On a conjecture of Gentner and Rautenbach

TL;DR

This paper disproves a conjecture by Gentner and Rautenbach by constructing connected graphs with maximum degree 3 that have a zero forcing number exceeding the conjectured bound, showing the bound is not universally valid.

Contribution

The authors provide a counterexample collection of graphs with maximum degree 3 that have a zero forcing number larger than previously conjectured, refuting the original conjecture.

Findings

Counterexamples with zero forcing number ≥ 4/9 of vertices

Disproof of the conjecture for all large graph orders

Construction of graphs with arbitrarily large size and high zero forcing number

Abstract

Gentner and Rautenbach conjectured that the size of a minimum zero forcing set in a connected graph on $n$ vertices with maximum degree $3$ is at most $\frac{1}{3}n+2$. We disprove this conjecture by constructing a collection of connected graphs $\{G_n\}$ with maximum degree 3 of arbitrarily large order having zero forcing number at least $\frac{4}{9}|V(G_n)|$.