Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1296 
https://internationalpubls.com 

Closely Connected Domination Number in Corona Product of Graphs 

Gurusamy P1, * Angel Joy R𝟐, 

1 Research Scholar, Department of Mathematics, Sri G.V.G Visalakshi College for Women, Udumalpet. 

*Assistant Professor, Department of Mathematics. Government Arts and Science College, Kangeyam. 

2 Assistant Professor, Department of Mathematics, Sri G.V.G Visalakshi College for Women, Udumalpet 

E.Mail : 1gurusamymathsgasckgm@gmail.com ,E. Mail : 2angeljoy@gvgvc.ac.in 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  

    Let 𝐺 = ሺ𝑉, 𝐸ሻ be a simple connected graph. The vertices 𝑢, 𝑣 ∈ 𝑉 are closely connected if atleast one 

of the shortest paths connecting them is not a cut path. A set 𝑆 of vertices of a simple graph 𝐺 = ሺ𝑉, 𝐸ሻ 

is a CC-dominating set (closely connected dominating set) if for every vertex 𝑣 ∈ 𝑉\𝑆 there exist a 

vertex 𝑢 ∈ 𝑆 such that 𝛤𝑐𝑐ሺ𝑢, 𝑣ሻ ≥ 1, where 𝛤𝑐𝑐ሺ𝑢, 𝑣ሻ is the number of shortest paths connecting 𝑢 and 

𝑣 except the cut paths. The minimum cordiality of a CC-dominating set is called the CC-domination 

number, denoted by 𝛾𝑐𝑐ሺ𝐺ሻ. This paper evaluates CC- domination Number in Corona product of some 

standard graphs. 

Objectives: Closely connected domination is new direction of Domination in Graphs, here find closely 

connected domination number for corona product of path, cycle with some graphs  

Methods: Consider the Graph G is undirected connected simple graph. The closely connected 

domination number (CC- domination number) for product of graphs, which is represented as  𝛾𝑐𝑐ሺ𝐺ሻ is 

the minimum cardinality of Closely connected dominating set. 

Keywords: closely-connected vertices, CC-degree of a vertex, CC-Domination number, Corona product 

of graphs 

2020 Mathematics Subject Classification: 05C40, 05C07, 05C69, 05C76. 

1. Introduction  

       In our day-to-day life, shortest route is a path between two destinations which traverses the 

minimal distance over the network. Shortest route between different location cannot always be 

preferable and we prefer routes that optimize the cost and time. This is possible only if another path 

exists between the locations, even if the shortest route is not approachable. In graph theoretically, 

places are considered as vertices and paths are edges in a graph. Then our aim is to find for those vertex 

pairs which does not alter the connectivity of the graph. More precisely the vertex pairs do not 

disconnect the graph even if the shortest path between them is deleted. For this idea K. Priya and V. 

Anilkumar introduced closely connected vertices in [4]. Let 𝐺 = ሺ𝑉, 𝐸ሻ, the vertices 𝑢, 𝑣 ∈ 𝑉 are 

closely connected if atleast one of the shortest paths connecting them is not a cut path. 

The concept of domination discussed in [2,3] entirely depending upon the adjacency property of 

vertices in a graph. But adjacency is not at all sufficient to characterize the vertex pairs in a graph as 

the deletion of the edges linked by adjacent vertices may or may not disconnect the graph. A set 𝑆 ⊆

𝑉ሺ𝐺ሻ is called a dominating set of 𝐺 if every vertex in 𝑉ሺ𝐺ሻ − 𝑆 is adjacent to some vertex in 𝑆. The 

domination number 𝛾ሺ𝐺ሻ of 𝐺 is the minimum cardinality of its dominating sets. 

          The concept of closely connected domination introduced by K. Priya and V. Anilkumar in [4] . 

mailto:gurusamymathsgasckgm@gmail.com
mailto:angeljoy@gvgvc.ac.in
https://d.docs.live.net/dfc4d13c8eba2327/Desktop/CC-%20Domination%20zip%20file.zip
https://d.docs.live.net/dfc4d13c8eba2327/Desktop/CC-%20Domination%20zip%20file.zip
https://d.docs.live.net/dfc4d13c8eba2327/Desktop/CC-%20Domination%20zip%20file.zip
https://d.docs.live.net/dfc4d13c8eba2327/Desktop/CC-%20Domination%20zip%20file.zip
https://d.docs.live.net/dfc4d13c8eba2327/Desktop/CC-%20Domination%20zip%20file.zip
https://d.docs.live.net/dfc4d13c8eba2327/Desktop/CC-%20Domination%20zip%20file.zip
https://d.docs.live.net/dfc4d13c8eba2327/Desktop/CC-%20Domination%20zip%20file.zip


Communications on Applied Nonlinear Analysis 

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A subset 𝑆 of 𝑉 is closely connected dominating set(abbreviated as CC-dominating set) if for every 

vertex 𝑣 ∈ 𝑉\𝑆 there exist a vertex 𝑢 ∈ 𝑆 such that 𝛤𝐶𝐶ሺ𝑢, 𝑣ሻ ≥ 1 where 𝛤𝐶𝐶ሺ𝑢, 𝑣ሻ is cardinality of 

{𝑝|𝑃ሺ𝑢, 𝑣ሻ|𝑝 is a not a cut path in 𝐺}, where 𝑃ሺ𝑢, 𝑣ሻ is the set of all shortest paths linking 𝑢 and 𝑣 in 

𝐺. The minimum cardinality of CC-dominating set is called CC-domination number, denoted by 

𝛾𝑐𝑐ሺ𝐺ሻ. The open CC-neighbourhood of a vertex 𝑣 ∈ 𝑉 is the set 𝑁𝑐𝑐ሺ𝑣ሻ = {𝑢 ∈ 𝑉: 𝛤𝑐𝑐ሺ𝑢, 𝑣ሻ ≥ 1}, 

whereas the closed CC-neighbourhood of 𝑉 is defined as 𝑁𝑐𝑐[𝑣] = 𝑁𝐶𝐶ሺ𝑣ሻ ∪ {𝑣}. The cardinality of 

𝑁𝐶𝐶ሺ𝑣ሻ is the CC-degree of 𝑣, denoted by deg𝐶𝐶ሺ𝑣ሻ. The vertex 𝑣 is said to be CC-isolated if 𝑁𝑐𝑐ሺ𝑣ሻ =

𝜙. 

2. Definitions and Previous Results 

Definition 2.1 [4] 

        A subset 𝑆 of 𝑉 is closely connected dominating set(abbreviated as CC-dominating set) if for 

every vertex 𝑣 ∈ 𝑉\𝑆 there exist a vertex 𝑢 ∈ 𝑆 such that 𝛤𝐶𝐶ሺ𝑢, 𝑣ሻ ≥ 1 where 𝛤𝐶𝐶ሺ𝑢, 𝑣ሻ is cardinality 

of {𝑝|𝑃ሺ𝑢, 𝑣ሻ|𝑝 is a not a cut path in 𝐺}, where 𝑃ሺ𝑢, 𝑣ሻ is the set of all shortest paths linking 𝑢 and 𝑣 

in 𝐺. 

Definition 2.2[4] 

The minimum cardinality of CC-dominating set is called CC-domination number, denoted by 𝛾𝑐𝑐ሺ𝐺ሻ. 

     In this paper, we investigate the CC-domination number of Corona product of paths, cycles and 

some standard graphs. For a graph 𝐺 of order 𝑛, the following are some basic results of 𝛾𝑐𝑐ሺ𝐺ሻ in [4]. 

1. 1 ≤ 𝛾𝑐𝑐ሺ𝐺ሻ ≤ 𝑛. 

2. 𝛾𝑐𝑐ሺ𝐺ሻ = 𝑛 iff 𝐺 is acyclic. 

3. For path 𝑃𝑛, 𝛾𝑐𝑐ሺ𝑃𝑛ሻ = 𝑛. 

4. If 𝐺 has no cut edge, then 𝛾𝑐𝑐ሺ𝐺ሻ ≤ 𝛾ሺ𝐺ሻ. 

Theorem 2.3.[4]   Let 𝐺 be a graph and 𝑢 ∈ 𝑉ሺ𝐺ሻ be CC-isolated. Then 𝑢 belongs to every CC-

dominating set of 𝐺. 

Theorem 2.4.[4]   For the complete graph 𝐾𝑛. 𝛾𝑐𝑐ሺ𝐾𝑛ሻ = {
2  if 𝑛 = 2
1  if 𝑛 ≠ 2

 

Theorem 2.5.  For the cycle 𝐶𝑛, 𝛾𝑐𝑐ሺ𝐶𝑛ሻ = 𝛾ሺ𝐶𝑛ሻ = ⌈
𝑛

3
⌉ , ∀𝑛 ∈ 𝑁. 

Proof. In 𝐶𝑛 every vertex is closely connected only with its adjacent vertices, so we conclude the above 

result.    

3.  Closely connected Domination number in Corona product of graphs 

  The Corona of two graphs 𝐺1 and 𝐺2 has been defined by Frucht and Harary in [1] to be the graph 𝐺 

formed from one copy of 𝐺1 and |𝑉ሺ𝐺1ሻ| copies of 𝐺2, where the 𝑖th  vertex of 𝐺1 is adjacent to every 

vertex in the 𝑖th  copy of 𝐺2 and is denoted by 𝐺 = 𝐺1 ∘ 𝐺2. 

Theorem 3.1[2]   For the Path 𝐺 = 𝑃𝑛, 𝛾ሺ𝐺ሻ = ⌈
𝑛

2
⌉. 



Communications on Applied Nonlinear Analysis 

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Theorem 3.2.[4]   For the path 𝐺 = 𝑃𝑛, 𝛾𝑐𝑐ሺ𝐺ሻ = 𝑛. 

Theorem 3.3.  For 𝑛 ≥ 1, 𝛾𝑐𝑐ሺ𝑃𝑛 ∘ 𝐾1ሻ = 2𝑛. 

Proof. Let 𝐺 = 𝑃𝑛 ∘ 𝐾1, let 𝑉ሺ𝑃𝑛ሻ = {𝑠1, 𝑠2, … , 𝑠𝑛} and 𝑡𝑖 be the vertices of the 𝑖th  copy of 𝐾1 joined 

to 𝑠𝑖. Then 𝑉ሺ𝐺ሻ = {𝑠1, 𝑠2, … , 𝑠𝑛} ∪ {𝑡1, 𝑡2, ⋯ , 𝑡𝑛}. Every vertex in 𝑉ሺ𝐺ሻ is CC-isolated vertex, By the 

Theorem 1 we have all the CC-isolated vertices must belong to every CC-dominating set. The only 

one CC-dominating set is 𝑉ሺ𝐺ሻ. Therefore 𝛾𝑐𝑐ሺ𝐺ሻ = |𝑉ሺ𝐺ሻ| = 2𝑛.                                            

Theorem 3.4.  For 𝑛 ≥ 1, 𝛾𝑐𝑐ሺ𝑃𝑛 ∘ 𝐾2ሻ = 𝑛. 

Proof. Let 𝐺 = 𝑃𝑛 ∘ 𝐾2. Let 𝑉ሺ𝑃𝑛ሻ = {𝑠1, 𝑠2, … , 𝑠𝑛} and {𝑡𝑖1, 𝑡𝑖2} be the vertex set of the 𝑖th copy of 𝐾2 

attached with 𝑠𝑖. Then 𝑉ሺ𝐺ሻ = {𝑠1, 𝑠2, 𝑠3, … , 𝑠𝑛} ∪ {𝑡11, 𝑡12} ∪ {𝑡21, 𝑡22} ∪

⋯ {𝑡𝑛1, 𝑡𝑛2} 𝑎𝑛𝑑|𝑉ሺ𝐺ሻ| = 3𝑛. In Figure 1,Every vertex 𝑠𝑖 of 𝑃𝑛 is closely connected with 𝑡𝑖1 and 𝑡𝑖2 

but not closely connected with 𝑠𝑖−1 and 𝑠𝑖+1. i.e., 𝛤𝑐𝑐ሺ𝑠𝑖, 𝑡𝑖1ሻ ≥ 1, 𝛤𝑐𝑐ሺ𝑠𝑖, 𝑡𝑖2ሻ ≥ 1 and 𝛤𝑐𝑐ሺ𝑠𝑖, 𝑠𝑖−1ሻ =

0 and 𝛤𝑐𝑐ሺ𝑠𝑖, 𝑠𝑖+1ሻ = 0. So, one of the minimums 𝐶𝐶-dominating set 𝐷 = {𝑠1, 𝑠2, … , 𝑠𝑛}. Since all 

vertices in 𝐺 is closely connected to at least one vertex in 𝐷. On removing one vertex in 𝐷, it is not a 

CC-dominating set. Hence 𝛾𝑐𝑐ሺ𝐺ሻ = |𝐷| = 𝑛. 

.  

Figure 1 The Corona product 𝑷𝒏 ∘ 𝑲𝟐 

  

Theorem 3.6. The CC-domination number of 𝑃𝑛 ∘ 𝐾𝑚 is 𝛾𝑐𝑐ሺ𝑃𝑛 ∘ 𝐾𝑚ሻ = 𝑛 for  𝑛 ≥ 1 and 𝑚 > 1. 

Proof. Let 𝐺 = 𝑃𝑛 ∘ 𝐾𝑚, 𝑉ሺ𝑃𝑛ሻ = {𝑠1, 𝑠2, … , 𝑠𝑛}. Let {𝑡𝑖1, 𝑡𝑖2, 𝑡𝑖3, … , 𝑡𝑖𝑚} be the vertex set of the 𝑖th 

copy of 𝐾𝑚 joined with the vertex 𝑠𝑖. 

∴  𝑉ሺ𝑃𝑛 ∘ 𝐾𝑚ሻ = {𝑠1, 𝑠2, … 𝑠𝑛, 𝑡11, 𝑡12, … , 𝑡1𝑚, 𝑡21, 𝑡22, … , 𝑡2𝑚, ⋯ , 𝑡𝑛1, 𝑡𝑛2, 𝑡𝑛3, … 𝑡𝑛𝑚} 

For 𝑗 = 1 to 𝑚, 𝑆𝑗 = {𝑡1𝑗, 𝑡2𝑗 , … , 𝑡𝑛𝑗} and 𝑆 = {𝑠1, 𝑠2, … , 𝑠𝑛} are some CC-dominating sets of 𝐺 and 

|𝑆| = |𝑆𝑗| = 𝑛, since all vertices in 𝐺 is closely connected to atleast one vertex in 𝑆 (or) 𝑆𝑗. On 

removing one vertex in 𝑆 (or) 𝑆𝑗, it is not CC-dominating set. Hence, we conclude 𝛾𝑐𝑐ሺ𝐺ሻ = 𝑛.    

Theorem 3.7   The CC-domination number of 𝐶𝑚 ∘ 𝐾1 is 𝛾𝑐𝑐ሺ𝐶𝑚 ∘ 𝐾1ሻ = 𝑚 + ⌈
𝑚

3
⌉ 

Proof. Let 𝐺 = 𝐶𝑚 ∘ 𝐾1 and let 𝑉ሺ𝐺ሻ = {𝑢1, 𝑢2, … , 𝑢𝑚} ∪ {𝑣1, 𝑣2, … , 𝑣𝑚}. Let 𝑣𝑖 be the vertices of the 

𝑖th  copy of 𝐾1 is joined to 𝑢𝑖, then |𝑉ሺ𝐺ሻ| = 2𝑚. Let 𝐷 = {𝑣1, 𝑣2, … , 𝑣𝑚} be the set of all pendent 

vertices of 𝐺 and is also CC-isolated vertices. So 𝐷 is subset of every CC-dominating set of 𝐺. Further 



Communications on Applied Nonlinear Analysis 

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each vertex 𝑢𝑖 in 𝐺 is closely connected with its adjacent vertices. i.e., 𝑁𝑐𝑐ሺ𝑢𝑖ሻ = {𝑢𝑖−1, 𝑢𝑖+1} for 2 ≤

𝑖 ≤ 𝑛 − 1. So, from the Figure.2, the set 𝑆 = {𝑣1, 𝑣2, … , 𝑣𝑚, 𝑢2, 𝑢5, … , 𝑢𝑚−1} be the one of the 

minimum CC-dominating set of 𝐺 . By using theorem 1.3, the cardinality of S is 𝑚 + ⌈
𝑚

3
⌉. So, every 

vertex in 𝐺 is closely connected to atleast one of the vertices in 𝑆. Hence, we conclude 𝛾𝑐𝑐ሺ𝐺ሻ = 𝑚 +

⌈
𝑚

3
⌉ 

 

 

Theorem 3.8.  The CC-domination number of 𝐶𝑚 ∘ 𝐾2 is 𝛾𝑐𝑐ሺ𝐶𝑚 ∘ 𝐾2ሻ = ⌈
𝑚

3
⌉  for  𝑚 ≥ 2. 

Proof. Let 𝐺 = 𝐶𝑚 ∘ 𝐾2, 𝑉ሺ𝐶𝑚ሻ = {𝑢1, 𝑢2, … , 𝑢𝑚}. Let {𝑣𝑖1 , 𝑣𝑖2} be the vertex set of the 𝑖th copy of 𝐾2 

joined with the vertex 𝑢𝑖. 

∴ 𝑉ሺ𝐺ሻ = {𝑢1, 𝑢2, … , 𝑢𝑚} ∪ {𝑣11, 𝑣12} ∪ {𝑣21, 𝑣22} ∪ ⋯∪ {𝑣𝑚1, 𝑣𝑚2} 

and |𝑉ሺ𝐺ሻ| = 3𝑚. Each vertex 𝑢𝑖 of 𝐶𝑚 is closely connected with 

𝑢𝑖−1, 𝑣𝑖+1, 𝑣𝑖1, 𝑣𝑖2, 𝑣ሺ𝑖−1ሻ1, 𝑣ሺ𝑖−1ሻ2, 𝑣ሺ𝑖+1ሻ1 and 𝑣ሺ𝑖+1ሻ2. So |𝑁𝐶𝐶ሺ𝑢𝑖ሻ| = 8. ∀𝑖 = 1,2, …𝑚. Then the 

set 𝐷1 = {𝑢1, 𝑢4, ⋯ , 𝑢𝑚−2} and 𝐷2 = {𝑢2, 𝑢5, ⋯ , 𝑢𝑚 − 1} are some CC-dominating sets of 𝐺 and it is 

also minimum CC-dominating set. Hence 𝛾𝑐𝑐ሺ𝐺ሻ = ⌈
𝑚

3
⌉.                                                                    

Theorem 3.9.  For 𝑛 ≥ 1 and 𝑚 > 1, 𝛾𝑐𝑐ሺ𝐶𝑚 ∘ 𝐾𝑛ሻ = ⌈
𝑚

3
⌉. 

Proof. Let 𝐺 = 𝐶𝑚 ∘ 𝐾𝑛, and 𝑉ሺ𝐶𝑚ሻ = {𝑢1, 𝑢2, … , 𝑢𝑚}. Let {𝑣𝑖1, 𝑣𝑖2, 𝑣𝑖3, ⋯ , 𝑣𝑖𝑛} be the vertex set of 

the 𝑖th  copy of 𝐾𝑛 joined with the vertex 𝑢𝑖. So 𝑉ሺ𝐺ሻ =

{𝑢1, 𝑢2, 𝑢3, … , 𝑢𝑚, 𝑣11, 𝑣12, … , 𝑣1𝑛, 𝑣21, 𝑣22, … , 𝑣2𝑛, … , 𝑣𝑚1, 𝑉𝑚2, … , 𝑣𝑚𝑛} and |𝑉ሺ𝐺ሻ| = 𝑚𝑛 +𝑚. For 

each 𝑢𝑖 in 𝐶𝑚 is closely connected with the following vertices 𝑢𝑖−1 , 𝑢𝑖+1, ሺ𝑖 − 1ሻth  copy of 𝐾𝑛,ሺ𝑖ሻth  

copy of 𝐾𝑛,ሺ𝑖 + 1ሻth  copy of 𝐾𝑛,then |𝑁𝐶𝐶ሺ𝑢𝑖ሻ| = 3𝑛 + 2. Then the set 𝐷 = {𝑢2, 𝑢5, ⋯ , 𝑢𝑚−1} is CC-

dominating set, on removing one vertex in 𝐷 it is not 𝐶𝐶-dominating set. Hence of 𝐷 is the minimal 

CC-dominating set and |𝐷| = ⌈
m

3
⌉. So, we conclude 𝛾𝑐𝑐ሺ𝐺ሻ = ⌈

𝑚

3
⌉.    



Communications on Applied Nonlinear Analysis 

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Theorem 3.10.  Let 𝐻1 , 𝐻2 be two connected graphs and 𝐻2 ≠ 𝐾1, then 𝛾𝑐𝑐ሺ𝐻1 ∘ 𝐻2ሻ = 𝛾𝑐𝑐ሺ𝐻1ሻ. 

Proof. Let 𝐺 = 𝐻1 ∘ 𝐻2. In 𝐺, the 𝑗th  vertex of 𝐻1 is adjacent to every vertex in the 𝑗th  copy of 𝐻2 . 

𝐻2 is non-trivial connected graph, and every pair of vertices of 𝐻2 in the 𝑖th  copy of 𝐻2 are closely 

connected. So, every vertex in the 𝑖th  copy of 𝐻2 is closely connected with the 𝑗th  vertex of 𝐻1. 

Suppose 𝐷 is the minimum CC-dominating set of 𝐻1, every vertex in 𝑉ሺ𝐻1ሻ − 𝐷 is closely connected 

with at lest one vertex in 𝐷 and also every vertex in 𝑉ሺ𝐺ሻ − 𝐷 is closely connected with atleast one 

vertex in 𝐷. 

Hence, we conclude that 𝛾𝑐𝑐ሺ𝐺ሻ = |𝐷| = 𝛾𝑐𝑐ሺ𝐻1ሻ.                                                                                    

Theorem 3.11.  Let 𝐻1 be a connected graph and 𝐻2 = 𝐾‾𝑛 (complement of complete graph 𝐾𝑛). Then 

𝛾𝑐𝑐ሺ𝐻1 ∘ 𝐻2ሻ = 𝛾𝑐𝑐ሺ𝐻1ሻ + |𝑉ሺ𝐻1ሻ||𝑉ሺ𝐻2ሻ|. 

Proof. Let 𝐺1 = 𝐻1 ∘ 𝐻2. In 𝐺 all vertices of 𝐻2 are pendent vertices and also CC-isolated vertices. ie, 

the 𝑗th vertex of 𝐻1 is adjacent to every vertex in the 𝑖th  copy of 𝐻2. since 𝐻2 is isolated graph, so we 

remove any edge between 𝐻1 and 𝐻2, then the graph 𝐺 is disconnected. Hence, we have 

|𝑉ሺ𝐻1ሻ||𝑉ሺ𝐻2ሻ| number of isolated vertices and also 𝐻2 does not alter the 𝐶𝐶-domination of 𝐻1. By 

theorem 1.1, every CC-isolated vertex must belong to every 𝐶𝐶-dominating set. Hence, we have 

𝛾𝑐𝑐ሺ𝐻1 ∘ 𝐻2ሻ = 𝛾𝑐𝑐ሺ𝐻1ሻ + |𝑉ሺ𝐻1ሻ||𝑉ሺ𝐻2ሻ|. 

   Corollary 3.12.  The following results are direct computation from Theorem 12. 

1. 𝛾𝑐𝑐ሺ𝐶𝑚 ∘ 𝑃𝑛ሻ = ⌈
𝑚

3
⌉ = 𝛾𝑐𝑐ሺ𝐶𝑚ሻ 

2. 𝛾𝑐𝑐ሺ𝑃𝑛 ∘ 𝑃𝑚ሻ = 𝑛 = 𝛾𝑐𝑐ሺ𝑃𝑛ሻ 

3. 𝛾𝑐𝑐ሺ𝐶𝑛 ∘ 𝐶𝑚ሻ = ⌈
𝑛

3
⌉ = 𝛾𝑐𝑐ሺ𝐶𝑛ሻ 

4. 𝛾𝑐𝑐ሺ𝑃𝑛 ∘ 𝐶𝑚ሻ = 𝑛 = 𝛾𝑐𝑐ሺ𝑃𝑛ሻ 

4. Applications 

        In traditional network is operated by all hosts for two-way communications. Every host has a 

service area and within this close-transmission covers the range, link failure of the network does not 

disconnect or disrupt the communication of the entire network. A pair of such hosts that are 

communication with one another are called as closely connected neighbors. Thus, closely connected 

neighbors serves as a new approach to design networks in [5]. 

5. Conclusion 

     In this paper, the concept of CC-domination number is studied using closely connected vertices of 

graphs for Corona product of graphs. Investigation of 𝐶𝐶-domination of various other products is a 

significant direction for further research. Also determine 𝛾𝑐𝑐ሺ𝐺ሻ for the family of graphs. 

 

 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1301 
https://internationalpubls.com 

References 

[1] Frucht, R., and Harary, F. “On the corona of two graphs.” Aeq. Math. 4, 322-325 (1970). 

[2] Haynes, Teresa W., Stephen Hedetniemi, and Peter Slater. Fundamentals of domination in graphs. 

CRC press, 2013. 

[3] Haynes, Teresa W. Domination in graphs: Volume 2: advanced topics. Routledge, 2017. 

[4] Priya, K., and Anil Kumar. CC-Domination in graphs, Palestine Journal of Mathematics, Vol - 

12(3), 2023. 

[5] Dekker, A.H, and Colbert, Networks and Robustness and Graph Topology, Proceedings of the 

27th  Australasian conference on Computer science, 26(2004), 359 - 368.