Forcing Parameters in Fully Connected Cubic Networks

: Domination in graphs has been extensively studied and adopted in many real life applica-tions. 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. validation, S.K., J.A. and I.R.; formal analysis, H.R. and I.R.; investigation, Y.R., J.A., I.R. and H.R.; data curation, Y.R., S.K., J.A., I.R. and H.R.; writing—original draft preparation, Y.R., S.K., J.A., I.R. and H.R.; writing—review and editing, J.A., I.R. and H.R.; visualization, Y.R., S.K. and H.R.;


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(G) together with a set of edges E = E(G) 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 ∈ V, the set of all vertices adjacent to u are said to be in the neighbors of u and is denoted by N(u). Then, the closed neighborhood of u is defined as N[u] = N(u) ∪ {u}.
A subset S ⊆ 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 ⊆ V. Vertices M i (K) monitored by K at level i, i ≥ 0, inductively are as follows: 1.
M 0 (K) = N[K]. 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 γ p (G) [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 ζ(G) [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.

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 . . . a(n times). FCCN of level r, r ≥ 1 denoted by FCCN r , is defined recursively as follows [15]:

2.
When r ≥ 2, FCCN r is built from eight node-disjoint copies of FCCN r−1 by adding 28 edges. Specifically, if, for 0 ≤ k ≤ 7, we let kFCCN r−1 denote a copy of FCCN r−1 with each node being prefixed with k, then FCCN r is defined by: For 0 ≤ k ≤ 7, kFCCN r−1 is named an (r − 1)-level sub-FCCN of FCCN r , or simply a sub-FCCN of FCCN r , if there is no uncertainty.

3.
Given an FCCN r , r ≥ 2, a boundary node is a node of the form k r . An ICE is an edge of the form (pq n−1 , qp n−1 ). Each ICV of FCCN is of degree 4 except the boundary nodes of degree 3. Obviously, kFCCN r−1 has seven ICV and one boundary node for 0 ≤ k ≤ 7 and r ≥ 2 [15] (see Figure 1).

Note:
In what follows, we denote FCCN r by H r r ≥ 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 ≤ i ≤ r, each level i contains 8 i−k node disjoint copies of H j , 1 ≤ j ≤ i. This implies that there are 8 r−2 node disjoint copies of H 2 in H r .

Main Results
In this section, we compute PDN and ZFN for H r , r ≥ 2.

Power Domination in FCCNs
We obtain lower bounds for the PDN of H r , r ≥ 2, and prove that the bounds are sharp. Proof. H 2 contains eight node disjoint copies of H 1 , say 0H 1 , 1H 1 , . . . , 7H 1 . Let S be a PDS of G. We claim that |S| ≥ 4. Suppose S ⊆ 0H 1 , any node in iH 1 , 1 ≤ i ≤ 7 is neighbor to at most one node of 0H 1 . This implies that every node in iH 1 , 1 ≤ i ≤ 7 is neighbor to at least three unmonitored nodes, a contradiction. Therefore, S ⊆ 0H 1 (see Figure 2a). Suppose S ⊆ 0H 1 ∪ 1H 1 . Then, S can monitor at most two nodes of iH 1 , 2 ≤ i ≤ 7, each of which is neighbor to at least two unmonitored nodes, a contradiction. Therefore, S ⊆ 0H 1 ∪ 1H 1 (see Figure 2b).
Suppose S ⊆ 0H 1 ∪ 1H 1 ∪ 2H 1 . Then, S can monitor at most three nodes of each iH 1 , 3 ≤ i ≤ 7. No three nodes in iH 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 ∪ 1H 1 ∪ 2H 1 . Hence, nodes in S are in at least 4 copies of H 1 . Therefore, |S| ≥ 4.
Consider the case when r = k + 1. Let S be a PDS H k+1 . We have to prove that γ p (H k+1 ) ≥ 2 3k−1 . Suppose not, let |S| < 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 ∈ 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 i 1 , 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, |S| ≥ 2 3k−1 . Therefore, γ p (H k ) ≥ 2 3k−4 .
The following is the Algorithm 1.

Proof of Correctness. Let
PD Algorithm (Algorithm 1) together with Lemma 3 imply the following theorem.

Zero Forcing in FCCNs
The PD process on a graph G is choosing a set S ⊆ V(G) and applying the ZF process to the closed neighborhood N[S] of S. The set S is a PDS of G if and only if N[S] 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 , . . . , u t } be a PDS for G. Then In what follows, we obtain a sharp lower bound for the zero forcing number of FCCNs. Proof. H 2 contains eight node disjoint copies of H 1 , say 0H 1 , 1H 1 , . . . , 7H 1 . Let S be a ZFS of G. We claim that |S| ≥ 16. Suppose S ⊆ 0H 1 ∪ 1H 1 . Then, S can monitor at most two nodes of iH 1 , 2 ≤ i ≤ 7, each of which is neighbor to at least 2 unmonitored nodes, a contradiction. Therefore, S ⊆ 0H 1 ∪ 1H 1 (see Figure 2b). Suppose S ⊆ 0H 1 ∪ 1H 1 ∪ 2H 1 . Then, S can monitor at most three nodes of each iH 1 , 3 ≤ i ≤ 7. No three nodes in iH 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). 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 ∪ A ∪ B (see Figure 4a). Then, S can induce a path say, uvw in iH 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 ∪ B ∪ C ∪ D. Therefore, |S| ≥ 16 (see Figure 4b). Hence, nodes in S are in consecutive 4 cycle of at least 4 consecutive copies of H 1 . Even if all nodes of 0H 1 , 1H 1 and 3H 1 are monitored and S induces a path, say uvw in iH 1 , 4 ≤ i ≤ 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, |S| ≥ 16 (see Figure 5a). Even if all nodes of 1H 1 and 2H 1 are monitored and suppose S induces an independent set in iH 1 , 3 ≤ i ≤ 7. Then, the independent node is neighbor to 3 unmonitored nodes, a contradiction. Therefore, |S| ≥ 16 (see Figure 5b). Even if, all nodes of 0H 1 and 1H 1 are monitored and S induces an edge in iH 1 , 2 ≤ i ≤ 7. Then, the node labeled as 23 in 2H 1 ∈ S, and end nodes of a path say, uvw in 3H 1 ∈ S and each end node of an edge in iH 1 , 4 ≤ i ≤ 7 is neighbor to at least 2 unmonitored nodes, a contradiction. Therefore, |S| ≥ 16 (see Figure 5c).
Consider the case when r = k + 1. Let S be a ZFS of H k+1 . We have to prove that ζ(H k+1 ) ≥ 2 3k+1 . Suppose not, let |S| < 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 ∈ 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 iH 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, |S| ≥ 2 3k+1 . Therefore, ζ(H k ) ≥ 2 3k−2 .
The following is the Algorithm 2.

Conclusions
In this paper, we have obtained the PDN and ZFN for the fully connected cubic networks H r , r ≥ 2, identifying classes of graphs for which ζ(G) ∆(G) = γ p (G) 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.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: