Zero Forcing with Random Sets

TL;DR

This paper studies the probability that a randomly chosen vertex set in a graph is a zero forcing set, providing bounds and confirming a conjecture related to the number of such sets under certain conditions.

Contribution

It introduces probabilistic bounds for zero forcing sets in graphs and proves a conjecture on their quantity given a minimum degree condition.

Findings

Probability bounds for zero forcing sets in trees and paths

Upper bounds on the probability for large trees

Confirmation of a conjecture on the number of zero forcing sets

Abstract

Given a graph $G$ and a real number $0\le p\le 1$, we define the random set $B_p(G)\subset V(G)$ by including each vertex independently and with probability $p$. We investigate the probability that the random set $B_p(G)$ is a zero forcing set of $G$. In particular, we prove that for large $n$, this probability for trees is upper bounded by the corresponding probability for a path graph. Given a minimum degree condition, we also prove a conjecture of Boyer et.\ al.\ regarding the number of zero forcing sets of a given size that a graph can have.