Various Characterizations of Throttling Numbers

TL;DR

This paper explores various characterizations of throttling numbers in zero forcing on graphs, introducing methods to extend graphs via PSD zero forcing and characterizing graphs with certain throttling bounds through minors and forbidden subgraphs.

Contribution

It presents new characterizations of PSD zero forcing throttling numbers using graph minors and forbidden subgraph families, extending the understanding of zero forcing variants.

Findings

Graphs with PSD throttling number ≤ t are minors of specific graph products.

Characterization of graphs with high throttling numbers via forbidden subgraphs.

Extension method for PSD zero forcing applied to graph characterization.

Abstract

Zero forcing can be described as a combinatorial game on a graph that uses a color change rule in which vertices change white vertices to blue. The throttling number of a graph minimizes the sum of the number of vertices initially colored blue and the number of time steps required to color the entire graph. Positive semidefinite (PSD) zero forcing is a commonly studied variant of standard zero forcing that alters the color change rule. This paper introduces a method for extending a graph using a PSD zero forcing process. Using this extension method, graphs with PSD throttling number at most $t$ are characterized as specific minors of the Cartesian product of complete graphs and trees. A similar characterization is obtained for the minor monotone floor of PSD zero forcing. Finally, the set of connected graphs on $n$ vertices with throttling number at least $n-k$ is characterized by forbidding a finite family of induced subgraphs. These forbidden subgraphs are constructed for standard throttling.