Various Characterizations of Throttling Numbers

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.