Sharp bounds for product and sum throttling numbers

TL;DR

This paper introduces new techniques to establish sharp upper bounds for sum and product throttling numbers in graphs, particularly for power domination and zero forcing, and analyzes how these bounds change under graph operations.

Contribution

It provides the first sharp bounds for sum and product throttling numbers in power domination and examines their behavior under various graph operations.

Findings

Sharp upper bounds for sum throttling in power domination

Sharp bounds on changes in product throttling due to graph operations

Analysis of throttling bounds for zero forcing and positive semidefinite forcing

Abstract

Throttling in graphs optimizes a sum or product of resources used, such as the number of vertices in an initial set, and time required, such as the propagation time, to complete a given task. We introduce a new technique to establish sharp upper bounds in terms of graph order for sum throttling and initial cost product throttling for power domination. Furthermore, we establish sharp bounds on possible changes of the product throttling number, both with and without initial cost, caused by certain graph operations for standard zero forcing, positive semidefinite forcing, and power domination.