for random graphs and graphs with linear minimum degree. We also obtain tight bounds on the size of a minimal separating path system in the case of trees.