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. We give a universal definition of throttling for variants of zero forcing and introduce the study of throttling for the minor monotone floor of zero forcing. For standard zero forcing and its floor, we characterize graphs with throttling number \leq t as certain minors of cartesian products of complete graphs and paths. We apply these characterizations to power domination and extreme throttling numbers.
Keywords Zero forcing, propagation time, throttling, minor monotone floor, power domination
AMS subject classification 05C57, 05C15, 05C50
1 Introduction
Zero forcing is a process on graphs in which an initial set of vertices is colored blue (with the remaining vertices colored white) and vertices can force white vertices to become blue according to a color change rule. Typically, the goal in zero forcing is to use the color change rule to change the color of every vertex in the graph to blue. Zero forcing can be used to model graph searching [12], the spread of information on graphs [6], and control of quantum systems [5, 11]. Naturally, we may want to know the smallest possible size of an initial set that can be used to color all vertices in the graph blue. We may also be interested in the time it takes to complete this process (often called propagation time). The idea of throttling is to study the relationship between the size of the initial set and its propagation time. In particular, the problem of throttling is to minimize the sum of these two quantities.
Unless otherwise stated, the graphs in this paper are simple, undirected, and finite. The (standard) color change rule is that a blue vertex u can force a white vertex w to become blue if w is the only white neighbor of u. In this case, we say u forces w and we write u\rightarrow w. A vertex is active if it is blue and has not yet performed a force. Let G be a graph with B\subseteq V(G) colored blue and V(G)\setminus B colored white. If we can force every vertex in V(G) to become blue by repeatedly applying the standard color change rule, then B is a (standard) zero forcing set of G. The (standard) zero forcing number, \operatorname{Z}(G), is the minimum size of a standard zero forcing set of G. In [1], it is shown that the zero forcing number can be used to bound the minimum rank of a matrix associated with a graph.
Zero forcing propagation is studied in [10]. The idea is to simultaneously perform all possible forces at each time step. Define B^{(0)}=B and for each t\geq 0, define B^{(t+1)} to be the set of vertices w for which there exists a vertex b\in\bigcup_{s=0}^{t}B^{(s)} such that w is the only neighbor of b not in \bigcup_{s=0}^{t}B^{(s)}. The (standard) propagation time of B in G, denoted \operatorname{pt}(G,B), is the smallest integer t_{0} such that V(G)=\bigcup_{t=0}^{t_{0}}B^{(t)}. Propagation time is particularly important in the control of quantum systems (see [11]).
Throttling for standard zero forcing was first studied by Butler and Young in [6]. If B is a zero forcing set of a graph G, the throttling of B is \operatorname{th}(G,B)=B+\operatorname{pt}(G,B). The (standard) throttling number of G is defined as \operatorname{th}(G)=\min\{\operatorname{th}(G,B)\ \ B\text{ is a zero % forcing set of }G\}. Since there are many variations of zero forcing, we can have many variations of throttling. In Section 2, we give a universal definition of propagation and throttling.
Commonly studied variants of zero forcing include positive semidefinite zero forcing and loop zero forcing. Suppose B is a set of blue vertices in a graph G. Let W_{1},\ldots,W_{k} be the sets of (white) vertices of the k components of GB. The positive semidefinite color change rule is that we can apply the standard color change rule in G[W_{i}\cup B] for any 1\leq i\leq k. The positive semidefinite zero forcing number of a graph G is denoted \operatorname{Z}_{+}(G). The positive semidefinite throttling number is defined analogously to standard throttling and is studied in [7]. Loop zero forcing [3] arises by considering a graph where every vertex has a loop. Applied to a simple graph, the loop color change rule is that we can apply the standard color change rule, or if every neighbor of a white vertex w is blue, then w can force itself to become blue. The loop zero forcing number of a graph G is denoted \operatorname{Z}_{\ell}(G).
If G and H are graphs and G is a subgraph of H, we write G\leq H. If G\leq H and V(G)=V(H), G is a spanning subgraph of H and H is a spanning supergraph of G. If G is a minor of H, we write G\preceq H. Suppose p is a graph parameter whose range is wellordered. Given a graph G, the minor monotone floor of p is defined as \lfloor p\rfloor(G)=\min\{p(H)\ \ G\preceq H\}. In [3], it was shown that \operatorname{\lfloor\operatorname{Z}\rfloor}, \lfloor\operatorname{Z}_{+}\rfloor, and \lfloor\operatorname{Z}_{\ell}\rfloor are zero forcing parameters and can be defined in terms of color change rules. In particular, the \operatorname{\lfloor\operatorname{Z}\rfloor} color change rule is that we can apply the standard color change rule, or if a vertex v is active and all neighbors of v are blue, then we can choose a white vertex w and v can force w to become blue. The latter condition of the \operatorname{\lfloor\operatorname{Z}\rfloor} color change rule is called “hopping”. If we use this condition, then we say that v forces w by a hop. It was also shown in [3] that the minor monotone floors of various zero forcing parameter are related to treewidth, pathwidth, and proper pathwidth. In Section 3, we study \operatorname{\lfloor\operatorname{Z}\rfloor} throttling and characterize graphs with \operatorname{\lfloor\operatorname{Z}\rfloor} throttling number at most t for a fixed positive integer t. Then we give a similar characterization for standard throttling in Section 4. Finally, in Section 5, we apply these characterizations to power domination [9, 4] and extreme throttling numbers.
2 Universal Propagation Time and Throttling
In this section, we give a universal definition of propagation time and throttling for general color change rules. To avoid unnecessary abstraction, we suppose we are given an accepted color change rule with the understanding that there are conditions under which vertices can perform forces by forcing white vertices to become blue. Let G be a graph with B\subseteq V(G) colored blue and V(G)\setminus B colored white. Let R be a given color change rule. Repeatedly apply R to G until it is no longer possible to do so and write down the forces u\rightarrow w in the order in which they are performed. This list of forces is called a chronological list of R forces of B and the unordered set of forces that appear in the list is a set of R forces of B. Suppose G is a graph and \mathcal{F} is a set of R forces of B\subseteq V(G). An R forcing chain of \mathcal{F} is a sequence of vertices (v_{1},v_{2},\ldots,v_{k}) in G such that (v_{i}\rightarrow v_{i+1})\in\mathcal{F} for each 1\leq i\leq k1. An R forcing chain of \mathcal{F} is maximal if it is not properly contained in any other R forcing chain of \mathcal{F}. The set of vertices in G that are blue after all forces in \mathcal{F} have been performed is an R final coloring of B.
Remark 2.1.
Since an R final coloring of a set B\subseteq V(G) is exactly B together with the vertices w that appear in a particular set of R forces of B as u\rightarrow w, an R final coloring of B does not depend on the order in which the R forces were performed. Instead, it depends only on the set of R forces used.
Let G be a graph and let R be a given color change rule. An R forcing set of G is a subset B of vertices such that V(G) is an R final coloring of B for some set of R forces. The R forcing parameter, R(G), is the minimum size of an R zero forcing set of G. If B is an R forcing set of G and B=R(G), we say B is a minimum R forcing set of G.
For a set of R forces \mathcal{F} of B\subseteq V(G), define \mathcal{F}^{(0)}=B and for t\geq 0, \mathcal{F}^{(t+1)} is the set of vertices w such that the force v\rightarrow w appears in \mathcal{F} and w can be R forced by v if the vertices in \bigcup_{i=0}^{t}\mathcal{F}^{(i)} are colored blue and the vertices in V(G)\setminus\left(\bigcup_{i=0}^{t}\mathcal{F}^{(i)}\right) are colored white. The R propagation time of \mathcal{F} in G, denoted \operatorname{pt}_{R}(G;\mathcal{F}), is the least t_{0} such that V(G)=\bigcup_{i=0}^{t_{0}}\mathcal{F}^{(i)}. If the R final coloring induced by \mathcal{F} is not V(G), then define \operatorname{pt}_{R}(G;\mathcal{F})=\infty. For 1\leq t\leq\operatorname{pt}_{R}(G;\mathcal{F}), the vertices in \mathcal{F}^{(t)} are the vertices that are forced in time step t. Note that B is colored blue at time 0, and for each 1\leq t\leq\operatorname{pt}_{R}(G;\mathcal{F}), time step t takes place between time t1 and time t in \mathcal{F}. A vertex in G is active at time t if it is blue at time t and has not performed a force in time step s for any s\leq t.
Note that the definition of standard propagation time of a set of vertices does not use sets of forces. This is because final colorings in standard zero forcing are unique and depend only on the initial set of blue vertices (see [1]). However, it is shown in [3] that there are variants of zero forcing that do not have unique final colorings. For example, consider the \operatorname{\lfloor\operatorname{Z}\rfloor} color change rule. If we are able to perform a force by hopping, there are many choices for the white vertex that gets forced. These choices allow a set of vertices to have multiple final colorings. It is easy to see that the standard propagation process defined in Section 1 yields a set of forces that has minimum propagation time. This motivates the following definition.
Definition 2.2.
Let G be a graph with B\subseteq V(G) and let R be a given color change rule. The R propagation time of B is defined as
\displaystyle\operatorname{pt}_{R}(G;B)=\min\{\operatorname{pt}_{R}(G;\mathcal% {F})\ \ \mathcal{F}\text{ is set of $R$ forces of }B\}. 
Definition 2.3.
Let G be a graph and let R be a given color change rule. The R propagation time of G is defined as
\displaystyle\operatorname{pt}_{R}(G)=\min\{\operatorname{pt}_{R}(G;B)\ \ B% \text{ is a minimum $R$ forcing set of }G\}. 
Definition 2.4.
Let G be a graph with B\subseteq V(G) and let R be a given color change rule. The R throttling of B is
\displaystyle\operatorname{th}_{R}(G;B)=B+\operatorname{pt}_{R}(G;B). 
Definition 2.5.
Let G be a graph and let R be a given color change rule. The R throttling number of G is defined as
\displaystyle\operatorname{th}_{R}(G)=\underset{B\subseteq V(G)}{\min}\{% \operatorname{th}_{R}(G;B)\}. 
When comparing propagation time and throttling for various color change rules, we will use \operatorname{Z} (i.e., \operatorname{pt_{\operatorname{Z}}} and \operatorname{th_{\operatorname{Z}}}) to denote the standard zero forcing color change rule.
3 Throttling for the Minor Monotone Floor of Z.
In this section, we investigate propagation and throttling for the \operatorname{\lfloor\operatorname{Z}\rfloor} color change rule. The following proposition shows that we can calculate the \operatorname{\lfloor\operatorname{Z}\rfloor} propagation time of a subset B\subseteq V(G) by minimizing the standard zero forcing propagation time of B on spanning supergraphs of G.
Proposition 3.1.
If G is a graph and B\subseteq V(G), then
\displaystyle\operatorname{pt}_{\operatorname{\lfloor\operatorname{Z}\rfloor}}% (G;B)=\min\{\operatorname{pt_{\operatorname{Z}}}(H;B)\ \ G\leq H\text{ and }% G=H\}.  (1) 
Proof.
Let \mathcal{F} be a set of \operatorname{\lfloor\operatorname{Z}\rfloor} forces of B such that \operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G;B)=% \operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G;\mathcal{F}). Note that every force in \mathcal{F} is either a \operatorname{Z} force or a force by a hop. Let G^{\prime} be the graph obtained from G by adding the edges uw such that u\rightarrow w appears in \mathcal{F} and u\rightarrow w by a hop. Note that for each edge uw\in E(G^{\prime})\setminus E(G), w is the only white neighbor of u in G^{\prime} and u is active at the time that u\rightarrow w in \mathcal{F}. This means that u\rightarrow w is a valid \operatorname{Z} force in G^{\prime} for each such edge. Thus, \mathcal{F} is a set of \operatorname{Z} forces of B in G^{\prime} and \operatorname{pt}_{Z}(G^{\prime};\mathcal{F})=\operatorname{pt_{\operatorname{% \lfloor\operatorname{Z}\rfloor}}}(G;\mathcal{F}). Therefore, we have that
\displaystyle\operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}% (G;B)=\operatorname{pt}_{Z}(G^{\prime};\mathcal{F})\geq\min\{\operatorname{pt_% {\operatorname{Z}}}(H;B)\ \ G\leq H\text{ and }G=H\}. 
Now let H^{\prime} be a spanning supergraph of G such that the right hand side of (1) is equal to \operatorname{pt_{\operatorname{Z}}}(H^{\prime},B). Let \mathcal{F} be a set of Z forces of B such that \operatorname{pt_{\operatorname{Z}}}(H^{\prime},\mathcal{F})=\operatorname{pt_% {\operatorname{Z}}}(H^{\prime},B). Let E be the set of edges E(H^{\prime})\setminus E(G). If uw\in E and (u\rightarrow w)\in\mathcal{F}, then u can \operatorname{\lfloor\operatorname{Z}\rfloor} force w in H^{\prime}uw by a hop when u\rightarrow w in \mathcal{F}. If uw\in E and (u\rightarrow w)\notin\mathcal{F}, then clearly \mathcal{F} is still a set of \operatorname{Z} forces of B in H^{\prime}uw. This means that \mathcal{F} is a set of \operatorname{\lfloor\operatorname{Z}\rfloor} forces of B in G with \operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G;\mathcal{F% })=\operatorname{pt_{\operatorname{Z}}}(H^{\prime};\mathcal{F}). Thus,
\displaystyle\operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}% (G;B)\leq\operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G;% \mathcal{F})=\operatorname{pt_{\operatorname{Z}}}(H^{\prime},B)=\min\{% \operatorname{pt_{\operatorname{Z}}}(H;B)\ \ G\leq H\text{ and }G=H\}.\qed 
We can use Definitions 2.3, 2.4, and 2.5 to extend the above proposition and give similar results for the \operatorname{\lfloor\operatorname{Z}\rfloor} propagation time of a graph and \operatorname{\lfloor\operatorname{Z}\rfloor} throttling.
Corollary 3.2.
Let G be a graph. Then
\displaystyle\operatorname{pt}_{\operatorname{\lfloor\operatorname{Z}\rfloor}}% (G)=\min\{\operatorname{pt_{\operatorname{Z}}}(H)\ \ G\leq H\text{ with }G=% H\text{ and }\operatorname{\lfloor\operatorname{Z}\rfloor}(G)=\operatorname{% Z}(H)\}. 
Proof.
Suppose G is a graph. If H is a spanning supergraph of G and B is a standard zero forcing set of H, then \operatorname{Z}(H)\leqB. Note that \operatorname{\lfloor\operatorname{Z}\rfloor}(G)\leq\operatorname{\lfloor% \operatorname{Z}\rfloor}(H)\leq\operatorname{Z}(H). So if we add the condition that B=\operatorname{\lfloor\operatorname{Z}\rfloor}(G), then B\leq\operatorname{Z}(H) and we have that B is minimum zero forcing set of H. By Proposition 3.1, it follows that
\displaystyle\operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}% (G)  \displaystyle=  \displaystyle\min\{\operatorname{pt_{\operatorname{\lfloor\operatorname{Z}% \rfloor}}}(G;B)\ \ \operatorname{\lfloor\operatorname{Z}\rfloor}(G)=B\}  
\displaystyle=  \displaystyle\min\{\min\{\operatorname{pt_{\operatorname{Z}}}(H;B)\ \ G\leq H% \text{ and }G=H\}\ \ \operatorname{\lfloor\operatorname{Z}\rfloor}(G)=B\}  
\displaystyle=  \displaystyle\min\{\operatorname{pt_{\operatorname{Z}}}(H;B)\ \ G\leq H\text{% with }G=H\text{ and }\operatorname{\lfloor\operatorname{Z}\rfloor}(G)=B\}  
\displaystyle=  \displaystyle\min\{\operatorname{pt_{\operatorname{Z}}}(H)\ \ G\leq H\text{ % with }G=H\text{ and }\operatorname{\lfloor\operatorname{Z}\rfloor}(G)=% \operatorname{Z}(H)\}.\qed 
Corollary 3.3.
If G is a graph and B\subseteq V(G), then
\displaystyle\operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}% (G;B)=\min\{\operatorname{th_{\operatorname{Z}}}(H;B)\ \ G\leq H\text{ and }% G=H\}. 
Corollary 3.4.
Let G be a graph. Then
\displaystyle\operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}% (G)=\min\{\operatorname{th_{\operatorname{Z}}}(H)\ \ G\leq H\text{ and }G=% H\}. 
Theorem 3.5.
The \operatorname{\lfloor\operatorname{Z}\rfloor} throttling number is subgraph monotone. In particular, if G and H are graphs with G\leq H, then \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq% \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(H).
Proof.
Let H be a graph. By Corollary 3.4, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G^{\prime})% \leq\operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(H) for any spanning subgraph G^{\prime} of H. Let v\in V(H) and let E(v) be the set of all edges in H incident with v. Define G^{\prime}=HE(v). So we have that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G^{\prime})% \leq\operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(H). Choose B^{\prime}\subseteq V(G^{\prime}) such that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G^{\prime};B% ^{\prime})=\operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G% ^{\prime}). Let \mathcal{F}^{\prime} be a set of \operatorname{\lfloor\operatorname{Z}\rfloor} forces of G^{\prime} with \operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G^{\prime};% \mathcal{F}^{\prime})=\operatorname{pt_{\operatorname{\lfloor\operatorname{Z}% \rfloor}}}(G^{\prime};B^{\prime}). We will produce a set B\subseteq V(G^{\prime}v) and a set of \operatorname{\lfloor\operatorname{Z}\rfloor} forces, \mathcal{F}, of B such that B\leqB^{\prime} and \operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G^{\prime}v% ,\mathcal{F})\leq\operatorname{pt_{\operatorname{\lfloor\operatorname{Z}% \rfloor}}}(G^{\prime};\mathcal{F}^{\prime}). Let v_{1}\rightarrow v_{2}\rightarrow\cdots\rightarrow v_{k} be the maximal \operatorname{\lfloor\operatorname{Z}\rfloor} forcing chain of \mathcal{F}^{\prime} that contains v. If k=1, then it suffices to choose B=B^{\prime}\setminus\{v\} and \mathcal{F}=\mathcal{F}^{\prime}. Now assume k>1. Note that v=v_{i} for some 1\leq i\leq k. Define B and \mathcal{F} as
\displaystyle B=\begin{cases}(B^{\prime}\setminus\{v_{i}\})\cup\{v_{i+1}\}&% \text{if }i=1,\\ B^{\prime}&\text{otherwise},\end{cases} 
and
\displaystyle\mathcal{F}=\begin{cases}\mathcal{F}^{\prime}\setminus\{v_{i}% \rightarrow v_{i+1}\}&\text{if }i=1,\\ (\mathcal{F}^{\prime}\setminus\{v_{i1}\rightarrow v_{i},v_{i}\rightarrow v_{i% +1}\})\cup\{v_{i1}\rightarrow v_{i+1}\}&\text{if }1<i<k,\\ \mathcal{F}^{\prime}\setminus\{v_{i1}\rightarrow v_{i}\}&\text{if }i=k.\end{cases} 
Recall that v is an isolated vertex in G^{\prime}. So when 1<i<k, v_{i1}\rightarrow v_{i} and v_{i}\rightarrow v_{i+1} by hopping in G^{\prime}. This means at the time that v_{i1}\rightarrow v_{i} in G^{\prime}, we can have v_{i1}\rightarrow v_{i+1} by a hop in G^{\prime}v. In the other cases, we simply remove the appropriate force from \mathcal{F}^{\prime}. So in all cases, it is clear that B\leqB^{\prime} and \operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G^{\prime}v% ;\mathcal{F})\leq\operatorname{pt_{\operatorname{\lfloor\operatorname{Z}% \rfloor}}}(G^{\prime};\mathcal{F}^{\prime}). Also note that G^{\prime}v=Hv. Thus, we have that for all 1\leq i\leq k,
\displaystyle\operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}% (Hv)  \displaystyle\leq  \displaystyleB+\operatorname{pt_{\operatorname{\lfloor\operatorname{Z}% \rfloor}}}(G^{\prime}v;\mathcal{F})\leqB^{\prime}+\operatorname{pt_{% \operatorname{\lfloor\operatorname{Z}\rfloor}}}(G^{\prime};\mathcal{F}^{\prime% })=\operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G^{\prime% })\leq\operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(H). 
Since v was chosen arbitrarily, it follows that removing vertices from H will not increase the \operatorname{\lfloor\operatorname{Z}\rfloor} throttling number. ∎
Since \operatorname{\lfloor\operatorname{Z}\rfloor} is minor monotone by definition, it is natural to ask if Theorem 3.5 can be strengthened to say that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}} is minor monotone. We will answer this question negatively (see Example 3.18) once we prove a characterization of \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}. Note that we can extend Theorem 3.5 in other ways. For each p\in\{\operatorname{Z}_{+},\operatorname{Z}_{\ell}\}, the color change rule for \lfloor p\rfloor takes the color change rule for p and allows hopping. This leads to the following remark.
Remark 3.6.
Suppose G is a graph and B\subseteq V(G). Then for each p\in\{Z_{+},Z_{\ell}\},
\displaystyle\operatorname{pt}_{\lfloor p\rfloor}(G;B)  \displaystyle=  \displaystyle\min\{\operatorname{pt}_{p}(H;B)\ \ G\leq H\text{ and }G=H\},  
\displaystyle\operatorname{th}_{\lfloor p\rfloor}(G;B)  \displaystyle=  \displaystyle\min\{\operatorname{th}_{p}(H;B)\ \ G\leq H\text{ and }G=H\}, 
and \operatorname{th}_{\lfloor p\rfloor} is subgraph monotone.
Note that if B is a standard zero forcing set of a graph G, then B is also a \operatorname{\lfloor\operatorname{Z}\rfloor} forcing set of G with \operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G;B)\leq% \operatorname{pt_{\operatorname{Z}}}(G;B). Thus, it is immediate that for any graph G, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G) is bounded above by \operatorname{th_{\operatorname{Z}}}(G). Butler and Young showed in [6] that for any graph G of order n, \operatorname{th_{\operatorname{Z}}}(G) is at least \left\lceil 2\sqrt{n}1\right\rceil. By Corollary 3.4, this lower bound holds for \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G) as well.
Corollary 3.7.
If G is a graph of order n, then
\displaystyle\operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}% (G)=\min\{\operatorname{th_{\operatorname{Z}}}(H)\ \ G\leq H\text{ and }G=% H\}\geq\left\lceil 2\sqrt{n}1\right\rceil. 
Since the \operatorname{\lfloor\operatorname{Z}\rfloor} throttling number is bounded above by the standard throttling number, any graph G that achieves \operatorname{th_{\operatorname{Z}}}(G)=\left\lceil 2\sqrt{n}1\right\rceil also achieves \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=\left% \lceil 2\sqrt{n}1\right\rceil. It was shown in [6] that \operatorname{th_{\operatorname{Z}}}(P_{n})=\left\lceil 2\sqrt{n}1\right\rceil. Thus, we can conclude that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(P_{n})=\left% \lceil 2\sqrt{n}1\right\rceil. The standard throttling number of a cycle was determined in [7] as follows.
Theorem 3.8.
[7] Let C_{n} be a cycle on n vertices. Define m to be the largest integer such that m^{2}\leq n and n=m^{2}+r. Then
\displaystyle\operatorname{th_{\operatorname{Z}}}(C_{n})=\begin{cases}2m1&% \text{if }r=0\text{ and }m\text{ is even},\\ 2m&\text{if }0<r\leq m\text{ or }(r=0\text{ and }m\text{ is odd}),\\ 2m+1&\text{if }m<r<2m+1.\end{cases} 
We can use Theorem 3.8 to determine the \operatorname{\lfloor\operatorname{Z}\rfloor} throttling number of a cycle.
Proposition 3.9.
Let C_{n} be a cycle on n vertices. Then \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(C_{n})=\left% \lceil 2\sqrt{n}1\right\rceil.
Proof.
Define m to be the largest integer such that m^{2}\leq n and n=m^{2}+r. Note that if m is even or r>0, then the conditions in Theorem 3.8 are equivalent to the conditions for \operatorname{th_{\operatorname{Z}}}(P_{n}) in [6]. So in this case, we have that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(C_{n})=% \operatorname{th_{\operatorname{Z}}}(P_{n})=\left\lceil 2\sqrt{n}1\right\rceil. Now suppose m is odd and r=0. So n=m^{2} and \operatorname{th_{\operatorname{Z}}}(C_{n})=2m=\left\lceil 2\sqrt{n}1\right% \rceil+1. In this case, we construct a \operatorname{\lfloor\operatorname{Z}\rfloor} forcing set B with B=m and \operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(C_{n};B)\leq m1. Draw C_{n} by arranging the vertices in an m by m array and adding the edges as in Figure 1. Let B be the set of vertices in the left column. Note that in each time step, every active vertex can force the vertex to its right to become blue (sometimes by a hop), so every vertex becomes blue one column at a time. Let \mathcal{F} be the set of \operatorname{\lfloor\operatorname{Z}\rfloor} forces of B obtained by this process. Clearly B=m and \operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(C_{n};B)\leq% \operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(C_{n};% \mathcal{F})=m1. Thus \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(C_{n})\leq 2% m1=\left\lceil 2\sqrt{n}1\right\rceil. ∎
The following example uses Theorem 3.5 to demonstrate that if \operatorname{th_{\operatorname{Z}}}(G)>\left\lceil 2\sqrt{n}1\right\rceil, then \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G) can differ greatly from \operatorname{th_{\operatorname{Z}}}(G).
Example 3.10.
Let G be the star K_{1,n1} on n vertices. It is easily seen that \operatorname{th_{\operatorname{Z}}}(G)=n. Consider the wheel W_{n1} on n vertices as a spanning supergraph of G. Obtain B\subseteq V(W_{n1}) by choosing the center vertex of the wheel and a set of vertices on the outside cycle that achieves optimal \operatorname{\lfloor\operatorname{Z}\rfloor} throttling for a cycle of order n1. By Theorem 3.5, we have that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq% \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(W_{n1})\leq% \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(C_{n1})+1% \leq\left\lceil 2\sqrt{n1}1\right\rceil+1. Recall that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\geq\left% \lceil 2\sqrt{n}1\right\rceil. It is easy to see that there are infinitely many integers n such that \left\lceil 2\sqrt{n1}1\right\rceil+1=\left\lceil 2\sqrt{n}1\right\rceil. So in these cases, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=\left% \lceil 2\sqrt{n}1\right\rceil.
The largeur de chemin of G, denoted by \operatorname{lc}(G), is defined in [3] as the minimum k for which G is a minor of the Cartesian product K_{k}\square P of a complete graph on k vertices with a path. The proper path width of a graph G, \operatorname{ppw}(G), is the smallest k such that G is a partial linear ktree (see [3]). These parameters are connected to \operatorname{\lfloor\operatorname{Z}\rfloor} by the following theorem.
Theorem 3.11.
[3] For every graph G having at least one edge, \operatorname{lc}(G)=\operatorname{ppw}(G)=\operatorname{\lfloor\operatorname{% Z}\rfloor}(G).
Theorem 3.11 exhibits a connection between \operatorname{\lfloor\operatorname{Z}\rfloor} and graphs of the form K_{k}\square P. We will capitalize on this connection in order to characterize \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G). The next definition constructs a graph from a given graph G, a standard zero forcing set B\subseteq V(G), and a set of forces \mathcal{F}. This construction is illustrated in Figure 3.
Definition 3.12.
Let G be a graph and let B\subseteq V(G) be a standard zero forcing set of G. Suppose \mathcal{F} is a set of \operatorname{Z} forces of B with \operatorname{pt_{\operatorname{Z}}}(G;B)=\operatorname{pt_{\operatorname{Z}}}% (G;\mathcal{F}). Let P_{1},P_{2},\ldots,P_{B} be the induced paths in G formed by the maximal forcing chains of \mathcal{F}. For each vertex v\in V(G), consider the path P_{i} that contains v and let \tau(v) be the number of times in the propagation process of \mathcal{F} at which v is active (possibly including time 0). Define the extension of G with respect to B and \mathcal{F}, denoted \mathcal{E}(G,B,\mathcal{F}), to be the graph obtained by the following procedure.

From each path P_{i} in G, construct a new path P_{i}^{\prime} so that for each v\in P_{i}, there are \tau(v) copies of v in P_{i}^{\prime}, and for each pair v_{a}, v_{b}\in P_{i} such that v_{a} is forced before v_{b} in P_{i}, every copy of v_{a} is to the left of every copy of v_{b} in P_{i}^{\prime}. Note that for each 1\leq i\leqB, V(P_{i}^{\prime})=\operatorname{pt_{\operatorname{Z}}}(G;B)+1 and we can arrange the paths \{P_{1}^{\prime},P_{2}^{\prime},\ldots,P_{B}^{\prime}\} into a B by \operatorname{pt}(G;B)+1 array of vertices.

For each edge uv\in E(G)\setminus\bigcup_{i=1}^{B}E(P_{i}), suppose P_{q} and P_{r} are the paths that contain u and v respectively. Since u and v must both be active before u or v can perform a force in G, there is at least one column in the B by \operatorname{pt}(G;B)+1 array such that a copy of u and a copy of v appear in that column. Draw an edge connecting the copy of u in P_{q}^{\prime} and the copy of v in P_{r}^{\prime} that are in the least such column.
Example 3.13.
Let G be the graph shown on the left in Figure 3. Choose B=\{v_{1},v_{4},v_{7}\} and let \mathcal{F} be the set of standard forces \mathcal{F}=\{v_{1}\rightarrow v_{2},v_{2}\rightarrow v_{3},v_{4}\rightarrow v% _{5},v_{5}\rightarrow v_{6},v_{7}\rightarrow v_{8},v_{8}\rightarrow v_{9}\}. Note that the forces in \mathcal{F} correspond to the horizontal edges in G as shown in Figure 3. The numbers above the vertices of G indicate the time step in \mathcal{F} when that vertex is forced (making that vertex active at the next time in the propagation process). For example, v_{7}\rightarrow v_{8} in time step 1 and v_{8}\rightarrow v_{9} in time step 3. Since there are two times in \mathcal{F} at which v_{8} active, there are two copies of v_{8} in \mathcal{E}(G;B;\mathcal{F}), which is shown on the right in Figure 3.
Suppose we are given the graph G=K_{a}\square P_{b}. Define the path edges of G to be the edges in G that correspond to edges of the factor P_{b}. Likewise, define the complete edges of G to be the edges in G that correspond to edges of the factor K_{a}. For example, if we draw G so that V(G) is arranged as an a by b array where each column induces a K_{a} and each row induces a P_{b}, then the path edges of G are the horizontal edges and the complete edges of G are the vertical edges. Given a graph G, an edge e\in E(G), a standard zero forcing set B\subseteq V(G), and a set \mathcal{F} of standard forces in G that uses e to perform a force, the following definition constructs a standard zero forcing set in G/e and a set of standard forces in G/e that mimic B and \mathcal{F} respectively.
Definition 3.14.
Let G be a graph with standard zero forcing set B\subseteq V(G) and suppose \mathcal{F} is a set of forces of B. Let e\in E(G) be an edge that is used to perform a force in \mathcal{F}. Define v_{1}\rightarrow v_{2}\rightarrow\cdots\rightarrow v_{k} to be the maximal forcing chain of \mathcal{F} that contains e. Note that k\geq 2. For each 1\leq j\leq k1, let e_{j} be the edge v_{j}v_{j+1} and let \vec{e}_{j} denote the force v_{j}\rightarrow v_{j+1}. So e=e_{i} for some 1\leq i\leq k1. Define v_{e} to be the vertex in G/e obtained by contracting e in G and define the sets B/e and \mathcal{F}/e as follows.
\displaystyle B/e=\begin{cases}(B\setminus\{v_{i}\})\cup\{v_{e}\}&\text{if }i=% 1,\\ B&\text{if }i>1,\\ \par\end{cases} 
and
\displaystyle\mathcal{F}/e=\begin{cases}(\mathcal{F}\setminus\{\vec{e}_{i1},% \vec{e}_{i},\vec{e}_{i+1}\})\cup\{v_{i1}\rightarrow v_{e},v_{e}\rightarrow v_% {i+2}\}&\text{if }k>2\text{ and }1<i<k1,\\ (\mathcal{F}\setminus\{\vec{e}_{i},\vec{e}_{i+1}\})\cup\{v_{e}\rightarrow v_{i% +2}\}&\text{if }k>2\text{ and }i=1,\\ (\mathcal{F}\setminus\{\vec{e}_{i1},\vec{e}_{i}\})\cup\{v_{i1}\rightarrow v_% {e}\}&\text{if }k>2\text{ and }i=k1,\\ \mathcal{F}\setminus\{\vec{e}_{i}\}&\text{if }k=2.\end{cases} 
The following lemma is used to prove Theorem 3.16 which exhibits a relationship between \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}} and graphs of the form K_{a}\square P_{b+1}.
Lemma 3.15.
Let G be a graph. Suppose B\subseteq V(G) is a standard zero forcing set of G with a set of standard forces \mathcal{F}. If e=uv is an edge in E(G) and (u\rightarrow v)\in\mathcal{F}, then \mathcal{F}/e is a set of standard forces of B/e in G/e such that \operatorname{pt_{\operatorname{Z}}}(G/e,\mathcal{F}/e)\leq\operatorname{pt_{% \operatorname{Z}}}(G;\mathcal{F}). Furthermore, if \mathcal{F} and B satisfy \operatorname{pt_{\operatorname{Z}}}(G;\mathcal{F})=\operatorname{pt_{% \operatorname{Z}}}(G;B) and \operatorname{th_{\operatorname{Z}}}(G)=\operatorname{th_{\operatorname{Z}}}(G% ;B), then \operatorname{th_{\operatorname{Z}}}(G/e)\leq\operatorname{th_{\operatorname{Z% }}}(G).
Proof.
Let G be a graph with standard zero forcing set B\subseteq V(G). Let \mathcal{F} be a set of forces of B and suppose we are given an edge e=uv\in E(G) that is used to perform a force in \mathcal{F}. Assume without loss of generality that (u\rightarrow v)\in\mathcal{F}. We proceed by induction on \operatorname{pt_{\operatorname{Z}}}(G;\mathcal{F}). If \operatorname{pt_{\operatorname{Z}}}(G;\mathcal{F})=0, then B=V(G) and no such edge e exists and there is nothing to prove. Suppose \operatorname{pt_{\operatorname{Z}}}(G;\mathcal{F})=1. In this case, it is clear that \mathcal{F}/e is a set of forces of B/e in G/e and \operatorname{pt_{\operatorname{Z}}}(G/e;\mathcal{F}/e)\leq 1=\operatorname{pt% _{\operatorname{Z}}}(G;\mathcal{F}).
Now suppose that for some k\geq 1, the result is true for any graph H and set of forces \mathcal{Q} with \operatorname{pt_{\operatorname{Z}}}(H;\mathcal{Q})\leq k. Again, let G be a graph with standard zero forcing set B\subseteq V(G). Now, suppose \mathcal{F} is a set of standard forces of B with \operatorname{pt_{\operatorname{Z}}}(G;\mathcal{F})=k+1. Let e=uv be a given edge in G such that (u\rightarrow v)\in\mathcal{F}. Define T(\mathcal{F}) to be all vertices in G that are forced last in \mathcal{F} (at time step k+1). For all vertices q\in T(\mathcal{F}), let q^{\prime} be the vertex in G that forces q at time step k+1. Note that for any q\in T(\mathcal{F}) and any neighbor y of q in G with y\neq q^{\prime}, y is also in T(\mathcal{F}). This is because if y\notin T(\mathcal{F}), then y cannot perform a force until q is forced. However, q is forced in time step k+1 which implies that y forces in a time step greater than \operatorname{pt_{\operatorname{Z}}}(G;\mathcal{F}), and this is a contradiction. Suppose uv=q^{\prime}q for some q\in T(\mathcal{F}). Since N(v)\setminus\{u\}\subseteq T(\mathcal{F}), \mathcal{F}/e is a set of forces of B/e in G/e such that \operatorname{pt_{\operatorname{Z}}}(G/e;\mathcal{F}/e)\leq k+1=\operatorname{% pt_{\operatorname{Z}}}(G;\mathcal{F}).
Finally, suppose u\rightarrow v in \mathcal{F} at a time step less than k+1. We can construct G/e by the following process. First, remove T(\mathcal{F}) from G to obtain H=GT(\mathcal{F}). Next, contract e in H to obtain H/e. Finally, add T(\mathcal{F}) to H so that the neighborhood in H of each q\in T(\mathcal{F}) is the same as the neighborhood of q in G (except that there may be a q\in T(\mathcal{F}) such that v_{e}\sim q in G/e whereas v\sim q in G). Let \mathcal{F}^{\prime}=\mathcal{F}\setminus\{q^{\prime}\rightarrow q\ \ q\in T(% \mathcal{F})\}. Clearly \operatorname{pt_{\operatorname{Z}}}(H;\mathcal{F}^{\prime})\leq k. So by the induction hypothesis, \operatorname{pt_{\operatorname{Z}}}(H/e;\mathcal{F}^{\prime}/e)\leq% \operatorname{pt_{\operatorname{Z}}}(H;\mathcal{F}^{\prime})\leq k. When we add T(\mathcal{F}) to H/e and consider the set of forces \mathcal{F}/e instead of \mathcal{F}^{\prime}/e, the propagation time will increase by at most 1. Thus, we have that \operatorname{pt_{\operatorname{Z}}}(G/e;\mathcal{F}/e)\leq\operatorname{pt_{% \operatorname{Z}}}(H/e;\mathcal{F}^{\prime}/e)+1\leq k+1=\operatorname{pt_{% \operatorname{Z}}}(G;\mathcal{F}). Note that if we choose \mathcal{F} and B such that \operatorname{pt_{\operatorname{Z}}}(G;\mathcal{F})=\operatorname{pt_{% \operatorname{Z}}}(G;B) and \operatorname{th_{\operatorname{Z}}}(G)=\operatorname{th_{\operatorname{Z}}}(G% ;B), then we have that
\displaystyle\operatorname{th_{\operatorname{Z}}}(G/e)\leqB/e+\operatorname{% pt_{\operatorname{Z}}}(G/e;\mathcal{F}/e)\leqB+\operatorname{pt_{% \operatorname{Z}}}(G;\mathcal{F})=B+\operatorname{pt_{\operatorname{Z}}}(G;B% )=\operatorname{th_{\operatorname{Z}}}(G).\qed 
Theorem 3.16.
Given a graph G and a positive integer t, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq t if and only if there exists integers a\geq 1 and b\geq 0 such that a+b=t and G can be obtained from K_{a}\square P_{b+1} by contracting path edges and deleting edges.
Proof.
First suppose \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq t. Let H be a spanning supergraph of G such that H has a standard zero forcing set B with \operatorname{th_{\operatorname{Z}}}(G;B)\leq t. Let \mathcal{F} be a set of \operatorname{Z} forces of B in H such that \operatorname{pt_{\operatorname{Z}}}(H;\mathcal{F})=\operatorname{pt_{% \operatorname{Z}}}(H;B). Let a=B, b^{\prime}=\operatorname{pt_{\operatorname{Z}}}(H;B)=\operatorname{th_{% \operatorname{Z}}}(G;B)a, and b=ta. Then b^{\prime}\leq b and
\displaystyle G\leq H\preceq\mathcal{E}(H,B,\mathcal{F})\leq K_{a}\square P_{b% ^{\prime}+1}\leq K_{a}\square P_{b+1}. 
Note that by the construction of H and \mathcal{E}(H,B,\mathcal{F}), we can obtain H from K_{a}\square P_{b+1} by contracting path edges. Then we can obtain G from H by deleting edges.
For the other direction, suppose G^{\prime}=K_{a}\square P_{b+1} with a+b=t and G can be obtained from G^{\prime} by contracting path edges and deleting edges. Choose B^{\prime}\subseteq V(G^{\prime}) such that B^{\prime} induces a copy of K_{a} in G^{\prime} that corresponds to an endpoint of P_{b+1}. Note that B^{\prime} is a standard zero forcing set of G^{\prime} with set of forces \mathcal{F}^{\prime} such that the set \{uv\ \ (u\rightarrow v)\in\mathcal{F}^{\prime}\} is the set of path edges in G^{\prime}. In other words, \mathcal{F}^{\prime} propagates along the path edges of G^{\prime}. Also note that \operatorname{pt_{\operatorname{Z}}}(G^{\prime};\mathcal{F}^{\prime})=b and B=a. Let D be a set of edges and let C be a set of path edges in G^{\prime} such that G can be obtained from G^{\prime} by first contracting the edges in C, then deleting the edges in D. Let H^{\prime} be the graph obtained from G^{\prime} by contracting the edges in C. Note that D\subseteq E(H^{\prime}). By repeated applications of Lemma 3.15, we can obtain a standard zero forcing set B\subseteq V(H^{\prime}) with set of forces \mathcal{F} such B\leqB^{\prime} and \operatorname{pt_{\operatorname{Z}}}(H^{\prime};\mathcal{F})\leq\operatorname{% pt_{\operatorname{Z}}}(G^{\prime};\mathcal{F}^{\prime})=b. Thus,
\displaystyle\operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}% (H^{\prime})\leq\operatorname{th_{\operatorname{Z}}}(H^{\prime})\leqB+% \operatorname{pt_{\operatorname{Z}}}(H^{\prime};\mathcal{F})\leqB^{\prime}+% \operatorname{pt_{\operatorname{Z}}}(G^{\prime};\mathcal{F}^{\prime})=a+b=t. 
By Theorem 3.5, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq% \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(H^{\prime})\leq t. ∎
Note that if we are given a fixed integer t\geq 1, then the graphs that have \operatorname{\lfloor\operatorname{Z}\rfloor} throttling number at most t are exactly the graphs given by Theorem 3.16. The next corollary is immediate from this observation.
Corollary 3.17.
If t is a fixed positive integer, then there are finitely many graphs with \operatorname{\lfloor\operatorname{Z}\rfloor} throttling number equal to t.
Given an integer t\geq 1, we can use Sage to generate a set, S(t), of all graphs given by Theorem 3.16 that satisfy \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq t. Then we can let S^{\prime}(t) be the set of subgraph maximal graphs in S(t). Since \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}} is subgraph monotone, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq t if and only if G\leq H for some H\in S^{\prime}(t). If t is sufficiently small, then it can be checked in reasonable time whether a given graph is a subgraph of a graph in S^{\prime}(t). The following example uses S^{\prime}(t) to illustrate that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}} is not minor monotone.
Example 3.18.
Let G=K_{3}\square P_{3} and let e\in E(G) as shown in Figure 4. Clearly \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G) is at most 5. However, it can be verified in Sage that G/e is not a subgraph of any graph in S^{\prime}(5). Thus, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G/e)\geq 6. In fact, if B\subseteq V(G/e) is the set of blue vertices shown below, then it is easy to see that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G/e;B)\leq 6 which implies that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G/e)=6.
In the next section, we modify the proof of Theorem 3.16 in order to characterize standard throttling.
4 A Characterization for Standard Throttling
Since there are graphs (e.g., stars) for which \operatorname{th_{\operatorname{Z}}}\neq\operatorname{th_{\operatorname{% \lfloor\operatorname{Z}\rfloor}}}, it is clear that the characterization in Theorem 3.16 does not also characterize \operatorname{th_{\operatorname{Z}}}. However, the only part of this characterization that does not work for standard throttling is the deletion of edges. In fact, Example 3.10 demonstrates that standard throttling is not spanning subgraph monotone. The next theorem shows how we can characterize \operatorname{th_{\operatorname{Z}}} if we are more careful about which edges we delete.
Theorem 4.1.
Given a graph G and a positive integer t, \operatorname{th_{\operatorname{Z}}}(G)\leq t if and only if there exists integers a\geq 0 and b\geq 1 such that a+b=t and G can be obtained from K_{a}\square P_{b+1} by contracting path edges and deleting complete edges.
Proof.
First suppose \operatorname{th_{\operatorname{Z}}}(G)\leq t. Let B\subseteq V(G) be a standard zero forcing set of G with \operatorname{th_{\operatorname{Z}}}(G;B)\leq t and let \mathcal{F} be a set of standard forces of B in G with \operatorname{pt_{\operatorname{Z}}}(G;\mathcal{F})=\operatorname{pt_{% \operatorname{Z}}}(G;B). Let a=B, b^{\prime}=\operatorname{pt_{\operatorname{Z}}}(G;B)=\operatorname{th_{% \operatorname{Z}}}(G;B)a, and b=ta. Then b^{\prime}\leq b and
\displaystyle G\preceq\mathcal{E}(G,B,\mathcal{F})\leq K_{a}\square P_{b^{% \prime}+1}\leq K_{a}\square P_{b+1}. 
Note that by the construction of \mathcal{E}(G,B,\mathcal{F}), we can obtain G from K_{a}\square P_{b+1} by contracting path edges and deleting complete edges.
For the other direction, suppose G^{\prime}=K_{a}\square P_{b+1} with a+b=t and G can be obtained from G^{\prime} by contracting path edges and deleting complete edges. Choose B^{\prime}\subseteq V(G^{\prime}) such that B^{\prime} induces a copy of K_{a} in G^{\prime} that corresponds to an endpoint of P_{b+1}. Note that B^{\prime} is a standard zero forcing set of G^{\prime} with set of forces \mathcal{F}^{\prime} such that the set \{uv\ \ (u\rightarrow v)\in\mathcal{F}^{\prime}\} is the set of path edges in G^{\prime}. In other words, \mathcal{F}^{\prime} propagates along the path edges of G^{\prime}. Also note that \operatorname{pt_{\operatorname{Z}}}(G^{\prime};\mathcal{F}^{\prime})=b and B^{\prime}=a. Let D be a set of complete edges in G^{\prime} and let C be a set of path edges in G^{\prime} such that G can be obtained from G^{\prime} by first deleting the edges in D, then contracting the edges in C. Let H^{\prime} be the graph obtained from G^{\prime} by deleting the edges in D. Since no edge in D is used to perform a force in \mathcal{F}^{\prime}, \mathcal{F}^{\prime} is still a set of forces of B^{\prime} in H^{\prime} with \operatorname{pt_{\operatorname{Z}}}(H^{\prime};\mathcal{F}^{\prime})\leq% \operatorname{pt_{\operatorname{Z}}}(G^{\prime};\mathcal{F}^{\prime})=b. Also, G can be obtained from H^{\prime} by contracting the edges in C. By repeated applications of Lemma 3.15, we can obtain a standard zero forcing set B\subseteq V(G) with set of forces \mathcal{F} such B\leqB^{\prime} and \operatorname{pt_{\operatorname{Z}}}(G;\mathcal{F})\leq\operatorname{pt_{% \operatorname{Z}}}(H^{\prime};\mathcal{F}^{\prime})\leq b. Thus,
\displaystyle\operatorname{th_{\operatorname{Z}}}(G)\leqB+\operatorname{pt_{% \operatorname{Z}}}(G;\mathcal{F})\leqB^{\prime}+\operatorname{pt_{% \operatorname{Z}}}(H^{\prime};\mathcal{F}^{\prime})\leqB^{\prime}+% \operatorname{pt_{\operatorname{Z}}}(G^{\prime};\mathcal{F}^{\prime})=a+b=t.\qed 
Corollary 4.2.
If t is a fixed positive integer, then there are finitely many graphs G with standard throttling number equal to t.
Suppose G is a graph on n vertices and t is a postive integer with \operatorname{th_{\operatorname{Z}}}(G)\leq t. Note that we can use t to bound the number of vertices in G. Since \left\lceil 2\sqrt{n}1\right\rceil\leq\operatorname{th_{\operatorname{Z}}}(G)\leq t, V(G)=n\leq\frac{(t+1)^{2}}{4}. By Corollary 3.7, this bound still holds when \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq t.
In order to construct forcing sets in paths and cycles that are optimal for throttling, it has been useful to “snake” the graph in some way. This idea was used for \operatorname{th_{\operatorname{Z}}}(P_{n}) in [6], and again for \operatorname{th_{\operatorname{Z}}}(C_{n}) in [7]. We also used a “snaking” construction for \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(C_{n}) in Proposition 3.9 (see Figure 1). Note that in most of these cases, the “snaked” graph is a spanning subgraph or a minor of a graph of the form K_{a}\square P_{b+1}. So it is clear that the “snaking” method is present in Theorems 3.16 and 4.1.
5 Applications
In this section, we use Theorems 3.16 and 4.1 to quickly obtain some additional results. First, we define and characterize throttling for power domination. Then, we characterize extreme values of the \operatorname{\lfloor\operatorname{Z}\rfloor} throttling number.
5.1 Throttling for Power Domination
Phase Measurement Units (PMUs) are often used to monitor electric power networks (see [2]). PMUs are placed at particular nodes and the way they monitor the power network is similar to the process of standard zero forcing. The power domination number was studied in [9] and its connection to zero forcing was made in [4]. Recall that in a graph G, the neighborhood of a set B\subseteq V(G) is defined as N(B)=\bigcup_{v\in B}N(v). In power domination, we start with a set of blue vertices B\subseteq V(G) and in the first time step we change the color of each vertex in N(B)\setminus B to blue. Each subsequent time step is a time step of standard zero forcing using the vertices that are blue at that time. The following definition of power dominating set makes use of the relationship between power domination and standard zero forcing rather than following the literature, although it is observed to be equivalent in [4].
Suppose G is a graph and B\subseteq V(G). We say that B is a power dominating set of G if N[B]=B\cup N(B) is a standard zero forcing set of G. The power domination number of a graph G, denoted \gamma_{P}(G), is the cardinality of a minimum power dominating set in G. The power propagation time of a set of vertices is the power domination analog of \operatorname{pt_{\operatorname{Z}}} and is studied in [8]. To be consistent with the notation in Section 2, we can define the power propagation time of B in G as \operatorname{pt}_{\gamma_{P}}(G;B)=\operatorname{pt_{\operatorname{Z}}}(G;N[B% ])+1. Note that we add 1 in order to count the first time step of power domination in which N[B] becomes the new set of blue vertices. Now, the definition of \gamma_{P} throttling immediately follows.
Definition 5.1.
Suppose G is a graph and B\subseteq V(G). The power domination (\gamma_{P}) throttling of B is defined as
\displaystyle\operatorname{th}_{\gamma_{P}}(G;B)=B+\operatorname{pt}_{\gamma% _{P}}(G;B). 
Definition 5.2.
Suppose G is a graph. The power domination (\gamma_{P}) throttling number of G is defined as
\displaystyle\operatorname{th}_{\gamma_{P}}(G)=\underset{B\subseteq V(G)}{\min% }\{\operatorname{th}_{\gamma_{P}}(G;B)\}. 
We will use Theorem 4.1 to give a characterization of \operatorname{th}_{\gamma_{P}}. The graphs in the following definition play an important role and are illustrated in Figure 5.
Definition 5.3.
Suppose a, s, and b are fixed positive integers and let G be the dijoint union K_{a}\cup(K_{s}\square P_{b}). Let H be the disjoint copy of K_{a} and let H^{\prime} be an induced K_{s} that corresponds to one of the endpoints of P_{b} in K_{s}\square P_{b}. Define D to be the complete bipartite graph with vertices V(H)\cup V(H^{\prime}) and define \operatorname{PD}(a,s,b) to be the graph G\cup D. We will refer to the edges in D as the domination edges of \operatorname{PD}(a,s,b) and the vertices in V(H^{\prime}) are the dominated vertices of \operatorname{PD}(a,s,b).
The path edges of \operatorname{PD}(a,s,b) are the path edges of the induced K_{s}\square P_{b} as defined in Section 3. Define the complete edges of \operatorname{PD}(a,s,b) to be the complete edges of the induced K_{s}\square P_{b} together with the edges of the induced K_{a}. Suppose G is a graph with A\subseteq V(G) and B\subseteq E(G). The vertices in A are saturated by the edges in B if for every vertex v\in A, there is an edge in B that is incident to v. Recall that for a graph G, \Delta(G) denotes the maximum degree of a vertex in V(G).
Theorem 5.4.
Given a graph G and a positive integer t, \operatorname{th}_{\gamma_{P}}(G)\leq t if and only if there exists positive integers a, s, and b such that a+b=t, s\leq a\Delta(G), and G can be obtained from \operatorname{PD}(a,s,b) by contracting path edges, deleting complete edges, and deleting domination edges provided the remaining domination edges saturate the dominated vertices.
Proof.
Suppose \operatorname{th}_{\gamma_{P}}(G)\leq t. Let B\subseteq V(G) be a power dominating set of G such that \operatorname{th}_{\gamma_{P}}(G;B)=\operatorname{th}_{\gamma_{P}}(G)\leq t. Let a=B, b^{\prime}=\operatorname{pt}_{\gamma_{P}}(G;B)=\operatorname{th}_{\gamma_{P}}(% G)a, and b=ta. Note that since \operatorname{th}_{\gamma_{P}}(G)\leq t, b\leq b^{\prime}. Define S=N(B)\setminus B and let s^{\prime}=s. Note that S is a standard zero forcing set of GB with \operatorname{pt_{\operatorname{Z}}}(GB;S)=\operatorname{pt}_{\gamma_{P}}(G;B% )1=b^{\prime}1. Let \mathcal{F} be a set of forces of S such that \operatorname{pt_{\operatorname{Z}}}(GB;\mathcal{F})=\operatorname{pt_{% \operatorname{Z}}}(GB;S)=b^{\prime}1. Then
\displaystyle GB\preceq\mathcal{E}(GB,S,\mathcal{F})\leq K_{s}\square P_{b^{% \prime}}\leq K_{s}\square P_{b} 
and GB can be obtained from K_{s}\square P_{b} by contracting path edges and deleting complete edges. Therefore, we can obtain G from \operatorname{PD}(a,s,b) by contracting path edges, deleting complete edges, and deleting domination edges. As we obtain G from \operatorname{PD}(a,s,b), it is easy to see that s\leq a\Delta(G).
Next, suppose G can be obtained from G^{\prime}=\operatorname{PD}(a,s,b) as described with a+b=t and s\leq a\Delta(G). Let B be the vertices of the induced K_{a} in G^{\prime}. Clearly B is a power dominating set of G^{\prime} with \operatorname{th}_{\gamma_{P}}(G^{\prime})\leqB+\operatorname{pt}_{\gamma_{P% }}(G^{\prime};B)\leq a+b\leq t. Again, let S=N(B)\setminus B with s=S. Applying Lemma 3.15 to K_{s}\square P_{b} shows that contracting path edges of G^{\prime} will not increase the \gamma_{P} propagation time. Note that in the power propagation process of B, complete edges are not used in the domination time step and are never used to perform forces. So deleting complete edges from G^{\prime} does not increase the \gamma_{P} propagation time. Also, when we delete domination edges to obtain G, we maintain the fact that every dominated vertex will be colored blue in the first time step. Thus, \operatorname{th}_{\gamma_{P}}(G)\leq\operatorname{th}_{\gamma_{P}}(G^{\prime}% )\leq t. ∎
Note that unlike \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}} and \operatorname{th_{\operatorname{Z}}}, if t is a fixed positive integer, there are infinitely many graphs G with \operatorname{th}_{\gamma_{P}}(G)=t. For example, \operatorname{th}_{\gamma_{P}}(K_{1,n1})=2 for all n\geq 2. In this regard, \operatorname{th}_{\gamma_{P}} more closely resembles positive semidefinite throttling (see [7]).
5.2 Extreme \operatorname{\lfloor\operatorname{Z}\rfloor} Throttling
For a fixed positive integer t, Theorem 3.16 characterizes all graphs G with \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq t. Clearly \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=t if and only if \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq t and \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\nleq t1. So we can characterize all graphs with \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=t by applying Theorem 3.16 and removing the graphs with \operatorname{\lfloor\operatorname{Z}\rfloor} throttling number at most t1. This is easily done by hand for t\leq 3.
Observation 5.5.
The graph G=K_{1} is the only graph with \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=1.
Proposition 5.6.
For a graph G, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=2 if and only if G=K_{2} or G=2K_{1}.
Proof.
By Theorem 3.16, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq 2 if and only if G can be obtained from K_{1}\square P_{2}=K_{2} or K_{2}\square P_{1}=K_{2} by deleting edges and contracting path edges. Thus, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq 2 if and only if G\in\{K_{1},K_{2},2K_{1}\}. Since G=K_{1} is the only graph that satisfies \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=1, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=2 if and only if G\in\{K_{2},2K_{1}\}. ∎
Proposition 5.7.
For a graph G, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=3 if and only if G\in\mathcal{G} where
\displaystyle\mathcal{G}=\{C_{4},P_{4},2K_{2},K_{1}\dot{\cup}P_{3},K_{2}\dot{% \cup}2K_{1},4K_{1},K_{3},P_{3},K_{1}\dot{\cup}K_{2},3K_{1}\}. 
Proof.
By Theorem 3.16, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq 3 if and only if G can be obtained from K_{3}\square P_{1}=K_{3}, K_{2}\square P_{2}=C_{4}, or K_{1}\square P_{3}=P_{3} by deleting edges and contracting path edges. Let \mathcal{H} be the set of all subgraphs of C_{4} and K_{3}. It is clear that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq 3 if and only if G\in\mathcal{H}. Note that \mathcal{G}=\mathcal{H}\setminus\{K_{1},K_{2},2K_{1}\}. ∎
Theorems 3.16 and 4.1 reinforce the fact that for any graph G, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq% \operatorname{th_{\operatorname{Z}}}(G). Let G be a graph. Since \operatorname{th_{\operatorname{Z}}} is bounded below by \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}, if we can find a subset B\subseteq V(G) with \operatorname{th_{\operatorname{Z}}}(G;B)=\operatorname{th_{\operatorname{% \lfloor\operatorname{Z}\rfloor}}}(G), then \operatorname{th_{\operatorname{Z}}}(G)=\operatorname{th_{\operatorname{% \lfloor\operatorname{Z}\rfloor}}}(G).
Corollary 5.8.
If t\in\{1,2,3\} and G\notin\{K_{1}\dot{\cup}P_{3},K_{2}\dot{\cup}2K_{1},4K_{1}\}, then \operatorname{th_{\operatorname{Z}}}(G)=t if and only if \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=t.
Proof.
Let \mathcal{J}=\{K_{1}\dot{\cup}P_{3},K_{2}\dot{\cup}2K_{1},4K_{1}\}. For each graph G with \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq 3 and G\notin\mathcal{J}, it is easy to construct a standard zero forcing set B\subseteq V(G) with \operatorname{th_{\operatorname{Z}}}(G;B)=\operatorname{th_{\operatorname{% \lfloor\operatorname{Z}\rfloor}}}(G). If G\in\mathcal{J}, then \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=3, but \operatorname{th_{\operatorname{Z}}}(G)=4 because we are no longer allowed to force by a hop. ∎
High \operatorname{\lfloor\operatorname{Z}\rfloor} throttling values are harder to characterize. Clearly, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(K_{n})=% \operatorname{th_{\operatorname{Z}}}(K_{n})=n. Let (K_{n})_{e} be the complete graph on n vertices minus a single edge. It is also clear that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}((K_{n})_{e})% =\operatorname{th_{\operatorname{Z}}}((K_{n})_{e})=n. More generally, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=n implies that \operatorname{th_{\operatorname{Z}}}(G)=n. For a given graph G, the following proposition gives an upper bound for \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G) in terms of the independence number, \alpha(G).
Proposition 5.9.
If G is a graph of order n, then \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq n% \alpha(G)+\left\lceil 2\sqrt{\alpha(G)}1\right\rceil.
Proof.
Suppose G is a graph with independent set A\subseteq V(G). Let B=V(G)\setminus A. Note that GB has no edges and by Theorem 3.5, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(GB)\leq% \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(C_{A})=% \left\lceil 2\sqrt{A}1\right\rceil. Choose C\subseteq A such that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(GB,C)=\left% \lceil 2\sqrt{A}1\right\rceil. Then B\cup C is a \operatorname{\lfloor\operatorname{Z}\rfloor} forcing set of G with \operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G;B\cup C)% \leq\operatorname{pt_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(GB,C). Thus, \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq nA% +\left\lceil 2\sqrt{A}1\right\rceil. If we choose A so that A=\alpha(G), we get the desired result. ∎
Corollary 5.10.
If G is a graph with \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=n, then \alpha(G)\leq 3.
Proof.
Let G be a graph and define f(x)=x\left\lceil 2\sqrt{x}1\right\rceil. So Proposition 5.9 says that \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq nf(% \alpha(G)). It is easily shown that if x\geq 4 is an integer, then f(x)\geq 1. So if \alpha(G)\geq 4, then \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)\leq nf(% \alpha(G))\leq n1. ∎
Note that the converse of Corollary 5.10 is false. For example, let G=P_{6}. Then \alpha(G)=3 and \operatorname{th_{\operatorname{\lfloor\operatorname{Z}\rfloor}}}(G)=\left% \lceil 2\sqrt{6}1\right\rceil=4<6=n.
Acknowledgment
Research supported in part by Holl Chair funds.
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