Vertex-separating path systems in random graphs

TL;DR

This paper investigates the minimal size of path-connected vertex sets that can separate all pairs of vertices in random graphs and regular graphs, revealing thresholds and optimal bounds for such separating systems.

Contribution

It determines the size of minimal separating systems in random graphs and regular graphs, introducing new bounds and threshold phenomena for vertex separation.

Findings

Minimal separating systems in $G(n,p)$ when $np o olinebreak \infty$

High-degree regular graphs can be separated by $oxed{ ext{log}_2 n}$ sets

Bounds for degree thresholds for optimal separation in general graphs

Abstract

A set $V$ is said to be separated by subsets $V_1,\ldots,V_k$ if, for every pair of distinct elements of $V$, there is a set $V_i$ that contains exactly one of them. Imposing structural constraints on the separating subsets is often necessary for practical purposes and leads to a number of fascinating (and, in some cases, already classical) graph-theoretic problems. In this work, we are interested in separating the vertices of a random graph by path-connected vertex sets $V_1,\ldots,V_k$, jointly forming a separating system. First, we determine the size of the smallest separating system of $G(n,p)$ when $np\to \infty$ up to lower order terms, and exhibit a threshold phenomenon around the sharp threshold for connectivity. Second, we show that random regular graphs of sufficiently high degree can typically be optimally separated by $\lceil \log_2 n\rceil$ sets. Moreover, we provide bounds for the minimum degree threshold for optimal separation of general graphs.