Product Thottling

The Hopping Forcing Rule

“…Instead, we can try to minimize the size of the hopping forcing set times its propagation time; this is known as product throttling. First introduced in [5] in the context of the graph theoretic game Cops and Robbers, product throttling was later applied to various types of forcing by Anderson et al in [2]. In this section, we consider product throttling for hopping forcing, and show that it behaves very similarly to product throttling for standard zero forcing.…”

“…This lemma helps give us the analogous result to Theorem 4.3 for hopping throttling. The proof is equivalent to that from [2], but using hopping forcing parameters instead of standard zero forcing ones. As such, we omit the proof here.…”

“…However, this idea of reversing a forcing process was first considered by Barioli et al in [3], who provide the following definition. Anderson et al use this notion of reversing a forcing set in [2]. As discussed there, a zero forcing set S can have many different reversals, any of which can be denoted by rev(S).…”

“…As an example, we will determine the no initial cost product hopping throttling number for the path graphs. Anderson et al show that, for no initial cost product throttling for standard zero forcing, paths have the minimum possible throttling number [2]. Under hopping throttling, paths achieve the minimum possible value of th * H (G) for connected graphs.…”