Certain Domination Parameters and Their Resolving Versions of Fractal Cubic Networks

Abstract

Networks are designed to communicate, operate, and allocate tasks to respective commodities. Operating supercomputers became challenging, which was handled by the network design commonly known as hypercube, denoted by $Q^n$. In a recent study, the hypercube networks were insufficient to hold supercomputers’ parallel processors. Thus, variants of hypercubes were discovered to produce an alternative to the hypercube. A new variant of the hypercube, the fractal cubic network, can be used as the best alternative in the case of hypercubes. Our research investigates that the fractal cubic network is a rooted product of two graphs. We try to determine its domination and resolving domination parameters, which could be applied to resource location and broadcasting-related problems.

1. Introduction and Motivation

Graph frameworks, consisting of a network of connections, are extensively employed in various dynamic, circuit-related, genetic, and chemical systems. They aid in modeling the transmitters of brain systems. The structure of a graph consists of vertices and edges. Each vertex describes a node in the network, and each edge represents a link between nodes. The interconnection network is a sophisticated linkage between the array of processors and the communication pathways connecting any two distinct processors. The network is employed to exchange data across processors in parallel network computation. Network dependability is the most critical component in designing a network’s geometry. The interconnection network is an essential subsystem for high-performance computing systems and data centers [1]. Consequently, contemporary suggestions for the interconnection network must ensure minimal latency overhead and maximal transmission bandwidth. Occasionally, it is impractical to execute and evaluate new designs physically; they must be examined and validated using data-driven software tools, such as network simulators for connectivity information graphs. Interconnection networks have numerous primary and intrinsic uses in system-designed architectures, although they are predominantly employed in parallel computing architecture.

Interconnection networks with multiprocessors are often crucial for connecting many reliably replicated processors. Message passing is predominantly employed in place of shared memory to provide comprehensive transmission and synchronization across processors for planned execution. The graph $\Gamma$ can be shown such that every pair of vertices is directly linked via transmission links. The metrics employed to assess the efficacy of the structure include bisection width, broadcasting duration, degree, diameter, and fault tolerance [1]. Let $\Gamma$ be a connected graph with $V(\Gamma )$ (vertex set) and $E(\Gamma )$ (edge set), respectively. For any $s\in V(\Gamma )$, $N_{\Gamma }\left(s\right)=\{t\in V(\Gamma )|st\in E(\Gamma )\}$ and $N_{\Gamma }\left[s\right]=N_{\Gamma }\left(s\right)\cup \left\{s\right\}$ as open and closed neighborhoods of s, respectively. If $\Gamma$ is understood we denote $N_{\Gamma }\left(s\right)\left(N_{\Gamma }\left[s\right]\right)$ as $N\left(s\right)\left(N\right[s\left]\right)$ respectively. The degree of s is defined by $d_{\Gamma }\left(s\right)=\left|N_{\Gamma }\left(s\right)\right|$. $\Delta$ and $\delta$ symbolizes the greatest degree and the least degree of $\Gamma$ respectively. For a k-regular graph, all the vertices have degree k. For a subset D of $V(\Gamma )$, $\Gamma \left[D\right]$ is the induced subgraph of $\Gamma$. For a graph $\Gamma$ with n vertices, the order is computed as n, which could be referred to as $\left|V\right(\Gamma \left)\right|=n$. An isolated vertex $v\in V(\Gamma )$ is a vertex with $d\left(v\right)=0$. $\Gamma -v$ is a graph obtained from $\Gamma$ by deleting a vertex $v\in V(\Gamma )$ and deleting all the edges incident to the vertex v. Denote ${\mathbb{N}}_n:=\{1,2,\dots ,n\}$. Two vertices e and f are said to be false twins if $N\left(e\right)=N\left(f\right)$ and true twins if $N\left[e\right]=N\left[f\right]$. Two vertices in the connected graph $\Gamma$ are said to be twins if they are either true or false twins. A set $T\subseteq V(\Gamma )$ is said to be an open (closed) twin set if every pair of vertices in T are false (true) twins in $\Gamma$.

Operation on graphs and design of new networks could be used as alternatives to multistage interconnection networks. Several operations on graphs deal with merging the vertex and adding additional edges. A few operations on graphs are listed here: Cartesian product, strong product, corona product, lexicographic product, tensor product, and rooted product. Here, we concentrate on the rooted product operation, which is defined as follows. A graph is usually called rooted if one of its nodes is designated as a root to set it apart from the other nodes. Let ${\Omega }_i$, $i\in {\mathbb{N}}_n$ be the n copies of $\Omega$, where $\Omega$ is any finite graph of order $n_1>0$, and let $\Gamma$ be an n ordered graph. The graph $\Gamma {\circ }_v\Omega$ is generated by assigning a unique vertex v to each ${\Omega }_i$ on the $i^{\mathrm{th}}$ node of $\Gamma$, and the graph obtained is the rooted product of $\Gamma$ and $\Omega$, respectively.

In Section 2, we discuss a new variant of hypercube known as fractal cubic network with specific definitions on domination and resolving parameters through illustrations, and certain existing theorems to be used in Section 3 are given. Section 3 is devoted to our findings as main results, and Section 4 concludes with our research findings with applications and future directions.

2. Fractal Cube: A Fascinating Variant of Hypercube

The examination of self-similarity and fractality in discrete systems, especially complex networks, has intensified. This increase in interest is driven by theoretical advancements in complex network theory and the practical requirements of real-world applications. Translating the ideas of fractal geometry from general topology, which addresses continuous or infinite objects, to finite structures in a mathematically rigorous manner presents a significant difficulty. The investigation of fractals enhances our comprehension of the intrinsic beauty and intricacy of the natural world while also possessing extensive applicability across multiple scientific fields, including biology, physical sciences, computer networks, and chemical graph theory. Sierpiński-type structures have been thoroughly investigated in fractal theory and application, with substantial research illustrating its significance and utility across various domains [2]. Additional recent work of fractal networks can be found in [3,4,5,6,7,8].

The cube ($Q^n$) is a prevalent architecture characterized by its regularity, transit efficiency, recursive configuration, symmetry, and elevated connectivity. See Figure 1 for the hypercubes of dimensions 2, 3, and 4. In recent years, hypercubes have been extensively studied for their diverse features [9]. In Intel’s hypercube, the new node functions as the cube manager, with direct links to all processors within the system, which is analogous to a physical machine. In this case, the bisection width of this architecture is $2^{n-1}$. In parallel architecture, hypercubes fail to have some properties. For example, they have a high bisection width and non-constant node degree. The literature presents numerous variations, as shown in

Table 1. The literature on variants of hypercubes.

S.No	Variants of Hypercubes	References
1	Exchanged hypercubes	[10]
2	Folded hypercubes	[11,12]
3	Crossed cubes	[13]
4	Exchanged crossed cubes	[14,15]
5	Twisted cubes	[16,17]
6	Möbius cubes	[18]
7	Spined cubes	[19]
8	Locally twisted cubes	[20]
9	Shuffle cubes	[21]
10	Augmented cubes	[22]
11	Hierarchical cubic networks	[23,24]
12	Folded hierarchical cubic networks	[25]

.

Although the hypercube versions listed above have been the subject of several research studies, none of them, except the fractal cubic network, have examined their problem and its resolving number. Several variations on hypercubes have been proposed in the literature, but unfortunately, none have a lower bisection width and a constant node degree. Fractal cubic network (FCN) is an ideal architecture for its constant node degree and low bisection width. While the concept of this structure is ambiguous in [26], it was rectified in [27]. Motivated by this, we recently examined power domination and resolving power domination [28] for this recently introduced hypercube variation fractal cubic network.

The authors of [27] characterize $FCN\left(0\right)=\left(V\right(0),E(0\left)\right)$ where $V\left(0\right)=\{00,01,10,11\}$ and $E\left(0\right)=\left\{\right\{00,01\},\{01,11\},\{11,10\},\{10,00\left\}\right\}$. For $l\ge 1$, define $FCN\left(l\right)$ as follows:

An l-dimensional FCN is defined as $FCN\left(l\right)=\left(V\right(l),E(l\left)\right)$, $l>0$, and can be designed as follows

$$FCN\left(l\right)=11\parallel FCN(l-1)\cup 01\parallel FCN(l-1)\cup 10\parallel FCN(l-1)\cup 00\parallel FCN(l-1),$$

where

$$V\left(l\right)=11\parallel V(l-1)\cup 01\parallel V(l-1)\cup 10\parallel V(l-1)\cup 00\parallel V(l-1)$$

and

$$\begin{array}{ccc}E\left(l\right)\hfill & =& 11\parallel E(l-1)\cup 01\parallel E(l-1)\cup 10\parallel E(l-1)\cup 00\parallel E(l-1)\hfill \\ & & \cup \left\{\right(00100101\dots 01,10100101\dots 01),(10100101\dots 01,11100101\dots 01\left)\right\}\hfill \\ & & \cup \left\{\right(11100101\dots 01,01100101\dots 01),(01100101\dots 01,00100101\dots 01\left)\right\}.\hfill \end{array}$$

Figure 2 denotes $FCN$ dimensions of 0, 1, and 2 separately, where $\parallel$ denotes the concatenation operator, and each string is of length $2l+2$ for the dimension of l.

Interestingly, the hypercube is operated by the Cartesian product of $K_2$ and its lower dimensional hypercube. That is, $Q^n=K_2\times Q^{n-1}$. The new variant of hypercube FCN is constructed by a rooted product of $C_4$ and its lower dimension of FCN. Thus, FCN can be represented as $FCN\left(l\right)=C_4{\circ }_{v}FCN(l-1),v\in \{0010{\left(01\right)}^{l-1},0110{\left(01\right)}^{l-1},1010{\left(01\right)}^{l-1},1110{\left(01\right)}^{l-1}\}$. Godsil and McKay [29] coined the rooted product operation of two graphs, and further, the readers could refer to [30,31,32,33] for detailed studies on domination variants in the rooted product of graphs.

Resolving sets provide a mechanism for identifying the origin of diffusion in a network. Identifying the source of a disease disseminated through a community is beneficial in numerous contexts. Although the resolving set (RS) provides a solution when inter-node intervals and starting spread time are established, resolvability must be broadened to include random start timings and random nodal communication delays.

For $R=\{m_1,m_2,\dots ,m_t\}\subseteq V(\Gamma )$, the code of $j\in V(\Gamma )$ with respect to R is defined as the t-vector

$$C_R\left(j\right)=\left(d(j,m_1),d(j,m_2),\dots ,d(j,m_t)\right),$$

where $d(j,k)$ denotes the distance between j and k. A set R is an RS for $\Gamma$ if and only if $C_R\left(i\right)\ne C_R\left(j\right)$ for any pair $\{i,j\}\subset V(\Gamma )$. Among all potential RS for $\Gamma$, the ones with the smallest size are of particular interest, referred to as a basis. The size of the smallest RS is referred to as the metric dimension of $\Gamma$, indicated by $dim(\Gamma )$. The problem of finding resolving sets remains NP-complete for general graphs [34].

A dominating subset D of $V(\Gamma )$ is a set in which every member of $V(\Gamma )$ is in D or adjacent to at least one vertex of D. The smallest size of D is known as the domination number and is represented as $\gamma$ [35]. By imposing constraints on a D, several domination parameters are laid. Some of them are independent, total, connected, double, and 2-domination. A dominating set (DS) is an independent dominating set (IDS) if $\Gamma \left[D\right]$ is a null graph. We say DS is a total dominating set (TDS) if each member in $V(\Gamma )$ is connected to some member belonging to D and a connected dominating set (CDS) if $\Gamma \left[D\right]$ is connected. If $\left|N\right[v]\cap D|\ge 2$ for any $v\in V(\Gamma )$, then D is a double dominating set (DDS) of $\Gamma$ and if $\left|N\right(v)\cap D|\ge 2$ for $v\in V(\Gamma )\setminus D$, then D is a 2-dominating set (2DS) of $\Gamma$. The smallest size of these sets were, respectively, denoted as ${\gamma }_i$, ${\gamma }_t$, ${\gamma }_c$, ${\gamma }_{\times 2}$, and ${\gamma }_2$. Let $D_2$-set and $D_{\times 2}$-set be the collections of all 2DS and DDS of $\Gamma$, respectively. Let $U,V\subseteq V(\Gamma )$, where $(U,V)$ is an ordered pair of disjoint sets U, V, and also a quasi-double dominating pair of $\Gamma$ if $U\cup V\in D_2$-set and $V\in D_{\times 2}$-set of $\Gamma -U$. Then, the quasi-double domination number denoted by ${\gamma }_{q\times 2}$ is $min\left\{\right|U|+|V|:U\cup V\in D_2$-set and $V\in D_{\times 2}$-set of $(\Gamma -U)\}$. Domination remains NP-complete and has application in numerous areas like communication systems [36], resource location problems [37,38], social networks [39], and models of biological networks [40,41,42]. Determining the domination number of an r-dimensional hypercube ($Q^r$) is fundamental in coding, graph theory, and circuit-related sciences.

In the graph theory literature, various parameters on resolving sets and dominating sets were identified and extensively studied. They are applied to parallel computing architectures and neural networks for solving resource location, image processing, and chemical-related problems. Specific variants of resolving sets are the path resolving set, connected resolving set, independent resolving set, one-factor resolving set, and one-size resolving set.

In the technological era, combining the concepts and logic creates new variations for solving higher-level problems, and one such idea is the resolving domination (RD) in which the set $D\subseteq V(\Gamma )$ is both a resolving and dominating set. The resolving domination number denoted by ${\gamma }_r$ is the smallest size of resolving dominating set [43,44]. Specific variants of resolving domination are discussed, including resolving independent domination, resolving total domination and resolving connected domination. The smallest cardinalities of these sets was represented as ${\gamma }_{ri},{\gamma }_{rt}$, and ${\gamma }_{rc}$, respectively. We abbreviate the resolving dominating set as RDS, resolving independent dominating set as RIDS, resolving total dominating set as RTDS, and resolving connected dominating set as RCDS. Since domination, and resolving set problems remain NP-complete, resolving domination problems is also a class of NP-complete problems for general graphs. Brigham et al., in 2003, introduced the concept of RD and provided the lower and upper bounds, RD number of standard graphs, relationship with the diameter, order, and clique number and characterized graphs with ${\gamma }_r=n-1$ [44]. Monsanto and Rara established RD number of certain graphs under some binary operations [45]. In 2015, resolving connected domination was introduced by Naji and Soner and proved primary results on resolving connected domination [46]. For further resolving and domination-related problems, the readers could refer to [28,43,47,48,49,50]. The illustrations to the definitions of RS, DS, IDS, TDS, CDS, DDS, 2DS, RDS, RIDS, RTDS, and RCDS are depicted in Figure 3.

In Figure 3, the graph $\Gamma$ exhibits the following sets: RS is $\{v_3,v_{12}\}$, DS, IDS, RDS, RIDS is $\{v_1,v_4,v_8,v_{13}\}$, TDS is $\{v_5,v_6,v_8,v_9,v_{13}\}$, CDS is $\{v_5,v_6,v_8,v_9,v_{10}\}$, DDS is $\{v_1,v_4,v_5,v_6,v_8,v_{11},v_{12},v_{13},v_{14}\}$, 2DS is $\{v_1,v_5,v_6,v_7,v_{11},v_{12},v_{14}\}$, RTDS is $\{v_1,v_5,v_6,v_8,v_9,v_{13}\}$, and RCDS is $\{v_1,v_5,v_6,v_8,v_9,v_{10}\}$. A few theorems from the literature related to resolving and dominating sets are listed below, which will be used to prove theorems in Section 3.

Theorem 1

([47]). Let Γ be a connected graph with twin sets $T_k$, $1\le k\le p$, then $dim(\Gamma )\ge {\sum }_{i=1}^p\left|T_k\right|-p$.

Theorem 2

([44]). For an isolate-free graph Γ, $max\{\gamma (\Gamma ),dim(\Gamma )\}\le {\gamma }_r(\Gamma )\le \gamma (\Gamma )+dim(\Gamma )$.

Theorem 3

([27]). For $l>0$, $dim\left(FCN\left(l\right)\right)=4^l$.

Theorem 4

([30]). Let Γ be an isolate-free graph of order $n\ge 2$. Then, for any graph Ω with root v and $\left|V\right(\Omega \left)\right|\ge 2$, $\gamma (\Gamma {\circ }_v\Omega )\in \{n\gamma (\Omega ),n\gamma (\Omega )-n+\gamma (\Gamma )\}$.

Theorem 5

([30]). Let Γ be an isolate-free graph of order $n\ge 2$. Then, for any graph Ω with root v and $\left|V\right(\Omega \left)\right|\ge 2$, ${\gamma }_i(\Gamma {\circ }_v\Omega )\in \{n{\gamma }_i(\Omega ),n{\gamma }_i(\Omega )-n+{\gamma }_i(\Gamma )\}$.

Theorem 6

([31]). Let Γ be an isolate-free graph of order $n\ge 2$ and Ω be any graph with $\left|V\right(\Omega \left)\right|\ge 2$. For any $v\in V(\Omega )$, ${\gamma }_t(\Gamma {\circ }_v\Omega )\in \{n{\gamma }_t(\Omega )-n,\gamma (\Gamma )+n{\gamma }_t(\Omega )-n,{\gamma }_t(\Gamma )+n{\gamma }_t(\Omega )-n,n{\gamma }_t(\Omega )\}$.

Theorem 7

([30]). Let Γ be an isolate-free graph of order $n\ge 2$. Then, for any graph Ω with root v and $\left|V\right(\Omega \left)\right|\ge 2$, ${\gamma }_c(\Gamma {\circ }_v\Omega )\in \{n{\gamma }_c(\Omega ),n{\gamma }_c(\Omega )+n\}$.

Theorem 8

([32]). Let Γ be an isolate-free graph of order $n\ge 2$ and Ω be any graph with $\left|V\right(\Omega \left)\right|\ge 2$. If $v\in V(\Omega )$, then ${\gamma }_{\times 2}(\Gamma {\circ }_v\Omega )\in \{n{\gamma }_{\times 2}(\Omega ),{\gamma }_{q\times 2}(\Gamma )+n{\gamma }_{\times 2}(\Omega )-n,{\gamma }_2(\Gamma )+n{\gamma }_{\times 2}(\Omega )-n,\gamma (\Gamma )+n{\gamma }_{\times 2}(\Omega )-n,n{\gamma }_{\times 2}(\Omega )-n,{\gamma }_{\times 2}(\Gamma )+n{\gamma }_{\times 2}(\Omega )-2n\}$.

Theorem 9

([32]). Let Γ be an isolate-free graph of order $n\ge 2$, then ${\gamma }_{\times 2}(\Gamma {\circ }_v{C}_4)=n{\gamma }_{\times 2}\left(C_4\right)$.

Theorem 10

([33]). Let Γ be an isolate-free graph of order $n\ge 2$ and Ω be any graph with $\left|V\right(\Omega \left)\right|\ge 2$. For any vertex $v\in V(\Omega )$, ${\gamma }_2(\Gamma {\circ }_v\Omega )\in \{\gamma (\Gamma )+n{\gamma }_2(\Omega )-n,{\gamma }_2(\Gamma )+n{\gamma }_2(\Omega )-n,n{\gamma }_2(\Omega )\}$.

Theorem 11

([33]). Let Γ be an isolate-free graph of order $n\ge 2$ such that ${\gamma }_2(\Gamma )<n$ and let Ω be any graph with $\left|V\right(\Omega \left)\right|\ge 2$ and $v\in V(\Omega )$. Then, (a) and (b) are equivalent:

(a) 

${\gamma }_2(\Gamma {\circ }_v\Omega )=n{\gamma }_2(\Omega )$.

(b) 

${\gamma }_2(\Omega -v)\ge {\gamma }_2(\Omega )$.

3. Main Results

The following result is a Corollary of Theorem 2.

Corollary 1.

For an isolate-free graph Γ, $max\{{\gamma }_i(\Gamma ),dim(\Gamma )\}\le {\gamma }_{ri}(\Gamma )\le {\gamma }_i(\Gamma )+dim(\Gamma )$.

Theorem 12.

If ${\gamma }_t(\Gamma )\ge dim(\Gamma )$, then ${\gamma }_t(\Gamma )\le {\gamma }_{rt}(\Gamma )\le {\gamma }_t(\Gamma )+dim(\Gamma )$.

Proof. 

Since ${\gamma }_t(\Gamma )\ge dim(\Gamma )$, so ${\gamma }_t(\Gamma )\le {\gamma }_{rt}(\Gamma )$. The upper bound could be exhibited by the graph, which exists as $P_2{\circ }_v{K}_3$. Consult Figure 4. □

Theorem 13.

If ${\gamma }_t(\Gamma )\le dim(\Gamma )$, then $dim(\Gamma )\le {\gamma }_{rt}(\Gamma )\le {\gamma }_t(\Gamma )+dim(\Gamma )$.

Proof. 

Since ${\gamma }_t(\Gamma )\le dim(\Gamma )$, then $dim(\Gamma )\le {\gamma }_{rt}(\Gamma )$. The upper bound could be exhibited by the graph, which exists as $P_2{\circ }_v{K}_3$. Consult Figure 4. □

The following Corollary is a result obtained from Theorem 12 and 13.

Corollary 2.

For every graph Γ, $max\{{\gamma }_t(\Gamma ),dim(\Gamma )\}\le {\gamma }_{rt}(\Gamma )\le {\gamma }_t(\Gamma )+dim(\Gamma )$.

Theorem 14.

If ${\gamma }_c(\Gamma )\ge dim(\Gamma )$, then ${\gamma }_c(\Gamma )\le {\gamma }_{rc}(\Gamma )\le {\gamma }_c(\Gamma )+dim(\Gamma )$.

Proof. 

Since ${\gamma }_c(\Gamma )\ge dim(\Gamma )$, then ${\gamma }_c(\Gamma )\le {\gamma }_{rc}(\Gamma )$. The upper bound could be exhibited by the graph, which exists as $P_2{\circ }_v{K}_3$. Consult Figure 4. □

Theorem 15.

If ${\gamma }_c(\Gamma )\le dim(\Gamma )$, then $dim(\Gamma )\le {\gamma }_{rc}(\Gamma )\le {\gamma }_c(\Gamma )+dim(\Gamma )$.

Proof. 

Since ${\gamma }_c(\Gamma )\le dim(\Gamma )$, then $dim(\Gamma )\le {\gamma }_{rc}(\Gamma )$. The upper bound could be exhibited by the graph, which exists as $P_2{\circ }_v{K}_3$. Consult Figure 4. □

The following Corollary is a result obtained from Theorem 14 and 15.

Corollary 3.

For every graph Γ, $max\{{\gamma }_c(\Gamma ),dim(\Gamma )\}\le {\gamma }_{rc}(\Gamma )\le {\gamma }_c(\Gamma )+dim(\Gamma )$.

In Figure 4, we see that the resolving total domination and resolving connected domination numbers attain the upper bound. In contrast, in the case of resolving domination, it attains the lower bound. This led to an interesting theorem where we try to relate the resolving domination parameters for any graph $\Gamma$.

Theorem 16.

For any graph Γ, ${\gamma }_r(\Gamma )\le {\gamma }_{rt}(\Gamma )\le {\gamma }_{rc}(\Gamma )$.

Proof. 

Since TDS is a DS, we have

$${\gamma }_{rt}(\Gamma )\ge {\gamma }_r(\Gamma ).$$

(1)

And CDS is a DS implies

$${\gamma }_{rc}(\Gamma )\ge {\gamma }_r(\Gamma ).$$

(2)

Also, as CDS is a TDS, we have

$${\gamma }_{rc}(\Gamma )\ge {\gamma }_{rt}(\Gamma ).$$

(3)

From Equations (1)–(3), we obtain ${\gamma }_r(\Gamma )\le {\gamma }_{rt}(\Gamma )\le {\gamma }_{rc}(\Gamma )$. □

Theorem 17.

$\gamma \left(FCN\right(l\left)\right)=4\left(\gamma \right(FCN(l-1)\left)\right)-2$ for $l\ge 1$.

Proof. 

By Theorem 4, we obtain $\gamma \left(FCN\right(l\left)\right)\in \left\{4\right(\gamma \left(FCN\right(l-1\left)\right))-2,4(\gamma \left(FCN\right(l-1\left)\right)\left)\right\}$ for $l\ge 1$. Since $4\left(\gamma \right(FCN(l-1)\left)\right)-2$ is minimum, we obtain $\gamma \left(FCN\right(l\left)\right)=4\left(\gamma \right(FCN(l-1)\left)\right)-2$ for $l\ge 1$. Let $D_{FCN\left(l\right)}=\left\{ab\right||D_{FCN(l-1)}:a,b\in \{0,1\}\}\setminus \{1110{\left(01\right)}^{(l-1)},0010{\left(01\right)}^{(l-1)}\}$ for $l\ge 1$, where $D_{FCN\left(1\right)}=\{1101,1010,1001,0110,0101,0001\}$. It is to be noted that ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left(x\right)=V\left(FCN\left(l\right)\right)\setminus D_{FCN\left(l\right)}$ and ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left[x\right]=V\left(FCN\left(l\right)\right)$. Therefore, $D_{FCN\left(l\right)}$ is a DS and $D_{FCN\left(2\right)}$ is depicted in Figure 5. □

Note that the DS with cardinality $4\left(\gamma \right(FCN(l-1)\left)\right)-2$ shown in Figure 5 is also an IDS. In consequence, we have the following immediate Corollary of Theorem 17.

Corollary 4.

${\gamma }_i\left(FCN\left(l\right)\right)=4\left({\gamma }_i\left(FCN(l-1)\right)\right)-2$ for $l\ge 1$.

Theorem 18.

${\gamma }_t\left(FCN\left(1\right)\right)=8$.

Proof. 

By Theorem 6, we obtain ${\gamma }_t\left(FCN\left(1\right)\right)\in \{4,6,6,8\}$. Let $I\left(FCN\right(1\left)\right)$ be the set of all possible subgraphs of $FCN\left(1\right)$ with at least one isolated vertex. If ${\gamma }_t\left(FCN\left(1\right)\right)\in \{4,6\}$, then ∃$v\in V\left(FCN\right(1\left)\right)$ but $v\notin N\left[D_{FCN\left(1\right)}\right]$ or $FCN\left(1\right)\left[D_{FCN\left(1\right)}\right]\subseteq I\left(FCN\left(1\right)\right)$. This is a contradiction to the definition of the total dominating set. Thus, ${\gamma }_t\left(FCN\left(1\right)\right)=8$. Let $D_{FCN\left(1\right)}=\{1111,1110,1010,1011,0111,0110,0011,0010\}$. It is to be noted that ${\bigcup }_{x\in D_{FCN\left(1\right)}}N\left(x\right)=V\left(FCN\left(1\right)\right)\setminus D_{FCN\left(1\right)}$, ${\bigcup }_{x\in D_{FCN\left(1\right)}}N\left[x\right]=V\left(FCN\left(1\right)\right)$ and the induced graph of $D_{FCN\left(1\right)}$ does not contain any isolated vertex. Therefore, $D_{FCN\left(1\right)}$ is a TDS and $D_{FCN\left(1\right)}$ is depicted in Figure 6. □

Theorem 19.

${\gamma }_t\left(FCN\left(l\right)\right)=4\left({\gamma }_t\left(FCN(l-1)\right)\right)-2$ for $l\ge 2$.

Proof. 

By Theorem 6, we obtain ${\gamma }_t\left(FCN\left(l\right)\right)\in \{4({\gamma }_t\left(FCN(l-1)\right)-1),4\left({\gamma }_t\left(FCN(l-1)\right)\right)-2,4\left({\gamma }_t\left(FCN(l-1)\right)\right)-2,4\left({\gamma }_t\left(FCN(l-1)\right)\right)\}$. Let $I\left(FCN\right(l\left)\right)$ be the set of all possible subgraphs of $FCN\left(l\right)$ with at least one isolated vertex. Since they are three minimum values of the TDS, we have ${\gamma }_t\left(FCN\left(l\right)\right)\in \{4({\gamma }_t\left(FCN(l-1)\right)-1),4\left({\gamma }_t\left(FCN(l-1)\right)\right)-2,4\left({\gamma }_t\left(FCN(l-1)\right)\right)-2\}$. If ${\gamma }_t\left(FCN\left(l\right)\right)=4({\gamma }_t\left(FCN(l-1)\right)-1)$, then ∃$v\in V\left(FCN\right(l\left)\right)$ but $v\notin N\left[D_{FCN\left(l\right)}\right]$ or $FCN\left(l\right)\left[D_{FCN\left(l\right)}\right]\subseteq I\left(FCN\left(l\right)\right)$. This is a contradiction to the definition of the total dominating set. Thus, ${\gamma }_t\left(FCN\left(l\right)\right)=4\left({\gamma }_t\left(FCN(l-1)\right)\right)-2$ for $l\ge 2$. Let $D_{FCN\left(l\right)}=\left\{ab\right||D_{FCN(l-1)}:a,b\in \{0,1\}\}\cup \{1010{\left(01\right)}^{l-1},1110{\left(01\right)}^{l-1}\}\setminus \{\left(1010\right){\left(01\right)}^{l-2}\left(11\right),\left(1110\right){\left(01\right)}^{l-2}\left(11\right),\left(0110\right){\left(01\right)}^{l-2}\left(11\right),\left(0010\right){\left(01\right)}^{l-2}\left(11\right)\}$ for $l\ge 2$, where $D_{FCN\left(1\right)}=\{1111,1110,1010,1011,0111,0110,0011,0010\}$. It is to be noted that ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left(x\right)=V\left(FCN\left(l\right)\right)\setminus D_{FCN\left(l\right)}$, ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left[x\right]=V\left(FCN\left(l\right)\right)$ and the induced graph of $D_{FCN\left(l\right)}$ does not contain any isolated vertex. Therefore, $D_{FCN\left(l\right)}$ is a TDS and $D_{FCN\left(2\right)}$ is depicted in Figure 7. □

Theorem 20.

${\gamma }_c\left(FCN\left(l\right)\right)=4({\gamma }_c\left(FCN(l-1)\right)+1)$ for $l\ge 1$.

Proof. 

By Theorem 7, ${\gamma }_c\left(FCN\left(l\right)\right)\in \{4\left({\gamma }_c\left(FCN(l-1)\right)\right),4({\gamma }_c\left(FCN(l-1)\right)+1)\}$. If ${\gamma }_c\left(FCN\left(l\right)\right)=4\left({\gamma }_c\left(FCN(l-1)\right)\right)$, then $FCN\left(l\right)\left[D_{FCN\left(l\right)}\right]$ will be an induced disconnected graph or ∃$v\in V\left(FCN\right(l\left)\right)$ but not in $N\left[D_{FCN\left(l\right)}\right]$. This is a contradiction to the definition of the connected dominating set. Thus, ${\gamma }_c\left(FCN\left(l\right)\right)=4({\gamma }_c\left(FCN(l-1)\right)+1)$ for $l\ge 1$. Let $D_{FCN\left(l\right)}=\left\{ab\right||D_{FCN(l-1)}:a,b\in \{0,1\}\}\cup \{0010{\left(01\right)}^{l-1},0110{\left(01\right)}^{l-1},1010{\left(01\right)}^{l-1},1110{\left(01\right)}^{l-1}\}$ for $l\ge 1$, where $D_{FCN\left(1\right)}=\{1111,1110,1011,1010,0111,0110,0011,0010\}$. It is to be noted that ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left(x\right)=V\left(FCN\left(l\right)\right)\setminus D_{FCN\left(l\right)}$, ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left[x\right]=V\left(FCN\left(l\right)\right)$ and $D_{FCN\left(l\right)}$ is connected. Therefore, $D_{FCN\left(l\right)}$ is a CDS and $D_{FCN\left(2\right)}$ is depicted in Figure 8. □

Theorem 21.

${\gamma }_{\times 2}\left(FCN\left(1\right)\right)=12$.

Proof. 

It is obvious that for $FCN\left(0\right)$, the double domination number is 3 since it is isomorphic to $C_4$. By Theorem 9, we determine ${\gamma }_{\times 2}\left(FCN\left(1\right)\right)=4\left({\gamma }_{\times 2}\left(FCN\left(0\right)\right)\right)=12$. Let $D_{FCN\left(1\right)}=\{1111,1110,1101,1011,1010,1001,0111,0110,0101,0011,0010,0001\}$. It is to be noted that ${\bigcup }_{x\in D_{FCN\left(1\right)}}N\left(x\right)=V\left(FCN\left(1\right)\right)\setminus D_{FCN\left(1\right)}$, ${\bigcup }_{x\in D_{FCN\left(1\right)}}N\left[x\right]=V\left(FCN\left(1\right)\right)$ and for every $x\in V\left(FCN\right(1\left)\right)$, $|N\left[x\right]\cap D_{FCN\left(1\right)}| \ge 2$. Therefore, $D_{FCN\left(1\right)}$ is a DDS and $D_{FCN\left(1\right)}$ is depicted in Figure 9. □

Theorem 22.

${\gamma }_{\times 2}\left(FCN\left(l\right)\right)=4({\gamma }_{\times 2}\left(FCN(l-1)\right)-1)$ for $l\ge 2$.

Proof. 

By Theorem 8, we obtain ${\gamma }_{\times 2}\left(FCN\left(l\right)\right)\in \{4{\gamma }_{\times 2}\left(FCN(l-1)\right),4\left({\gamma }_{\times 2}\left(FCN(l-1)\right)\right),4\left({\gamma }_{\times 2}\left(FCN(l-1)\right)\right)-2,4\left({\gamma }_{\times 2}\left(FCN(l-1)\right)\right)-2,4({\gamma }_{\times 2}\left(FCN(l-1)\right)-1),4\left({\gamma }_{\times 2}\left(FCN(l-1)\right)\right)-5\}$. Here, there are two minimum values for the DDS. That is, ${\gamma }_{\times 2}\left(FCN\left(l\right)\right)\in \{4({\gamma }_{\times 2}\left(FCN(l-1)\right)-1),4\left({\gamma }_{\times 2}\left(FCN(l-1)\right)\right)-5\}$. If ${\gamma }_{\times 2}\left(FCN\left(l\right)\right)=4\left({\gamma }_{\times 2}\left(FCN(l-1)\right)\right)-5$, then ∃$v\in V\left(FCN\right(l\left)\right)$ where $|N\left[v\right]\cap D_{FCN\left(l\right)}| \le 1$. Thus, ${\gamma }_{\times 2}\left(FCN\left(l\right)\right)=4({\gamma }_{\times 2}\left(FCN(l-1)\right)-1)$ for $l\ge 2$. Let $D_{FCN\left(l\right)}=\left\{ab\right||D_{FCN(l-1)}:a,b\in \{0,1\}\}\setminus \{\left(1110\right){\left(01\right)}^{l-2}\left(11\right),\left(1010\right){\left(01\right)}^{l-2}\left(11\right),\left(0110\right){\left(01\right)}^{l-2}\left(11\right),\left(0010\right){\left(01\right)}^{l-2}\left(11\right)\}$ for $l\ge 2$, where $D_{FCN\left(1\right)}=\{1111,1110,1101,1011,1010,1001,0111,0110,0101,0011,0010,0001\}$. It is to be noted that ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left(x\right)=V\left(FCN\left(l\right)\right)\setminus D_{FCN\left(l\right)}$, ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left[x\right]=V\left(FCN\left(l\right)\right)$ and for every $x\in V\left(FCN\right(l\left)\right)$, $|N\left[x\right]\cap D_{FCN\left(l\right)}| \ge 2$. Therefore, $D_{FCN\left(l\right)}$ is a DDS and $D_{FCN\left(2\right)}$ is depicted in Figure 10. □

Theorem 23.

${\gamma }_2\left(FCN\left(l\right)\right)=4\left({\gamma }_2\left(FCN(l-1)\right)\right)$ for $l\ge 1$.

Proof. 

From Theorem 10 and Theorem 11, we obtain ${\gamma }_2\left(FCN\left(l\right)\right)=4\left({\gamma }_2\left(FCN(l-1)\right)\right)$ for $l \ge 1$. Let $D_{FCN\left(l\right)}=\{N(s_1\dots s_{2l}01):s_1,s_2,\dots ,s_{2l}\in \{0,1\}\}$ for $l \ge 1$. It is to be noted that ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left(x\right)=V\left(FCN\left(l\right)\right)\setminus D_{FCN\left(l\right)}$, ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left[x\right]=V\left(FCN\left(l\right)\right)$ and for every $x\in V\left(FCN\left(l\right)\right)\setminus D_{FCN\left(l\right)}$, $|N\left(x\right)\cap D_{FCN\left(l\right)}| \ge 2$. Therefore, $D_{FCN\left(l\right)}$ is a 2DS and $D_{FCN\left(2\right)}$ is depicted in Figure 11. □

Theorem 24.

${\gamma }_r\left(FCN\left(l\right)\right)=4\left({\gamma }_2\left(FCN(l-1)\right)\right)$ for $l\ge 1$.

Proof. 

From Theorem 3, Theorem 2, and Theorem 17, we have $max\{4^l,4\left(\gamma \left(FCN(l-1)\right)\right)-2\}\le {\gamma }_r\left(FCN\left(l\right)\right)\le 4^l+4\left(\gamma \left(FCN(l-1)\right)\right)-2$. Suppose ${\gamma }_r\left(FCN\left(l\right)\right)=4^l$, then ∃$v\in V\left(FCN\right(l\left)\right)$, which is a two-degree twin vertex, such that $v\notin N\left[D_{FCN\left(l\right)}\right]$. This is a contradiction to the definition of resolving domination. Suppose ${\gamma }_r\left(FCN\left(l\right)\right)=4\left(\gamma \left(FCN(l-1)\right)\right)-2$, by Theorem 1 and 17, the vertices of the set does not contain twin vertices, which are necessary for resolving the graph. This is a contradiction to the definition of the resolving domination. Thus, the set must contain all twin vertices except one from each twin set and must dominate the vertices of $FCN\left(l\right)$. Thus, in every disjoint $C_4$ of $FCN\left(l\right)$, we need at least two vertices and one must be from the twin set. This implies that ${\gamma }_r\left(FCN\left(l\right)\right)=4\left({\gamma }_2\left(FCN(l-1)\right)\right)$ for $l\ge 1$. Let $D_{FCN\left(l\right)}=\{N(s_1\dots s_{2l}01):s_1,s_2,\dots ,s_{2l}\in \{0,1\}\}$ for $l\ge 1$. It is to be noted that ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left(x\right)=V\left(FCN\left(l\right)\right)\setminus D_{FCN\left(l\right)}$, ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left[x\right]=V\left(FCN\left(l\right)\right)$ and $x\in D_{FCN\left(l\right)}$, $d(x,y)\ne d(x,z)$ for every pair of $y,z\in V\left(FCN\right(l\left)\right)$. Therefore, $D_{FCN\left(l\right)}$ is an RDS and $D_{FCN\left(2\right)}$ is depicted in Figure 11. □

Note that the RDS with cardinality $4\left({\gamma }_2\left(FCN(l-1)\right)\right)$ shown in Figure 11 is also an RIDS. In consequence, we have the following immediate Corollary of Theorem 24.

Corollary 5.

${\gamma }_{ri}\left(FCN\left(l\right)\right)=4\left({\gamma }_2\left(FCN(l-1)\right)\right)$ for $l\ge 1$.

Theorem 25.

${\gamma }_{rt}\left(FCN\left(1\right)\right)=8$.

Proof. 

From Theorem 3, Corollary 2, and Theorem 18 we obtain ${\gamma }_{rt}\left(FCN\left(1\right)\right)=8$, since it must be both a resolving and total dominating set. Let $D_{FCN\left(1\right)}=\{1111,1110,1011,1010,0111,0110,0011,0010\}$. It is to be noted that ${\bigcup }_{x\in D_{FCN\left(1\right)}}N\left(x\right)=V\left(FCN\left(1\right)\right)\setminus D_{FCN\left(1\right)}$, ${\bigcup }_{x\in D_{FCN\left(1\right)}}N\left[x\right]=V\left(FCN\left(1\right)\right)$, the induced graph of $D_{FCN\left(1\right)}$ does not contain any isolated vertex, and $x\in D_{FCN\left(1\right)}$, $d(x,y)\ne d(x,z)$ for every pair of $y,z\in V\left(FCN\right(1\left)\right)$. Therefore, $D_{FCN\left(1\right)}$ is an RTDS and $D_{FCN\left(1\right)}$ is depicted in Figure 6. □

Theorem 26.

${\gamma }_{rt}\left(FCN\left(l\right)\right)=4\left({\gamma }_{rt}\left(FCN(l-1)\right)\right)$ for $l\ge 2$.

Proof. 

From Theorem 3, Corollary 2, and Theorem 19, we have $max\{4^l,4\left({\gamma }_t\left(FCN(l-1)\right)\right)-2\}\le {\gamma }_{rt}\left(FCN\left(l\right)\right)\le 4^l+4\left({\gamma }_t\left(FCN(l-1)\right)\right)-2$. The same argument constructed in Theorem 24 holds here. But we need to choose the twins except one from each twin set including its one neighbor. Thus, ${\gamma }_{rt}\left(FCN\left(l\right)\right)=4\left({\gamma }_{rt}\left(FCN(l-1)\right)\right)$ for $l\ge 2$ and the set is a total dominating and resolving set. Let $D_{FCN\left(l\right)}=\left\{ab\right||D_{FCN(l-1)}:a,b\in \{0,1\}\}$ for $l\ge 2$, where $D_{FCN\left(1\right)}=\{1111,1110,1011,1010,0111,0110,0011,0010\}$. It is to be noted that ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left(x\right)=V\left(FCN\left(l\right)\right)\setminus D_{FCN\left(l\right)}$, ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left[x\right]=V\left(FCN\left(l\right)\right)$, the induced graph of $D_{FCN\left(l\right)}$ does not contain any isolated vertex, and $x\in D_{FCN\left(l\right)}$, $d(x,y)\ne d(x,z)$ for every pair of $y,z\in V\left(FCN\right(l\left)\right)$. Therefore, $D_{FCN\left(l\right)}$ is an RTDS and $D_{FCN\left(2\right)}$ is depicted in Figure 12. □

Theorem 27.

${\gamma }_{rc}\left(FCN\left(l\right)\right)=4({\gamma }_c\left(FCN(l-1)\right)+1)$ for $l\ge 1$.

Proof. 

From Theorem 3, Corollary 3, and Theorem 20, we obtain ${\gamma }_{rc}\left(FCN\left(l\right)\right)=4({\gamma }_c\left(FCN(l-1)\right)+1)$ for $l\ge 1$. Let $D_{FCN\left(l\right)}=\left\{ab\right||D_{FCN(l-1)}:a,b\in 0,1\}\cup \{0010{\left(01\right)}^{l-1},0110{\left(01\right)}^{l-1},1010{\left(01\right)}^{l-1},1110{\left(01\right)}^{l-1}\}$ for $l\ge 1$ where $D_{FCN\left(1\right)}=\{1111,1110,1011,1010,0111,0110,0011,0010\}$. It is to be noted that ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left(x\right)=V\left(FCN\left(l\right)\right)\setminus D_{FCN\left(l\right)}$, ${\bigcup }_{x\in D_{FCN\left(l\right)}}N\left[x\right]=V\left(FCN\left(l\right)\right)$, the induced graph of $D_{FCN\left(l\right)}$ is connected, and $x\in D_{FCN\left(l\right)}$, $d(x,y)\ne d(x,z)$ for every pair of $y,z\in V\left(FCN\right(l\left)\right)$. Therefore, $D_{FCN\left(l\right)}$ is an RCDS and $D_{FCN\left(2\right)}$ is depicted in Figure 8. □

4. Conclusions

Studying domination in the context of rooted product graphs has provided new insights into domination parameters, such as the domination number and the change when two graphs are combined through a rooted product operation. This work has contributed to understanding the original properties of the graph influencing the domination properties of resultant graphs. Specifically, for rooted product graphs, the domination behavior has been linked to the properties of the root vertex and interaction with other vertices. In satellite technologies, the space information network is constructed by the satellites (fixed as the node of the network), and the links of the satellites are used for communication and environment monitoring (commonly known as domination). Considering fractal cubic networks in satellite technologies will enhance environmental monitoring by fixing the satellites on the root vertex. These insights can help design more efficient algorithms and solve graph-theoretical problems in network theory, biology, computer science, and telecommunication systems. Future research includes developing algorithms, root selection strategies, and parameter generalization. By addressing these future research directions, a more comprehensive understanding of domination in rooted product graphs can be developed, leading to theoretical advancements and practical applications across various fields.