Throttling positive semidefinite zero forcing propagation time on graphs

Abstract

Zero forcing is a process on a graph that colors vertices blue by starting with some of the vertices blue and applying a color change rule. Throttling minimizes the sum of the size of the initial blue vertex set and the number of the time steps needed to color the graph. We study throttling for positive semidefinite zero forcing. We establish a tight lower bound on the positive semidefinite throttling number as a function of the order, maximum degree, and positive semidefinite zero forcing number of the graph, and determine the positive semidefinite throttling numbers of paths, cycles, and full binary trees. We characterize the graphs that have extreme positive semidefinite throttling numbers.

Throttling for the game of Cops and Robbers on graphs

“…If T is a tree with radius r, then Observe that if we denote the order of T B (h) by n, then th c (T B (h)) = h + 1 = Θ(log n). It is shown in [13] that th + (G) = Ω(log n) for a graph G of order n. By contrast, th c (P n ) = Θ( √ n), and we now show that th c (T ) = O( √ n) for all trees T of order n. This is established by providing an algorithm to construct a set S with |S| ≤ √ n and pt + (S) = √ n .…”

“…The PSD propagation time of graph G is pt + (G) = min{pt + (G; S) : S is a minimum PSD zero forcing set of G} [27]. In [13] Carlson et al define th + (G; S) = |S| + pt + (G; S) and the PSD throttling number of a graph G as th + (G) = min S⊆V (G) th + (G; S).…”

“…Butler and Young define the throttling number as the minimum of the sum of |S| and the propagation time of S over sets S of vertices. It was this study, and a subsequent study of throttling for positive semidefinite zero forcing [13], that led to our interest in throttling for the game of Cops and Robbers.…”

Power domination throttling

“…We conclude this section by deriving tight bounds on the power domination throttling numbers of trees; some ideas in the following results are adapted from [18]. N [S ]).…”

The zero forcing polynomial of a graph

“…The set of minimum zero forcing sets of a graph has been alluded to previously in the context of propagation time [7,36,45], where the objective is to find the largest or smallest number of timesteps it takes for the graph to be colored by a minimum zero forcing set. Similarly, the set of all zero forcing sets of a graph has been used in the context of throttling [19,20], where the objective is to minimize the sum of the size of a zero forcing set and the number of timesteps it takes for that zero forcing set to color the graph. In order to study the collection of zero forcing sets of a graph in a more general framework, we introduce the zero forcing polynomial of a graph, which counts the number of distinct zero forcing sets of a given size.…”