Positive Semidefinite Initial Cost Product Throttling

TL;DR

This paper investigates the initial cost product throttling number in positive semidefinite zero forcing on graphs, providing bounds, exact values for cycles, and results for various graph families.

Contribution

It introduces bounds and exact values for the initial cost product throttling number in positive semidefinite zero forcing, expanding understanding of resource-time trade-offs in graph processes.

Findings

Lower bound of 1+rad(G) for the throttling number

Exact initial cost throttling number for cycles

Results for various graph families

Abstract

Product throttling answers the question of minimizing the product of the resources needed to accomplish a task, and the time in which it takes to accomplish the task. In product throttling for positive semidefinite zero forcing, task that we wish to accomplish is positive semidefinite zero forcing. Positive semidefinite zero forcing is a game played on a graph $G$ that starts with a coloring of the vertices as white and blue. At each step any vertex colored blue with a unique white neighbor in a component of the graph formed by deleting the blue vertices from $G$ forces the color of the white neighbor to become blue. We give various results and bounds on the initial cost product throttling number, including a lower bound of $1+rad(G)$ and the initial cost product throttling number of a cycle. We also include a table with results on the initial cost and no initial cost product throttling number for various graph families.