Throttling for Zero Forcing and Variants

Abstract

Zero forcing is a process on a graph in which the goal is to force all vertices to become blue by applying a color change rule. Throttling minimizes the sum of the number of vertices that are initially blue and the number of time steps needed to color every vertex. This paper provides a new general definition of throttling for variants of zero forcing and studies throttling for the minor monotone floor of zero forcing. The technique of using a zero forcing process to extend a given graph is introduced. For standard zero forcing and its floor, these extensions are used to characterize graphs with throttling number ≤ t as certain minors of cartesian products of complete graphs and paths. Finally, these characterizations are applied to determine graphs with extreme throttling numbers.

Power domination throttling

“…Thus, the power domination process of G(s, a, b) using S after the domination step is identical to the zero forcing process of K a P b using A. It follows from Lemma 3.15 of [17] that contracting path edges of K a P b does not increase the zero forcing propagation time using A. Thus, contracting path edges of G(s, a, b) does not increase the power propagation time using S.…”

“…Other variants of propagation time arising from similar dynamic graph coloring processes have also been studied; these include zero forcing propagation time [7,23,27,28] and positive semidefinite propagation time [40]. Throttling for other problems such as zero forcing [16], positive semidefinite zero forcing [18], minor monotone floor of zero forcing [17], and the game of Cops and Robbers [9] has been studied as well.…”