Communications on Applied Nonlinear Analysis 

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Strong Regular Domination in Litact Graphs 

 

G. Shankarajyothi1*, G. Upender Reddy2  
1*Research Scholar, Department of Mathematics, Osmania University, Hyderabad, India. 

2Department of Mathematics, Nizam College(A), Osmania University, Hyderabad, India.  

1E-mail: shankarajyothi.maths@gmail.com* 

2E-mail: yuviganga@gmail.com 

 

Article History: 

Received: 10-02-2024 

Revised: 12-04-2024 

Accepted: 26-04-2024 

 

Abstract: 

Strong regular domination in a litact graph is a novel domination 

parameter that has been introduced in this paper. A dominating set 𝐷 ⊆

𝑉(𝐺) is known as Strong regular dominating set of 𝐺, if for each point 

𝑥 ∈ 𝑉(𝐺) − 𝐷 there is a vertex 𝑦 ∈ 𝐷 with an edge 𝑥𝑦 ∈ 𝐸(𝐺) and 

deg⁡(𝑥) ≤ deg⁡(𝑦) and all vertices of 〈𝐷〉 holds the equal degree. The 

lowest cardinality of such vertices of 𝐷 is known as strong regular 

domination number of 𝐺 which is represented by 𝛾𝑠𝑡𝑟(𝐺). The current 

study aims by taking strong regular domination on a litact graph 𝑚(𝐺) 

denoted by  𝛾𝑠𝑡𝑟[𝑚(𝐺)]  and to obtain some bounds on 𝛾𝑠𝑡𝑟[𝑚(𝐺)]  in 

terms of various parameters of 𝐺 such as vertices, edges, maximum 

degree, diameter and so on and also in terms of various domination 

parameters of 𝐺 such as total domination of 𝐺, connected domination of 

𝐺 and so on. Furthermore, outcomes resembling those of Nordhaus-

Gaddum were also obtained. 

Keywords: litact graph, strong domination number, regular domination 

number, strong regular domination number. 

2020 AMS Subject Classification: 05C72. 

 
 

1. Introduction 

One area of graph theory that has been studied in great detail is domination. The theory of 

domination has multiple origins. According to historical accounts, the first domination-type problem 

originated with a chess board problem in 1850 that C.F. De Jaenisch mathematically explained in 

1862. Domination in graphs has implications beyond the chess board problem, including facility 

location problems, electric networks, power grids, land surveying, and more. The topic of 

domination was first introduced by C. Berge in ‘The Theory of Graphs and its Application’, and it 

was formalized mathematically by Ore in 1962. Berge referred to the number of domination as the 

coefficient of external stability and the domination as the external stability. The term "domination" 

was first used by Ore in his well-known 1962 book “Theory of Graphs”. 

Certain real-world scenarios naturally give rise to both strong and weak domination. Take a road 

network that connects several places, for instance, the degree of such a network is determined by ‘the 

number of roads that meet at a vertex 𝑣’. Let deg⁡(𝑢) ≥ deg⁡(𝑣).  It seems that, ‘traffic at 𝑢 appears 

mailto:shankarajyothi.maths@gmail.com
mailto:yuviganga@gmail.com


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to be heavier than at 𝑣’. Vehicles travelling from 𝑢 to 𝑣 should be given preference when it comes to 

traffic between 𝑢 and 𝑣. Therefore, ‘𝑢 strongly dominates 𝑣’ and ‘𝑣 weakly dominates 𝑢’ in a certain 

sense. 

In 1996, ‘Sampathkumar and Pushpa Latha’ presented the ideas of ‘strong and weak domination’. It 

was E. Sampath kumar who first proposed the concept of regular domination number. Numerous 

scholars have investigated these ideas, including (1,2,3,6,7). 

2. Notations and Definitions 

Definition-2.1: If each point in 𝑉 − 𝑋 is adjacent to a point in 𝑋, then 𝑋 is considered a ‘dominating 

set’ of 𝐺(𝑉, 𝐸). The domination number of 𝐺, represented by 𝛾(𝐺), is the ‘minimal cardinality of 

such a set’. 

Definition-2.2: Suppose 𝐺 is a connected graph with an edge 𝑥𝑦𝜖𝐸(𝐺). Then, ‘𝑥 strongly dominates 

𝑦’ (‘𝑦 weakly dominates 𝑥’) if ‘deg⁡(𝑥) ≥ deg⁡(𝑦)’. It is clear that each point in 𝑉(𝐺) has the ability 

to strongly dominate itself. A set 𝐷 ⊆ 𝑉(𝐺) is a ‘strong dominating set’ of G, if for each point 

𝑦𝜖𝑉(𝐺) − 𝐷 there is another point 𝑥𝜖𝐷 such that 𝑥𝑦𝜖𝐸(𝐺) and deg⁡(𝑥) ≥ deg⁡(𝑦). The ‘minimum 

cardinality’ of such set is known as ‘strong domination number’ of G and it is represented by 

𝛾𝑠𝑡(𝐺). 

Definition-2.3: A ‘graph’ 𝐺 is called a ‘regular graph’ if each point in it holds the equal degree. A 

‘dominating set 𝐷’ is called a ‘regular dominating set’ of 𝐺 if 〈𝐷〉 is regular. The ‘regular 

domination number’ of 𝐺, represented by 𝛾𝑟(𝐺), is the ‘minimal cardinality’ of such a set. 

Definition-2.4: A ‘dominating set D’ of a ‘graph G’ is known as ‘strong regular dominating set’ of 

G if (i) for each point 𝑦𝜖𝑉(𝐺) − 𝐷 there is another point 𝑥𝜖𝐷 such that 𝑥𝑦𝜖𝐸(𝐺) and deg⁡(𝑥) ≥

deg⁡(𝑦) and (ii) an induced subgraph 〈𝐷〉 is a ‘regular dominating set of G’. The smallest cardinality 

of such a set is known as ‘strong regular domination number’ of G and which is represented by 

𝛾𝑠𝑡𝑟(𝐺). 

Definition-2.5: The point set of a lit act graph 𝑚(𝐺) is made up of the edges and cut vertices of a 

graph 𝐺. If edges and cut vertices are incident or adjacent in 𝐺, then the two vertices in 𝑚(𝐺) are 

adjacent. 

Example. 

Figure 1 below shows a graph 𝐺 alongwith it’s litact graph 𝑚(𝐺). 

 

Fig. 1: A Graph 𝐺 and it’s Litact Graph 𝑚(𝐺) with 𝛾𝑟(𝑚(𝐺)) = 𝛾𝑠𝑡(𝑚(𝐺)) = 𝛾𝑠𝑡𝑟(𝑚(𝐺)) = 2 

3. Main Results  

The current study exclusively considers simple, finite, non-trivial, connected and undirected graphs. 

All the following results are hold for any graph on which strong regular domination is found. 

𝐺 𝑚(𝐺) 



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The following section evaluates a graph G’s and it’s litact graph 𝑚(𝐺)’s strong regular domination 

number for few standard graphs, such as a ‘cycle graph’, ‘star graph’, ‘wheel graph’, ‘complete 

graph’ and so on. 

Theorem-3.1: For any ‘cycle graph’ 𝐶𝑛⁡, 𝛾𝑠𝑡𝑟(𝐶𝑛⁡) = 𝛾𝑠𝑡𝑟(𝑚(𝐶𝑛⁡)). 

Theorem-3.2: For any ‘wheel graph’ 𝑊𝑛, ⁡𝛾𝑠𝑡𝑟(𝑊𝑛) = 1. 

Theorem-3.3: For any ‘complete graph’ 𝐾𝑛, ⁡𝛾𝑠𝑡𝑟(𝐾𝑛) = 1. 

Theorem-3.4: For any ‘complete bipartite graph’ 𝐾𝑚,𝑛, ⁡𝛾𝑠𝑡𝑟(𝐾𝑚,𝑛) = 𝑚 where 𝑚 ≤ 𝑛. 

Theorem-3.5: For any ‘star graph’ 𝐾1,𝑛, ⁡𝛾𝑠𝑡𝑟(𝐾1,𝑛) = ⁡𝛾𝑠𝑡𝑟(𝑚(𝐾1,𝑛)) = 1. 

An upper bound for 𝛾𝑠𝑡𝑟(𝑚(𝐺)) in terms of order of G has been determined in the following 

theorem. 

Theorem-3.6: If G is any ‘graph G’, then 𝛾𝑠𝑡𝑟(𝑚(𝐺)) < 𝑝. 

Proof: Suppose V represent G's vertex set, E represent G’s edge set and C represent G’s set of cut 

vertices. As stated in the definition of a litact graph 𝑚(𝐺), "𝑉(𝑚(𝐺)) = 𝐸(𝐺) ∪ 𝐶(𝐺)". If 𝑋 ⊆

𝑉(𝑚(𝐺)) is the smallest dominating set of 𝑚(𝐺) and deg⁡(𝑥𝑖) ≤ deg⁡(𝑥𝑗) where 𝑥𝑖𝜖𝑉(𝑚(𝐺)) − 𝑋 

and X is k-regular then X itself forms a strong regular dominating set. Then 𝛾𝑠𝑡𝑟(𝑚(𝐺)) = |𝑋|. 

Otherwise, there exists 𝑥𝑖 ∈ 𝑌 ⊆ 𝑉(𝑚(𝐺)) such that 〈𝑋 ∪ 𝑌〉 forms a minimal strong regular 

dominating set in 𝑚(𝐺). Therefore, |𝑋 ∪ 𝑌| = 𝛾𝑠𝑡𝑟(𝑚(𝐺)). Then, 𝑉(𝑚(𝐺)) < 2𝑝. Also,  

|𝑋 ∪ 𝑌| ≤ 𝑉(𝑚(𝐺)) − |𝑋 ∪ 𝑌| 

⟹ |𝑋 ∪ 𝑌| + |𝑋 ∪ 𝑌| ≤ 𝑉(𝑚(𝐺)) 

⟹ 2|𝑋 ∪ 𝑌| ≤ 𝑉(𝑚(𝐺)) < 2𝑝 

⟹ 2𝛾𝑠𝑡𝑟(𝑚(𝐺)) < 2𝑝 

⟹ 𝛾𝑠𝑡𝑟(𝑚(𝐺)) < 𝑝 

The relationship between 𝛾𝑠𝑡𝑟(𝑚(𝐺)), ⁡𝛾𝑠𝑡𝑟(𝐺), deg(𝑥) and deg⁡(𝑦)  is given below. 

Theorem-3.7: For any ‘graph G’, 𝛾𝑠𝑡𝑟(𝑚(𝐺)) ≤ 𝛾𝑠𝑡𝑟(𝐺) + deg(𝑥) + deg⁡(𝑦) where 𝑥, 𝑦 are the 

elements of a ‘strong regular dominating set of 𝑚(𝐺)’. 

Proof: Let 𝑚(𝐺) be the litact graph of G with V points and E edges. Suppose D is a ‘strong regular 

dominating set’ of 𝑚(𝐺). Assume that the points 𝑥, 𝑦 ∈ 𝐷 and x has the same degree as some of its 

neighbours (except y) and strongly dominates them, and that y does the same. Let 𝑣1 is adjacent to 

𝑥, 𝑣1 ≠ 𝑦 such that deg(𝑥) = deg⁡(𝑣1), and 𝑣1 is strongly dominated only by x. Hence, ‘D is a 

strong regular dominating set’, so that |𝐷| = 𝛾𝑠𝑡𝑟(𝑚(𝐺)). Let S be the strong regular dominating set 

in G such that |𝑆| = 𝛾𝑠𝑡𝑟(𝐺). From the above, it is easy to verify that |𝐷| ⊆ |𝑆| ∪ |𝑁(𝑥)| ∪

|𝑁(𝑦)| ⟹ 𝛾𝑠𝑡𝑟(𝑚(𝐺)) ≤ 𝛾𝑠𝑡𝑟(𝐺) + deg(𝑥) + deg⁡(𝑦). 

The relationship between 𝛾𝑠𝑡𝑟 (𝑚(𝐻𝑝)), ‘edges’ and ‘cut vertices’ of the ‘helm graph’ has been 

determined in the subsequent theorem. 



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Theorem-3.8: For any (𝑝, 𝑞) −Helm graph 𝐻𝑝, 𝛾𝑠𝑡𝑟 (𝑚(𝐻𝑝)) < 𝑞 + 𝑙 where 𝑙 is the number of cut 

vertices of  𝐻𝑝. 

Proof: Suppose 𝐺(𝑉, 𝐸) be a Helm graph with 𝑉 = {𝑣1, 𝑣2, 𝑣3, … . , 𝑣𝑝−1, 𝑣1
′ , 𝑣2

′ , 𝑣3
′ , … . , 𝑣𝑝−1

′ }  

vertices and 𝐸 = {𝑣1𝑣𝑖 , 2 ≤ 𝑖 ≤ 𝑝} ∪ {𝑣𝑖𝑣𝑖+1, 𝑣𝑖𝑣𝑖−1
′ , 2 ≤ 𝑖 ≤ 𝑝 − 1} ∪ {𝑣2𝑣𝑝} ∪ {𝑣1𝑣𝑖

′, 1 ≤ 𝑖 ≤ 𝑝 −

1} edges. Since the helm graph is formed by attaching one vertex by an edge to each of the 𝑝 − 1  

points in the wheel graph's ‘outer circuit’, the vertex 𝑣1 comes after each of the 𝑝 − 1   vertices in the 

wheel graph. Every 𝑝 − 1  vertex in the wheel network is adjacent to 𝑣1. This indicates that the 

strong regular domination for a wheel graph is 1 such that 𝛾𝑠𝑡𝑟(𝑊𝑝) = 1. But the vertex 𝑣1 strongly 

dominates only the wheel graph; the helm graph is not dominated by it. Given that the helm network 

contains 𝑝 − 1  pendent vertices, and for each pendant vertex 𝑣𝑗 , 2 ≤ 𝑗 ≤ 𝑝 is connected by an edge. 

The 𝑝 − 1  vertices in the outer circuit are therefore superior to both of the 𝑝 − 1  vertices in the 

helm graph. Thus, the ‘strong regular domination number’ of the helm graph is 𝑝 − 1, that is t 

𝛾𝑠𝑡𝑟(𝐶) = 𝑝 − 1. Since the litact graph of helm graph is formed by the cut vertices and edges of the 

helm graph, and it is clear that if D is the strong regular domination number of the helm graph, then 

|𝐷| < 𝑞(𝐻𝑝) + 𝑙(𝐻𝑝). Hence, 𝛾𝑠𝑡𝑟 (𝑚(𝐻𝑝)) < 𝑞 + 𝑙. 

The next theorem connects 𝛾𝑠𝑡𝑟(𝑚(𝐺)), 𝛾(𝐺), 𝑝(𝐺) and 𝛼0(𝐺). 

Theorem-3.9: For any connected (𝑝, 𝑞) graph G, 𝛾𝑠𝑡𝑟(𝑚(𝐺)) + 𝛾(𝐺) ≤ 𝑝 + ⌈
𝛼0

2
⌉. 

Proof: Let 𝑉 = {𝑣1, 𝑣2, 𝑣3, … . , 𝑣𝑖} be the smallest set of vertices that encompasses every edge in G, 

ensuring that |𝑉| = 𝛼0. Let D be the ‘dominating set of G’ going forward so that |𝐷| = 𝛾(𝐺). 

Assume that the ‘set of edges’ in G that form a minimal connected edge dominating set in G is 

represented by the set 𝐸 = {𝑒1, 𝑒2, 𝑒3, … . . , 𝑒𝑘}. Let 𝐶 ⊆ 𝑉(𝐺)  represent G's set of cut vertices. The 

minimal strong dominating set in 𝑚(𝐺) is then the set 𝑋𝜖𝑉(𝑚(𝐺))  where 𝑋 ⊆ 𝐹 ∪ 𝐶 and 𝑁(𝑋) =

𝑉(𝑚(𝐺)) − 𝑋. Further, let 𝑌 ⊆ 𝑉(𝑚(𝐺)) − 𝑋 and 𝑌 ∈ 𝑁(𝑋), then take a set 𝑌′ ⊂ 𝑌 so that 〈𝑋 ∪ 𝑌′〉 

is the minimal strong k-regular sub graph of 𝑚(𝐺), then 𝑋 ∪ 𝑌′  is the minimal strong regular 

dominating set of 𝑚(𝐺). Then, 

|𝑋 ∪ 𝑌′| ∪ |𝐷| ≤ 𝑝 + ⌈
𝛼0

2
⌉ ⟹ 𝛾𝑠𝑡𝑟(𝑚(𝐺)) + 𝛾(𝐺) ≤ 𝑝 + ⌈

𝛼0

2
⌉. 

The next outcome involving 𝛾𝑠𝑡𝑟(𝑚(𝐺)), 𝛾𝑐(𝐺), 𝛽0(𝐺) and 𝛾(𝐺) has been determined. 

Theorem-3.10: For any ‘graph G’, 𝛾𝑠𝑡𝑟(𝑚(𝐺)) + 𝛾𝑐(𝐺) ≤ 𝛽0(𝐺) + 𝛾(𝐺). 

Proof: Let 𝑋 ⊆ 𝑉(𝐺) be the set of points that cover all of the edges in G at a minimum distance of 

two and have deg(𝑣𝑖) ≥ 2, ∀𝑣𝑖 ∈ 𝑋, 1 ≤ 𝑖 ≤ 𝑘. Additionally, if 𝑁(𝑥) = 𝑉(𝐺) − 𝑋 for every vertex 

𝑥 ∈ 𝑋, then 𝑋 is an independent vertex collection. Otherwise, |𝑋1 ∪ 𝑋2| = 𝛽0(𝐺) creates a maximum 

independent set of G where 𝑋1 ⊆ 𝑋 and 𝑋2 ⊆ 𝑉(𝐺) − 𝑋. Let 𝑌 = 𝑋′ ∪ 𝑋"  represent the minimal set 

of points that covers all points in G, where 𝑋′ ⊆ 𝑋 and 𝑋" ⊆ 𝑉(𝐺) − 𝑋. 𝑌 undoubtedly makes up G's 

minimum dominating set. Assume that there is just one element in the sub graph 〈⁡𝑌 〉. Then, 𝑌 is a 

‘connected dominating set of G’ on its own. 



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Otherwise, attach the smallest number of vertices {𝑢𝑗} ∈ 𝑉(𝐺) − 𝑌 where deg⁡(𝑢𝑗) ≥ 2 that are 

between the vertices of 𝑌 such that 𝑌1 = 𝑌 ∪ {𝑢𝑗} forms precisely one component in the subgraph 

〈𝑌1〉. 𝑌1 undoubtedly constitutes a minimal connected dominating set within G. Let 𝐷 ⊆ 𝐶, where C 

is the set of points in G that correspond to the edges that meet 𝑌's vertices. A minimal set of vertices 

where ∀𝑣𝑘 ∈ 𝐷 and 𝑁(𝐷) = 𝑉(𝑚(𝐺)) − {𝑣𝑘} are found. D undoubtedly creates a strong regular 

dominating set in 𝑚(𝐺). 

Therefore, |𝐷| ∪ |𝑌1⁡| ≤ |𝑋1 ∪ 𝑋2| ∪ |𝑌| ⟹ 𝛾𝑠𝑡𝑟(𝑚(𝐺)) + 𝛾𝑐(𝐺) ≤ 𝛽0(𝐺) + 𝛾(𝐺). Hence, the 

proof. 

Using 𝛾𝑠𝑡𝑟(𝑚(𝐺) in terms of 𝑞 and ∆′(𝐺), the following theorem has been discovered. 

Theorem-3.11: For any ‘graph G’, ⌈
𝑞

2∆′(𝐺)+1
⌉ < 𝛾𝑠𝑡𝑟(𝑚(𝐺) + 1. 

Proof: Let ‘G be a graph’ with the set of edges 𝐸 = {𝑒1, 𝑒2, 𝑒3, …… , 𝑒𝑛}, such that |𝐸(𝐺)| = 𝑞 and 

∆′(𝐺) be the highest degree of an edge in G. Suppose 𝑋 ⊆ 𝑉(𝑚(𝐺))  is the smallest dominating set 

in 𝑚(𝐺), then 𝑋 creates a strong regular dominating set by itself. Next, it is clear that, according to 

the litact graph definition  𝑉(𝑚(𝐺)) = 𝐸(𝐺) ∪ 𝐶(𝐺), we get, 

(2∆′(𝐺) + 1)|𝑋| ≥ |𝐸(𝐺)| 

⟹ (2∆′(𝐺) + 1)|𝑋| ≥ 𝑞 

⟹ |𝑋| ≥
𝑞

2∆′(𝐺) + 1
 

⟹ ⌈
𝑞

2∆′(𝐺) + 1
⌉ ≤

𝑞

2∆′(𝐺) + 1
< |𝑋| + 1 

⟹ ⌈
𝑞

2∆′(𝐺) + 1
⌉ < 𝛾𝑠𝑡𝑟(𝑚(𝐺) + 1 

We derive the lower bound for 𝛾𝑠𝑡𝑟(𝑚(𝐺)) in terms of diameter of G in the subsequent theorem. 

Theorem-3.12: For any ‘graph G’, 𝑑𝑖𝑎𝑚(𝐺) − 1 ≤ 3𝛾𝑠𝑡𝑟(𝑚(𝐺)). 

Proof: Let the collection of edges in G that make up the diameteral path in G be denoted by 𝐸1 =

{𝑒𝑖; 1 ≤ 𝑖 ≤ 𝑛}. Therefore, |𝐸1| = 𝑑𝑖𝑎𝑚(𝐺). Let 𝐸2 ⊆ 𝐸(𝐺), ∀𝑒𝑖 ∈ 𝐸2 has the maximal edge degree 

in G. 𝐸2  becomes a minimum dominating set of 𝑚(𝐺) since 𝐸2 ⊆ 𝑉(𝑚(𝐺)), it is the smallest set of 

points which covers all of the vertices in 𝑚(𝐺). Moreover, 𝐸2  is a smallest strong regular 

dominating set in 𝑚(𝐺) if deg(𝑥) ≤ deg(𝑦) , ∀𝑥𝜖𝑉(𝑚(𝐺)) − 𝐸2 and 𝐸2 is k-regular. Consequently, 

|𝐸1| − 1 ≤ 3|𝐸2| ⟹ 𝑑𝑖𝑎𝑚(𝐺) − 1 ≤ 3𝛾𝑠𝑡𝑟(𝑚(𝐺). 

Hence, 𝑑𝑖𝑎𝑚(𝐺) − 1 ≤ 3𝛾𝑠𝑡𝑟(𝑚(𝐺).  

The subsequent theorem relates 𝛾𝑠𝑡𝑟(𝑚(𝐺)), 𝛾𝑐(𝐺), 𝑑𝑖𝑎𝑚(𝐺), 𝛼0(𝐺) and 𝛾(𝐺). 

Theorem-3.13: For any connected graph G, 𝛾𝑠𝑡𝑟(𝑚(𝐺)) + 𝛾𝑐(𝐺) < 𝑑𝑖𝑎𝑚(𝐺) + 𝛼0(𝐺) + 𝛾(𝐺). 

Proof: Define 𝐴 ⊆ 𝑉(𝐺)  as the smallest set of vertices that encompasses all edges in G, ensuring 

that |𝐴| equals 𝛼0(𝐺). Moreover, an edge set 𝐸 ⊆ 𝐸′  exists, where 𝐸′ is the collection of edges that 

coincide with the vertices of V to form the longest path in G, with |𝐸| equal to 𝑑𝑖𝑎𝑚(𝐺). To build a 

‘minimal dominating set of G’, let 𝑆 = {𝑣𝑖; 1 ≤ 𝑖 ≤ 𝑛} ⊆ 𝑉(𝐺). If 〈𝑆′〉  is a connected sub graph of 



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S, then ‘S is a connected dominating set’. In the absence of this, a ‘minimal connected dominating 

set of G’ is formed by at least one vertex, 𝑥 ∈ 𝑉(𝐺) − 𝑆′  and 𝑆" = 𝑆′ ∪ {𝑥}. 

Now, in 𝑚(𝐺), let 𝑈 = {𝑢𝑗; 1 ≤ 𝑗 ≤ 𝑛} ⊆ 𝑉(𝑚(𝐺)) such that {𝑢𝑗} = {𝑒𝑗} ∈ 𝐸(𝐺), 1 ≤ 𝑗 ≤ 𝑛, where 

{𝑒𝑗} are incident with the vertices of 𝑆′. Additionally, assume that D represents the dominant set of 

𝑚(𝐺) and that ∀𝑢𝑘 ∈ 〈𝑉(𝑚(𝐺)) − 𝐷〉, deg(𝑢𝑘) ≤ deg(𝑢𝑗) , ∀𝑢𝑗 ∈ 𝐷 and D is k-regular. So, D 

creates a strong regular dominating set in 𝑚(𝐺). Otherwise, deg(𝑢𝑘) > deg(𝑢𝑗) , ∀𝑢𝑗 ∈ 𝐷, exists for 

at least one vertex {𝑢} ∈ 𝑉(𝑚(𝐺)) − 𝐷. It is evident that 𝐷 ∪ {𝑢} constitutes a minimal strong 

regular dominating set within 𝑚(𝐺). Consequently, 

|𝐷 ∪ {𝑢}| ∪ |𝑆"| < |𝐸| + |𝑆′| + |𝐴| ⟹ 𝛾𝑠𝑡𝑟(𝑚(𝐺) + 𝛾𝑐(𝐺) < 𝑑𝑖𝑎𝑚(𝐺) + 𝛼0(𝐺) + 𝛾(𝐺) 

The following outcome has been ascertained. 𝛾𝑠𝑡𝑟(𝑚(𝑇)) concerning 𝛾(𝑇) and 𝛾𝑐(𝑇). 

Theorem-3.14: For any tree T, 𝛾𝑠𝑡𝑟(𝑚(𝑇)) ≤ 𝛾(𝑇) + 𝛾𝑐(𝑇). 

Proof: Assume that 𝑋 = {𝑣𝑖; 1 ≤ 𝑖 ≤ 𝑛} ⊆ 𝑉(𝑇)  is the smallest set of vertices that encompasses 

every edge in T. It is evident that 𝑋 makes up T's minimal dominating set, so that |𝑋| = 𝛾(𝑇). 

Assume that 〈𝑋′〉  is a connected sub graph of 𝑋. Then 𝑋′ is a connected dominating set in and of 

itself, so that |𝑋′| = 𝛾𝑐(𝑇). In the absence of this, 𝑋" = 𝑋′ ∪ {𝑥} forms a ‘minimal connected 

dominating set’ of T, and there exists at least one vertex 𝑥 ∈ 𝑉(𝑇) − 𝑋′. According to this, D creates 

a strong regular dominating set in 𝑚(𝑇), so that |𝐷| = 𝛾𝑠𝑡𝑟(𝑚(𝑇)). Consequently, 

|𝐷| ⊆ |𝑋| ∪ |𝑋′| ⟹ 𝛾𝑠𝑡𝑟(𝑚(𝑇)) ≤ 𝛾(𝑇) + 𝛾𝑐(𝑇). 

We require the following theorems in order to demonstrate our subsequent findings. 

Theorem-A[4]: For each graph G, 𝛾𝑚
−1(𝐺) < 𝛾𝑐(𝐺) + 𝛾𝑡(𝐺). 

Theorem-E[5]: In each graph G, 𝛾𝑚
′ (𝐺) + 𝑑𝑖𝑎𝑚(𝐺) ≤ 𝑝 + 𝛾(𝐺). 

Corollary-3.15: For each graph G, 𝛾𝑠𝑡𝑟(𝑚(𝐺)) + 𝛾𝑚
−1(𝐺) < 𝑝 + 𝛾𝑐(𝐺) + 𝛾𝑡(𝐺). 

Proof: Theorems 3.6 and A make it simple to demonstrate the aforementioned result.    

Corollary-3.16: For any ‘graph G’, 𝛾𝑠𝑡𝑟(𝑚(𝐺)) + 𝛾𝑐(𝐺) + 𝛾𝑚
′ (𝐺) < 2𝛼0(𝐺) + 𝛽0(𝐺) + 2𝛾(𝐺). 

Proof: Since 𝑝(𝐺) = 𝛼0(𝐺) + 𝛽0(𝐺), Theorem 3.13 and Theorem E lead to the above outcome. 

Finally, Nordhaus-Gaddum type outcomes are found at the end. 

Theorem-3.17: For any (𝑝, 𝑞)⁡connected graphs 𝐺 and �̅�, 

i) 𝛾𝑠𝑡𝑟(𝑚(𝐺)) + 𝛾𝑠𝑡𝑟(𝑚(�̅�)) ≤ 2𝑝 

ii) 𝛾𝑠𝑡𝑟(𝑚(𝐺)). 𝛾𝑠𝑡𝑟(𝑚(�̅�)) ≤ 𝑝𝑞 

4. Conclusion 

In the current study some results have been obtained on 𝛾𝑠𝑡𝑟(𝑚(𝐺)) in terms of several parameters 

of G, such as its vertices, edges, diameter and so on, as well as several domination parameters of G, 

such as edge domination, connected domination and total domination and many more. 



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