Upper Bounds for Positive Semidefinite Propagation Time

TL;DR

This paper establishes a tight upper bound for positive semidefinite propagation time based on the positive semidefinite zero forcing number, introduces transformation methods and algorithms, and explores related bounds and consequences.

Contribution

The paper introduces a new tight upper bound for positive semidefinite propagation time and provides transformation methods and algorithms to achieve it.

Findings

Established a tight upper bound for positive semidefinite propagation time.

Developed transformation methods and algorithms for zero forcing sets.

Derived a Nordhaus-Gaddum sum upper bound for propagation time.

Abstract

The tight upper bound $\operatorname{pt}_+(G) \leq \left\lceil \frac{\left\vert \operatorname{V}(G) \right\vert - \operatorname{Z}_+(G)}{2} \right\rceil$ is established for the positive semidefinite propagation time of a graph in terms of its positive semidefinite zero forcing number. To prove this bound, two methods of transforming one positive semidefinite zero forcing set into another and algorithms implementing these methods are presented. Consequences of the bound, including a tight Nordhaus-Gaddum sum upper bound on positive semidefinite propagation time, are established.