Forcing Parameters in Fully Connected Cubic Networks

Abstract

Domination in graphs has been extensively studied and adopted in many real life applications. The monitoring electrical power system is a variant of a domination problem called power domination problem. Another variant is the zero forcing problem. Determining minimum cardinality of a power dominating set and zero forcing set in a graph are the power domination problem and zero forcing problem, respectively. Both problems are $NP$-complete. In this paper, we compute the power domination number and the zero forcing number for fully connected cubic networks.

1. Introduction

Monitoring electrical power systems by placing as few Phase Measurement Units (PMUs) at selected regions in the system is modeled as a graph theoretic problem. The cost of such a synchronized devise is very high, and hence it is required to fetch the smallest set of devices while maintaining the ability to supervise the entire system. In 2002, Hayens et al. [1] considered this problem as the power domination problem in graphs, which is a variation of the domination problem. An electric power network is designed by a graph where the vertices represent the electric nodes and the edges are associated with the transmission lines joining two electrical nodes. In 2012, Paul Dorbec et al. [2] presented the idea of a k-power domination problem, which is a generalization of power domination problem in graphs.

A graph $G=(V,E)$ is defined as a nonempty set of vertices $V=V\left(G\right)$ together with a set of edges $E=E\left(G\right)$ joining certain pairs of vertices. Vertices u and v are said to be adjacent if u and v are the end vertices of an edge in G. For $u\in V$, the set of all vertices adjacent to u are said to be in the neighbors of u and is denoted by $N\left(u\right)$. Then, the closed neighborhood of u is defined as $N\left[u\right]=N\left(u\right)\cup \left\{u\right\}$.

A subset $S\subseteq V$ in a graph is said to be a domination set G if every vertex in V is either in S or is adjacent to some vertices in S [1]. In 2002, Haynes et al. introduced power domination by formulating propagation rules in terms of vertices and edges in a graph.

Let G be a graph with a vertex set V. Let $K\subseteq V$. Vertices $M^i\left(K\right)$ monitored by K at level i, $i\ge 0$, inductively are as follows:

• $M^0\left(K\right)=N\left[K\right]$.
• $M^{i+1}\left(K\right)=M^i\left(K\right)\cup \{v\in V:\exists u\in M^i\left(K\right)$, $N\left(u\right)\cap (V\left(G\right)\backslash M^i\left(K\right))=\left\{v\right\}\}$.

At some stage, if K monitors the entire vertex set V, we say that K is a power dominating set of G. The minimum cardinality of power dominating set of G is called the power domination number of G and is denoted by ${\gamma }_p\left(G\right)$ [3].

The zero forcing process can be treated as a coloring process on the vertices of the graph. If vertex x is colored red and exactly one neighbor y of x is green, then change the color of y to red, and we say that x forces y. A zero forcing set for G is a subset of vertices H such that if initially the vertices in H are colored red and the remaining vertices are colored green, then repeated application of the above process can color all vertices of G red. The cardinality of a minimum zero forcing set of G is represented by $\zeta \left(G\right)$ [3]. It is customary to address ’red vertices’ as monitored vertices and the ’green vertices’ as unmonitored vertices.

The power domination problem is NP-complete [1]. Bounds on the power domination number for any graph G were obtained in [1]. The power domination problem studied for trees [1] and grids [4], split graphs [5]. Barrera et al. [6] studied Power domination in petersen graphs. Cockayne et al. [7] presented new concepts of domination in graphs. Chen et al. [8] investigated networks with complete connection. Dorbec et al. [9] defined power domination in product graphs. Ho et al. [10] given hamiltonian connectivity in fully connected cubic networks. Kosari et al. [11,12] studied double roman domination and total domination in graphs. Xu et al. [13] introduced power domination in block graphs. Zhao et al. [14] presented new results of power domination in graph theory.

2. Fully Connected Cubic Networks (FCCNs)

$FCCNs$ are networks that provide excellent expandability. Optical and electronic technologies can be utilized in $FCCN$ to build a new hybrid computer architecture.

Let $a^n=aa\dots a\left(ntimes\right)$. $FCCN$ of level $r,r\ge 1$ denoted by $FCC{N}_r$, is defined recursively as follows [15]:

• $FCC{N}_1$ has the set of integers modulo 8 as the node set and all ordered pairs $x,y,x<y,x,y\in \{0,1,\dots ,7\}$ as the edge set.
• When $r\ge 2,FCC{N}_r$ is built from eight node-disjoint copies of $FCC{N}_{r-1}$ by adding 28 edges. Specifically, if, for $0\le k\le 7$, we let $kFCC{N}_{r-1}$ denote a copy of $FCC{N}_{r-1}$ with each node being prefixed with k, then $FCC{N}_r$ is defined by:
  $V\left(FCC{N}_r\right)={\displaystyle \bigcup _{k=0}^7}V\left(kFCC{N}_{r-1}\right)$,
  $E\left(FCC{N}_r\right)=\left({\displaystyle \bigcup _{k=0}^7}E\left(kFCC{N}_{r-1}\right)\right)\cup \left\{(p{q}^{r-1},q{p}^{r-1})\right|0\le p<q\le 7\}$
  For $0\le k\le 7,kFCC{N}_{r-1}$ is named an $(r-1)$-level sub-$FCCN$ of $FCC{N}_r$, or simply a sub-$FCCN$ of $FCC{N}_r$, if there is no uncertainty.
• Given an $FCC{N}_r,r\ge 2$, a boundary node is a node of the form $k^r$. An $ICE$ is an edge of the form $(p{q}^{n-1},q{p}^{n-1})$. Each $ICV$ of $FCCN$ is of degree 4 except the boundary nodes of degree 3. Obviously, $kFCC{N}_{r-1}$ has seven $ICV$ and one boundary node for $0\le k\le 7$ and $r\ge 2$ [15] (see Figure 1).
  

  Figure 1. FCCNs (a) $H_1$ (b) $H_2$ with $00,11,\dots ,77$ as boundary nodes.

Note: In what follows, we denote $FCC{N}_r$ by $H_{r}r\ge 1$.

Remark 1.

$H_r$ has $8^r$ nodes, eight of which degree 3 and the rest $8^r-8$ nodes of degree 4. $H_r$ has $2^{3r+1}-4$ edges. Its diameter is $2^{(r+1)}-1$. For $2\le i\le r$, each level i contains $8^{i-k}$ node disjoint copies of $H_j,1\le j\le i$. This implies that there are $8^{r-2}$ node disjoint copies of $H_2$ in $H_r$.

3. Main Results

In this section, we compute PDN and ZFN for $H_r,r\ge 2$.

3.1. Power Domination in $FCCNs$

We obtain lower bounds for the $PDN$ of $H_r,r\ge 2$, and prove that the bounds are sharp.

Lemma 1.

The power domination number of $H_2$ is at least 4. In other words ${\gamma }_p\left(H_2\right)\ge 4$.

Proof. 

$H_2$ contains eight node disjoint copies of $H_1$, say $0H_1,1H_1,\dots ,7H_1$. Let S be a $PDS$ of G. We claim that $\left|S\right|\ge 4$. Suppose $S\subseteq 0H_1$, any node in $i{H}_1,1\le i\le 7$ is neighbor to at most one node of $0H_1$. This implies that every node in $i{H}_1,1\le i\le 7$ is neighbor to at least three unmonitored nodes, a contradiction. Therefore, $S⊈0H_1$ (see Figure 2a). Suppose $S\subseteq 0H_1\cup 1H_1$. Then, S can monitor at most two nodes of $i{H}_1,2\le i\le 7$, each of which is neighbor to at least two unmonitored nodes, a contradiction. Therefore, $S⊈0H_1\cup 1H_1$ (see Figure 2b).

Figure 2. Circled vertices indicates in (a–d) be a PDS of $H_2$.

Suppose $S\subseteq 0H_1\cup 1H_1\cup 2H_1$. Then, S can monitor at most three nodes of each $i{H}_1,3\le i\le 7$. No three nodes in $i{H}_1$ induce an independent set. Hence, these three nodes induce an edge and an isolated node. Then, each end node of the edge is neighbor to two unmonitored nodes, and the independent node is neighbor to three unmonitored nodes, a contradiction (see Figure 2c). On the other hand, if three nodes induce a path say, $uvw$, then u and w are neighbor to two unmonitored nodes each and v is neighbor to exactly one unmonitored node, which in turn is neighbor to at least two unmonitred nodes, a contradiction (see Figure 2d). Therefore, $S⊈0H_1\cup 1H_1\cup 2H_1$. Hence, nodes in S are in at least 4 copies of $H_1$. Therefore, $\left|S\right|\ge 4$.    □

Lemma 2.

Let S be a PDS of $H_3$ and H be a subgraph of $H_3$ isomorphic to $H_2$. Then, $\left|V\right(H)\cap S|\ge 4$.

Proof. 

$H_3$ is composed of eight copies of $H_2$, denoted by $0H,1H,\dots ,7H$. Further, each $iH,1\le i\le 7$ contains 8 cubes as subgraphs each denoted by $Q^3$. Consider $0H$. Let $0H={\displaystyle \bigcup _{k=0}^7}k{Q}^3$. The worst case arises when all the 7 intercubic nodes $0ii,1\le i\le 7$, of $0H$ are monitored by nodes of S not in $0H$. Suppose $\left|V\right(0H)\cap S|=3$. Let $V\left(0H\right)\cap S=\{\alpha ,\beta ,\gamma \}$. Without loss of generality, let $\alpha ,\beta ,\gamma$ be in $0Q^3$. Suppose $\alpha =00i,\beta =00j$ and $\gamma =00k,i\ne j\ne k,1\le i,j,k\le 7$. Let $(\alpha ,\alpha {}^{\prime }),(\beta ,\beta {}^{\prime })$ and $(\gamma ,\gamma {}^{\prime })$ be the intercubic edges in $0H$ with $\alpha {}^{\prime }=0i0,\beta {}^{\prime }=0j0$ and $\gamma {}^{\prime }=0k0$ in $i{Q}^3,j{Q}^3$ and $k{Q}^3$, respectively, $1\le i,j,k\le 7$. Even if $0i0,0ii$ are neighbors, each of $0i0$ and $0ii$ is neighbor to two distinct unmonitored nodes in $i{Q}^3$, a contradiction (see Figure 3a).

Figure 3. Circled vertices indicates in (a–c) be a PDS of $0H_2$ of $H_3$.

Suppose $\alpha ,\beta$ are in $0Q^3$ and $\gamma$ is in $1Q^3$. Assume that $\alpha =00i,\beta =00j$ and $\gamma =01k,i\ne j\ne k,2\le i,j,k\le 7$. Let $(\alpha ,\alpha {}^{\prime }),(\beta ,\beta {}^{\prime })$ and $(\gamma ,\gamma {}^{\prime })$ be the intercubic edges in $0H$ with $\alpha {}^{\prime }=0i0,\beta {}^{\prime }=0j0$ and $\gamma {}^{\prime }=0k1$ in $i{Q}^3,j{Q}^3$ and $k{Q}^3$, respectively, $2\le i,j,k\le 7$. Even if $0j0,0jj$ and $0jk$ induce a path, then $0jj$ and $0jk$ are neighbors to two unmonitored nodes each and $0j0$ is neighbor to exactly one unmonitored node, which in turn is adjacent to at least two unmonitored nodes, a contradiction (see Figure 3b).

Suppose $\alpha \in 0Q^3,\beta \in 1Q^3$ and $\gamma \in 2Q^3$. Let $\alpha =00i,\beta =01j$ and $\gamma =02k,i\ne j\ne k,3\le i,j,k\le 7$. Let $(\alpha ,\alpha {}^{\prime }),(\beta ,\beta {}^{\prime })$ and $(\gamma ,\gamma {}^{\prime })$ be the intercubic edges in $0H$ with $\alpha {}^{\prime }=0i0,\beta {}^{\prime }=0j1$ and $\gamma {}^{\prime }=0k2$ in $i{Q}^3,j{Q}^3$ and $k{Q}^3$, respectively, $3\le i,j,k\le 7$. Even if $0k0,0kk$ and $0ik$ induce a path, the nodes $0ii$ and $0ik$ are neighbor to two unmonitored nodes each and $0k0$ is neighbor to exactly one unmonitored node, which in turn is neighbor to at least two unmonitored nodes, a contradiction. Therefore, $\left|V\right(H)\cap S|\ge 4$ (see Figure 3c).    □

Lemma 3.

The power domination number of $H_r,r\ge 3$ is at least $2^{3r-4}$. In other words, ${\gamma }_p\left(H_r\right)\ge 2^{3r-4}$.

Proof. 

We prove the result by induction on r. We consider the case when $r=3$. $H_3$ contains eight node disjoint copies of $H_2$, say $0H_2,1H_2,\dots ,7H_2$. Let S be a PDS of $H_r$. We claim that $\left|S\right|\ge 32$. By Lemma 1 and Lemma 2, there are at least 4 nodes in each copy of $H_2$ to monitor $H_3$. Hence, $\left|S\right|\ge 4\times 8$. Therefore, ${\gamma }_p\left(H_3\right)\ge 32$.

Assume the result is true for $r=k,r\ge 3$. That is, ${\gamma }_p\left(H_k\right)\ge 2^{3k-4}$. Consider the case when $r=k+1$. Let S be a PDS $H_{k+1}$. We have to prove that ${\gamma }_p\left(H_{k+1}\right)\ge 2^{3k-1}$. Suppose not, let $\left|S\right|<2^{3k-1}$. In $H_{k+1}$, there are $8^{k-1}$ node disjoint copies of $H_2$. With the deletion of one node from S in a copy of H, there is at least one node say, $u\in H_1$ monitored by intercubic edges. With this monitored node u, we claim that 3 nodes in $H_2$, one each in 3 copies of $H_1$ say $1H_1,2H_1,3H_1$, are not sufficient to monitor all nodes in any of $H_1^i,i=1,2,3$. In the worst case, suppose all the nodes in the copy of $H_1$ containing u are all already monitored, then, the saturated node in $1H_1$ has two unmonitored nodes neighbor to it, a contradiction. Thus, $\left|S\right|\ge 2^{3k-1}$. Therefore, ${\gamma }_p\left(H_k\right)\ge 2^{3k-4}$.    □

Radix-lexicographic ordering:

Name the nodes of $H_r,r\ge 1$ as follows:

(i)

Name the nodes of $H_1$ as 1 digit radix $Z_7$ number, say $0,1,\dots ,7$ by lexicographic order.

(ii)

Name the nodes of $H_2$, as 2 digit radix $Z_7$ number, $0H_1,1H_1,\dots ,7H_1$.

(iii)

Inductively, name the nodes of $H_r$, as r digit radix $Z_7$ number, $k{H}_{r-1},0\le k\le 7$ (see Figure 1).

The following is the Algorithm 1.

Algorithm 1 PD Algorithm

Input: $H_r,r\ge 2$, with radix-lexicographic ordering. Algorithm: $\left(i\right)$ Select $S_2=\{01,13,20,32\}$ in $H_2$ and let $S_3={\displaystyle \bigcup _{k=0}^7}k{S}_2$ in $H_3$. $\left(ii\right)$ Inductively select $S_r={\displaystyle \bigcup _{k=0}^7}k{S}_{r-1}$ in $H_r$. Output: ${\gamma }_p\left(H_r\right)\le 2^{3r-4}$, $r\ge 2$.

Proof of Correctness.

Let $S_r$ be a PDS of $H_r$. Consider $S_r={\displaystyle \bigcup _{k=0}^7}{k}^{r-2}{S}^{{}^{\prime }}$ where $S^{{}^{\prime }}=\{01,13,20,32,00,10,03,05,02,21,22,24,12,17,11,31,33,30,36,23\},k^{r-2}=kk\dots k(r-2)$ times. Then, node in $M^0\left(S_r\right)$ say, ${\displaystyle \bigcup _{k=0}^7}{k}^{r-2}{S}^{{}^{″}}$ where $S^{{}^{″}}=\{00,02,03,10,11,12,21,22,23,30,31,33\}$ is neighbor to exactly one unmonitored node say, ${\displaystyle \bigcup _{k=0}^7}{k}^{r-2}{S}^{{}^{‴}}$ where $S^{{}^{‴}}=\{04,06,07,14,15,16,25,26,27,34,35,37\}$. Now $M^1\left(S_r\right)=M^0\left(S_r\right)\cup$${\displaystyle \bigcup _{k=0}^7}{k}^{r-2}{S}^{{}^{‴}}$. Then, for every node $v\in M^1\left(S_r\right),|N\left[v\right]\backslash M^1\left(S_r\right)|=1$. Now $M^2\left(S_r\right)=M^1\left(S_r\right)\cup$ ${\displaystyle \bigcup _{k=0}^7}{k}^{r-2}ij,0\le i\le 3,4\le j\le 7$. Similarly, $M^3\left(S_r\right)=M^2\left(S_r\right)\cup {\displaystyle \bigcup _{k=0}^7}{k}^{r-2}ij,4\le i,j\le 7$. Thus, $M^3\left(S_r\right)=V\left(H_r\right)$. Inductively, we arrive at ${\gamma }_p\left(H_r\right)\le 2^{3r-4}$.    □

PD Algorithm (Algorithm 1) together with Lemma 3 imply the following theorem.

Theorem 1.

The power domination number of $H_r.r\ge 2$, is $2^{3r-4}$. In otherwords ${\gamma }_p\left(H_r\right)=2^{3r-4}$.

3.2. Zero Forcing in $FCCNs$

The PD process on a graph G is choosing a set $S\subseteq V\left(G\right)$ and applying the ZF process to the closed neighborhood $N\left[S\right]$ of S. The set S is a PDS of G if and only if $N\left[S\right]$ is a ZFS for G.

The following theorem was proved in 2015 by Ferrero et al. [3], which shows the relationship between ZFS and PDS.

Theorem 2

([3]). Let G be a graph with no isolated vertices, and let $S=\{u_1,u_2,\dots ,u_t\}$ be a PDS for G. Then $\zeta \left(G\right)\le {\sum }_{i=1}^{t}deg\left(u_i\right)$.

Theorem 3

([3]). Let G be a graph. Then, $⌈\frac{\zeta \left(G\right)}{\Delta \left(G\right)}⌉\le {\gamma }_p\left(G\right)$ and this bound is tight.

In what follows, we obtain a sharp lower bound for the zero forcing number of $FCCNs$.

Lemma 4.

The zeoro forcing number of $H_2$ is at least 16. In other words, $\zeta \left(H_2\right)\ge 16$.

Proof. 

$H_2$ contains eight node disjoint copies of $H_1$, say $0H_1,1H_1,\dots ,7H_1$. Let S be a ZFS of G. We claim that $\left|S\right|\ge 16$. Suppose $S\subseteq 0H_1\cup 1H_1$. Then, S can monitor at most two nodes of $i{H}_1,2\le i\le 7$, each of which is neighbor to at least 2 unmonitored nodes, a contradiction. Therefore, $S⊈0H_1\cup 1H_1$ (see Figure 2b).

Suppose $S\subseteq 0H_1\cup 1H_1\cup 2H_1$. Then, S can monitor at most three nodes of each $i{H}_1,3\le i\le 7$. No three nodes in $i{H}_1$ induce an independent set. Hence, these three nodes induce an edge and an isolated node. Each end node of the edge is neighbor to 2 unmonitored nodes and the independent vertex is neighbor to three unmonitored nodes, a contradiction (see Figure 2c). On the other hand, if three nodes induce a path, say $uvw$, then u and w are neighbor to 2 unmonitored nodes each and v is neighbor to exactly one node, which in turn is neighbor to at least 2 unmonitored nodes, a contradiction (see Figure 2d). Therefore, $S⊈0H_1\cup 1H_1\cup 2H_1$.

Suppose $S\subseteq 0H_1\cup A\cup B$, where $A=\{11,13,15,17\}$ and $B=\{31,32,33,36\}$. A induces four cycles and B induces two independent edges. Even if all nodes of $0H_1$ and $3H_1$ are monitored, S can color at most two independent nodes in $2H_1,4H_1,5H_1,6H_1,7H_1$ as red, and each node labeled as $10,12,16$ in $1H_1$ is neighbor to 2 unmonitored nodes, a contradiction. Therefore, $S⊈0H_1\cup A\cup B$ (see Figure 4a).

Figure 4. Circled vertices indicates in (a,b) be a zero forcing sets of $H_2$.

Suppose $S\subseteq A\cup B\cup C\cup D$, where $A=\{00,01,02,03\},B=\{10,11,12,13\},C=\{20,21,23,24\},D=\{34,35,36,37\}$. $A,B,C,D$ induces a four cycle. Then, S can induce a path say, $uvw$ in $i{H}_1,i=0,1,2,4$. Then, the end nodes of a path say, u and w are neighbor to 2 unmonitored nodes each and v is neighbor to exactly one unmonitored node which in turn is neighbor to at least 2 unmonitored nodes, a contradiction. This implies that, $S⊈A\cup B\cup C\cup D$. Therefore, $\left|S\right|\ge 16$ (see Figure 4b). Hence, nodes in S are in consecutive 4 cycle of at least 4 consecutive copies of $H_1$.    □

Lemma 5.

Let S be a zero forcing set of $H_3$. Let H be a subgraph of $H_3$ isomorphic to $H_2$. Then, $\left|V\right(H)\cap S|\ge 16$.

Proof. 

$H_3$ is composed of eight copies of $H_2$, denoted by $0H,1H,\dots ,7H$. Taking into account the symmetric nature of $H_3$, without loss of generality consider $0H$. Let $0H={\displaystyle \bigcup _{k=0}^7}k{Q}^3$. The worst case arises when all the 7 intercubic nodes $0ii,1\le i\le 7$, of $0H$ are monitored nodes of S not in $0H$. Suppose $\left|V\right(0H)\cap S|=15$.

Let $V\left(0H\right)\cap S=A\cup B\cup C\cup D\cup E\cup F\cup G\cup I$, where $A=\{000,001\},B=\{002,006\},C=\{003,007\},D=\{010,011\},E=\{013,017\},F=\{021,023\},G=\{031,033\},H=\left\{032\right\}$, each $A,B,C,D,E,F,G,I$ induces an edge and an independent set. Even if all nodes of $0H_1,1H_1$ and $3H_1$ are monitored and S induces a path, say $uvw$ in $i{H}_1,4\le i\le 7$ or $2H_1$, then u and w are neighbor to 2 unmonitored nodes each and v is neighbor to exactly one unmonitored node, which in turn is neighbor to at least 2 unmonitored nodes, a contradiction. Therefore, $\left|S\right|\ge 16$ (see Figure 5a).

Figure 5. (a–c) $H_3$ of $0H_2$ (d) Zero forcing sets of $H_2$.

Let $V\left(0H\right)\cap S=A\cup B\cup C\cup D\cup E$, where $A=\{000,001,002\},B=\{010,012,016\},C=\{011,013,017\},D=\{020,022,026\},E=\{021,023,027\}$, each $A,B,C,D,E$ induce a path say, $uvw$. Even if all nodes of $1H_1$ and $2H_1$ are monitored and suppose S induces an independent set in $i{H}_1,3\le i\le 7$. Then, the independent node is neighbor to 3 unmonitored nodes, a contradiction. Therefore, $\left|S\right|\ge 16$ (see Figure 5b).

Let $V\left(0H\right)\cap S=A\cup B\cup C\cup D$, where $A=\{000,001,002,003\},B=\{010,011,012,013\},C=\{020,021,022,023\},D=\{030,032,033\}$. Even if, all nodes of $0H_1$ and $1H_1$ are monitored and S induces an edge in $i{H}_1,2\le i\le 7$. Then, the node labeled as 23 in $2H_1\in S$, and end nodes of a path say, $uvw$ in $3H_1\in S$ and each end node of an edge in $i{H}_1,4\le i\le 7$ is neighbor to at least 2 unmonitored nodes, a contradiction. Therefore, $\left|S\right|\ge 16$ (see Figure 5c).    □

Lemma 6.

The zero forcing number of $H_r,r\ge 3$ is at least $2^{3r-2}$. In other words, $\zeta \left(H_r\right)\ge 2^{3r-2}$.

Proof. 

We prove the result by induction on r. We consider the case when $r=3$. $H_3$ contains eight node disjoint copies of $H_2$, say $0H_2,1H_2,\dots ,7H_2$. Let S be a ZFS of $H_r$. We claim that $\left|S\right|\ge 128$. By Lemmas 4 and 5, there are at least 16 nodes in each copy of $H_2$ to monitor all nodes of $H_3$. Hence, $\left|S\right|\ge 16\times 8$. Therefore, $\zeta \left(H_3\right)\ge 128$.

Assume the result is true for $r=k,r\ge 3$. That is, $\zeta \left(H_k\right)\ge 2^{3k-2}$. Consider the case when $r=k+1$. Let S be a ZFS of $H_{k+1}$. We have to prove that $\zeta \left(H_{k+1}\right)\ge 2^{3k+1}$. Suppose not, let $\left|S\right|<2^{3k+1}$. In $H_{k+1}$, there are $8^{k-1}$ node disjoint copies of $H_2$. With the deletion of one node from S in a copy of $H_2$, there is at least one node, say $u\in H_1$ monitored by intercubic edges. With this monitored node u, we claim that 15 nodes in $H_2$, four each in 3 copies of $H_1$ say $1H_1,2H_1,3H_1$, and 3 in $4H_1$ are not sufficient to monitor all nodes in any of $i{H}_1,i=1,2,3,4$. In the worst case, suppose all the nodes in the copy of $H_1$ containing u are all already monitored, then the monitored node in $1H_1$ has 2 unmonitored nodes adjacent to it, a contradiction. Thus, $\left|S\right|\ge 2^{3k+1}$. Therefore, $\zeta \left(H_k\right)\ge 2^{3k-2}$.    □

The following is the Algorithm 2.

Algorithm 2 ZF Algorithm

Input: $H_r,r\ge 2$, with radix-lexicographic ordering. Algorithm: $\left(i\right)$ Select $S_2=\{i0,i1,i2,i3\},0\le i\le 3$ in $H_2$ and let $S_3={\displaystyle \bigcup _{k=0}^7}k{S}_2$ in $H_3$. See Figure 5d. $\left(ii\right)$ Inductively select $S_r={\displaystyle \bigcup _{k=0}^7}k{S}_{r-1}$ in $H_r$. Output: $\zeta \left(G\right)\le 2^{3r-2}$.

Proof of Correctness.

Let $S_r$ be a ZF of $H_r$. Let $S_r={\displaystyle \bigcup _{k=0}^7}{k}^{r-2}{S}^{{}^{\prime }}$, where $S^{{}^{\prime }}=\{i0,i1,i2,i3\},0\le i\le 3,k^{r-2}=kk\dots k(r-2)$ times. Every node $v\in S_r$ is neighbor to exactly one unmonitored node which in turn forces exactly one unmonitored node say, $N\left(S_r\right)={\displaystyle \bigcup _{k=0}^7}{k}^{r-2}{S}^{{}^{″}}$ where $S^{{}^{″}}=\{i4,i5,i6,i7\},0\le i\le 3$. Then, every node $v\in N\left(S_r\right)$ is neighbor to exactly one unmonitored node, which in turn forces exactly one unmonitored node say, $A={\displaystyle \bigcup _{k=0}^7}{k}^{r-2}{S}^{{}^{‴}}$ where $S^{{}^{‴}}=\{i0,i1,i2,i3\},4\le i\le 7$. In the next step, each node in A is neighbor to exactly one green node, which in turn forces exactly one green node, say $B={\displaystyle \bigcup _{k=0}^7}{k}^{r-2}{S}^{{}^{⁗}}$ where $S^{{}^{⁗}}=\{i4,i5,i6,i7\},4\le i\le 7$. Now, $S_r={\displaystyle \bigcup _{k=0}^7}{k}^{r-2}{S}^{{}^{\prime }}$, where $S^{{}^{\prime }}=\{i0,i1,i2,i3\},0\le i\le 3,k^{r-2}=kk\dots k(r-2)$ times is a ZFS of $H_r,r\ge 2$. This implies that $\zeta \left(H_r\right)\le 2^{3r-2}$. □

ZF Algorithm (Algorithm 2) together with Lemma 6 imply the following theorem.

Theorem 4.

The zero forcing number of $H_r.r\ge 2$, is $2^{3r-2}$. In other words, $\zeta \left(H_r\right)=2^{3r-2}$.

4. Conclusions

In this paper, we have obtained the PDN and ZFN for the fully connected cubic networks $H_r,r\ge 2$, identifying classes of graphs for which $⌈\frac{\zeta \left(G\right)}{\Delta \left(G\right)}⌉={\gamma }_p\left(G\right)$ is an open problem. Another interesting line of research would be to determine the zero forcing number of networks such as hypercubes and circulant networks.