The Zero Forcing Numbers of Peony Graphs and Web Graphs

TL;DR

This paper investigates the zero forcing number in complex graph classes, using structural properties to determine minimal initial blue vertices needed to force entire graphs blue, expanding understanding of graph dynamics.

Contribution

It introduces new methods for analyzing zero forcing numbers in two infinite classes of graphs based on their structural properties.

Findings

Determined zero forcing numbers for specific classes of peony and web graphs.

Developed structural analysis techniques applicable to complex graph classes.

Enhanced understanding of how graph structure influences zero forcing processes.

Abstract

The concept of zero forcing involves a dynamic coloring process by which blue vertices cause white vertices to become blue, with the goal of forcing the entire graph blue while choosing as few as possible vertices to be initially blue. Past research in this area has focused on structural arguments, with approaches varying from graph substructures to the interplay between local and global graph structures. This paper explores the use of these structural concepts when determining the zero forcing number of complex classes of graphs, specifically two infinite classes of graphs each defined on multiple parameters.