Exploring the Influence of Graph Operations on Zero Forcing Sets

TL;DR

This paper investigates how graph operations influence the number of zero forcing sets, confirming a conjecture for certain graph classes like outerplanar and threshold graphs by analyzing their structural properties.

Contribution

It demonstrates that specific graph classes satisfy a conjecture relating zero forcing set counts, using analysis of graph operations and their effects.

Findings

Outerplanar graphs satisfy the conjecture.

Threshold graphs satisfy the conjecture.

Graph operations can predict zero forcing set counts.

Abstract

Zero forcing in graphs is a coloring process where a colored vertex can force its unique uncolored neighbor to be colored. A zero forcing set is a set of initially colored vertices capable of eventually coloring all vertices of the graph. In this paper, we focus on the numbers $z(G; i)$, which is the number of zero forcing sets of size $i$ of the graph $G$. These numbers were initially studied by Boyer et al. where they conjectured that for any graph $G$ on $n$ vertices, $z(G; i) \leq z(P_n; i)$ for all $i \geq 1$ where $P_n$ is the path graph on $n$ vertices. The main aim of this paper is to show that several classes of graphs, including outerplanar graphs and threshold graphs, satisfy this conjecture. We do this by studying various graph operations and examining how they affect the number of zero forcing sets.