The Hopping Forcing Rule

TL;DR

This paper explores the hopping forcing rule in zero forcing games on graphs, introducing new parameters and analyzing their relationships with graph properties to deepen understanding of information spread models.

Contribution

It studies the hopping forcing number and throttling number independently, revealing their connections to graph parameters and expanding zero forcing theory.

Findings

Hopping forcing number's relation to vertex connectivity.

Hopping throttling number's bounds and properties.

Comparison with classic zero forcing parameters.

Abstract

Zero forcing is a combinatorial game played on graphs that can be used to model the spread of information with repeated applications of a color change rule. In general, a zero forcing parameter is the minimum number of initial blue vertices that are needed to eventually color every vertex blue with a given color change rule. Furthermore, the throttling number minimizes the sum of the number of initial blue vertices and the time taken for all vertices to become blue. In 2013, Barioli et al. added a new rule, called hopping, to existing color change rules in order to demonstrate that the minor monotone floor of various zero forcing parameters is itself, a zero forcing parameter. In this paper, we examine the hopping color change rule independently from the other classic rules. Specifically, we study the hopping forcing number and the hopping throttling number. We investigate the ways in which these numbers are related to various graph theory parameters (such as vertex connectivity and independence number) as well as other zero forcing parameters.