New Structures and their Applications to Variants of Zero Forcing and Propagation Time

TL;DR

This paper introduces new structural concepts like relaxed chronologies and parallel increasing path covers to analyze and compare zero forcing and PSD forcing processes, revealing new bounds and connections.

Contribution

It proposes a generalized framework for zero forcing using relaxed chronologies and PIPs, and explores their properties and relationships with PSD forcing and rigid-linkage forcing.

Findings

Bounds comparing zero forcing and PSD propagation times

Introduction of parallel increasing path covers (PIPs)

Connection between PSD forcing and rigid-linkage forcing

Abstract

We introduce a generalization of the concept of a chronological list of forces, called a relaxed chronology. This concept is used to introduce a new way of formulating the standard zero forcing process, which we refer to as parallel increasing path covers, or PIPs. The combinatorial properties of PIPs are utilized to identify bounds comparing standard zero forcing propagation time to positive semidefinite propagation time. A collection of paths within a set of PSD forcing trees, called a path bundle, is used to identify the PSD forcing analog of the reversal of a standard zero forcing process, as well as to draw a connection between PSD forcing and rigid-linkage forcing.