A Lower Bound on the Failed Zero Forcing Number of a Graph

TL;DR

This paper establishes a lower bound on the failed zero forcing number for any graph, showing that it is at least half of the vertices minus one, which answers an open question in graph theory.

Contribution

The paper proves a universal lower bound on the failed zero forcing number for all graphs, advancing understanding of zero forcing properties.

Findings

Failed zero forcing number is at least (n-1)/2 for any n-vertex graph

Answers affirmatively an open question about bounds on failed zero forcing sets

Provides a theoretical lower bound applicable to all graphs

Abstract

Given a graph $G=(V,E)$ and a set of vertices marked as filled, we consider a color-change rule known as zero forcing. A set $S$ is a zero forcing set if filling $S$ and applying all possible instances of the color change rule causes all vertices in $V$ to be filled. A failed zero forcing set is a set of vertices that is not a zero forcing set. Given a graph $G$, the failed zero forcing number $F(G)$ is the maximum size of a failed zero forcing set. An open question was whether given any $k$ there is a an $\ell$ such that all graphs with at least $\ell$ vertices must satisfy $F(G)\geq k$. We answer this question affirmatively by proving that for a graph $G$ with $n$ vertices, $F(G)\geq \lfloor\frac{n-1}{2}\rfloor$.