Communications on Applied Nonlinear Analysis 

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Vol 32 No. 10s (2025) 

 

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More Results on Complementary Tree Domination Number of  

Semi Total Point Graph 

 

S. Jayalakshmi1 and P. Vidhya2 

1Research Scholar (Part Time), School of Mathematics 

Madurai Kamaraj University, Madurai - 625021, Tamilnadu, India 

E-mail: jayark83@gmail.com 
2Associate Professor, Department of Mathematics, 

EMG Yadava Women’s College 

Madurai - 625014, Tamilnadu, India 

E-mail: vidhyaramman@gmail.com, p.vidhya-mat@emgywomenscollege.ac.in 
 

 

Article History: 

Received: 02-01-2025 

Revised: 25-02-2025 

Accepted: 20-03-2025 

Abstract:  

A set D ⊆ V of a graph G =  (V, E) is a complementary tree dominating set if the induced 

subgraph < 𝑉(G) − D > is a tree. The complementary tree domination number γctd(G) is 

the minimum cardinality of a complementary tree dominating set (ctd-set) of G. The semi 

total point graph T2(G) is the graph G whose vertex set is V(G) ∪ E(G). Where two 

vertices are adjacent if and only if (i) they are adjacent vertices of G or (ii) one is a vertex 

and the other is an edge of G incident with it. In this paper complementary tree 

domination number of semi total point graph of graphs, its bounds and relation between 

γctd(G) and γctd(T2(G)) are obtained. 

Keywords: Dominating set, Complementary tree dominating set, Semi total point graph. 

 

1. Introduction 

A Graph 𝐺(𝑉, 𝐸) discussed in this paper be a simple, finite, undirected, connected graph with p 

vertices and q edges. A set of vertices in a graph G is independent, if no two vertices are adjacent. 

The largest number of vertices in such a set is called the independence number and is denoted by 

β0(𝐺). The corona 𝐺1 ∘ 𝐺2 of two graphs 𝐺1 and 𝐺2 are defined as the graph G obtained by taking 

one copy of 𝐺1 of order 𝑝1 and 𝑝1 copies of 𝐺2 and then joining the 𝑖𝑡ℎ vertex of 𝐺1 to every vertex 

in the 𝑖𝑡ℎ copy of 𝐺2. The Corona 𝐺1 ∘ 𝐺2 has 𝑝1(1 + 𝑝2) vertices and 𝑞1 + 𝑝1𝑞2 + 𝑝1𝑝2 edges. The 

graph 𝐶𝑛
(𝑡)

 is the one point union of t cycles of length n. A graph G is unicyclic if it contains exactly 

one cycle. A broom graph 𝐵𝑛,𝑚 is a graph of n vertices which have a path 𝑃𝑚 and 𝑛 − 𝑚 pendant 

vertices, all of these vertices are adjacent to either the origin u or the terminus v of the path.  Any 

undefined term in this paper may be found in Harary [2].  

The concept of domination in graphs was introduced by Ore [5]. A set 𝐷 ⊆ 𝑉 is said to be a 

dominating set of G, if every vertex in 𝑉 − 𝐷 is adjacent to some vertex in D. The minimum 

cardinality of a dominating set is called the domination number of G and is denoted by γ(𝐺). The 

complementary tree domination number of a graph was introduced byS. Muthammai, M. 

Bhanumathi and P. Vidhya [4] have established some results on complementary tree domination 

number of graphs. A set 𝐷 ⊆ 𝑉(𝐺) is said to be complementary tree dominating set (ctd-set) if the 



Communications on Applied Nonlinear Analysis 

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induced subgraph ⟨𝑉(𝐺)  − 𝐷⟩ is a tree. The minimum cardinality of a ctd-set is called the 

complementary tree domination number of G and is denoted by γ𝑐𝑡𝑑(𝐺). E. Sampath Kumar and 

S.B. Chikkodimath[3] introduced the concept of semi total point graphs of a graph. Also B. 

Basavanagoud, S.M. Hosamani and S.H. Malghan[1] have obtained some results on domination 

number of semi total point graph. Given a graph G, the semi total point graph 𝑇2(𝐺) of G is the 

graph whose point set is 𝑉(𝐺) ∪ 𝐸(𝐺) where two points are adjacent if and only if (i) they are 

adjacent points of G or (ii) one is a point of G and the other is a line of G, incident with it. 

For notation convenience, an edge (𝑢1, 𝑢2) ∈ 𝐸(𝐺) then its corresponding edge vertex is denoted by 

𝑢12
′  in 𝑇2(𝐺).  

In this paper complementary tree domination number of semi total point graph of graphs, its bounds 

and relation between γ𝑐𝑡𝑑(𝐺) and γ𝑐𝑡𝑑(𝑇2(𝐺)) are obtained. 

2. Prior Results 

Observation 2.1. [6] 

(i) For the path 𝑃𝑝, γ
𝑐𝑡𝑑

(𝑇2(𝑃𝑝)) = 𝑝 − 1, where (𝑝 ≥ 2). 

 

(ii) For the cycle 𝐶𝑝, γ
𝑐𝑡𝑑

(𝑇2(𝐶𝑝)) = 𝑝 − 1, where (𝑝 ≥ 3). 

 

(iii) For the star graph 𝐾1,𝑝−1, γ
𝑐𝑡𝑑

(𝑇2(𝐾1,𝑝−1)) = 𝑝 − 1 or 𝑞, 𝑝 ≥ 2. 

 

(iv) For a complete graph 𝐾𝑝, 𝑝 ≥ 4 then γ
𝑐𝑡𝑑

(𝑇2(𝐾𝑝)) =
𝑝2−5𝑝+12

2
. 

 

(v) For a wheel graph 𝑊𝑝, 𝑝 ≥ 4γ
𝑐𝑡𝑑

(𝑇2(𝑊𝑝)) = 𝑝 where 𝑊𝑝 = 𝐶𝑝−1 + 𝐾1 for (𝑝 ≥ 4). 

 

(vi) For a corona graph 𝑃𝑝 ∘ 𝐾1, 𝑝 ≥ 2 then γ
𝑐𝑡𝑑

(𝑇2(𝑃𝑝 ∘ 𝐾1)) = 2𝑝 − 1. 

 

(vii) For a corona graph 𝐶𝑝 ∘ 𝐾1, 𝑝 ≥ 3 then γ
𝑐𝑡𝑑

(𝑇2(𝐶𝑝 ∘ 𝐾1)) = 2𝑝. 

 

(viii) γ
𝑐𝑡𝑑

(𝑇2(𝐾1 + 𝑃𝑛)) = 𝑝, 𝑝 ≥ 3. 

 

Proposition 2.2. [6]  

Let 𝐺(𝑝, 𝑞) be a connected graph with δ(𝐺) ≥ 2 then atleast one vertex of G is the member of 

ctd-set of 𝑇2(𝐺). 

 

Proposition 2.3. [6] 

For any connected graph 𝐺(𝑝, 𝑞) with 𝑝 ≥ 2,⌈
𝑝

Δ(𝐺)+1
⌉ ≤ γ𝑐𝑡𝑑(𝑇2(𝐺)) ≤ 𝑝 + 𝑞 − 2. 

 

Theorem 2.4. [6] 

γ𝑐𝑡𝑑(𝑇2(𝐺)) = 1 if and only if 𝐺 ≅ 𝐾2. 

 

 

 



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3. Characterisation of Complementary Tree Dominating Set in Semi Total Point Graph T2(G) 

In the following, a necessary and sufficient condition for a ctd-set of a graph G to be a ctd-set of 

its semi total point graph 𝑇2(𝐺) is found. 

 

Theorem 3.1. 

A ctd-set D of a connected graph 𝐺 =  (𝑉, 𝐸) is also a ctd-set of 𝑇2(𝐺) if and only if  

 

(i) ⟨𝐷 ⟩ has an isolated vertices. 

 

(ii) For each 𝑣 ∈ 𝐷, 𝑁(𝑣) ∩ (𝑉 − 𝐷) ≠ and 

 

(iii) ⟨𝑉 − 𝐷⟩ ≅ 𝐾1 and if 𝑣 ∈ 𝑉 −  𝐷 then 𝑁(𝑣) ∩ 𝐷 ≠  . 

 

Proof. 

Let D be a ctd-set of both G and 𝑇2(𝐺). 

(i) Let 𝑣 ∈ 𝐷 be not an isolated vertex in ⟨𝐷 ⟩ then its edge vertex 𝑣′ in 𝑇2(𝐺) is isolated 

in⟨𝑉(T2(G))  −  D ⟩ which contradicts the ctd-set of 𝑇2(𝐺). Therefore, ⟨𝐷 ⟩ has an isolated 

vertices.  

 

(ii) Let there exists a vertex 𝑣 ∈ 𝐷 such that 𝑁(𝑣) ∩ (𝑉 − 𝐷) =  .  Then, its edge vertex 𝑣′ is 

isolated in ⟨𝑉(𝑇2(𝐺))  −  𝐷 ⟩.  
 

(iii) If 𝐾2 is an induced subgraph of ⟨𝑉 −  𝐷 ⟩. Then ⟨𝑉(𝑇2(𝐺))  −  𝐷 ⟩ contains a cycle. Therefore, 
⟨𝑉 − 𝐷⟩ ≅ 𝐾1. Let 𝑣 ∈ 𝐾1. Since G is connected so that remaining vertices are adjacent to v.  

There exists a edge vertices 𝑣𝑖
′ ∈ (𝑣, 𝑣𝑖) such that ⟨𝑣, 𝑣𝑖, 𝑣𝑖

′⟩ ≅ 𝐶3 in 𝑇2(𝐺).Hence            

𝑁(𝑣) ∩ 𝐷 ≠  . 

Conversely, if (i) is true, D is dominating set of 𝑇2(𝐺). If (ii) holds, then ⟨𝑉(𝑇2(𝐺))  −  𝐷 ⟩ is 

connected and if (iii) holds, then ⟨𝑉(𝑇2(𝐺))  −  𝐷 ⟩ is acyclic. Therefore, ⟨𝑉(𝑇2(𝐺))  −  𝐷 ⟩ is a tree.  

Hence, D is also a ctd-set of 𝑇2(𝐺).                                                                              □ 

 

In the following, exact values of complementary tree domination number of semi total point graph 

of some classes of graphs are given. 

 

Proposition 3.2. 

Let 𝐶𝑝
(𝑡)

, 𝑡 ≥ 2 be the one point union of t cycles of length p(𝑝 ≥ 3) thenγ𝑐𝑡𝑑 (𝑇2(𝐶𝑝
(𝑡)

)) =

(𝑝 − 1)𝑡, 𝑝 ≥ 3. 

Proof. 

Let 𝐺 = 𝐶𝑝
(𝑡)

 and u be the point of union of t cycles of length p. Let the vertex set of 𝑘𝑡ℎ cycle in 

𝐶𝑝
(𝑡)

 be 𝑉𝑘 = {𝑢, 𝑢𝑘1, 𝑢𝑘2, … , 𝑢𝑘,𝑝−1}𝑘 = 1,2, … , 𝑡(𝑡 ≥ 2)  𝑉𝑘 (𝑇2(𝐶𝑝
(𝑡)

)) = {𝑢, 𝑢𝑘1, 𝑢𝑘2, … , 𝑢𝑘,𝑝−1} ∪

{𝑢𝑘1
′ , 𝑢𝑘2

′ … 𝑢𝑘𝑝
′ } where 𝑢𝑘1

′ , 𝑢𝑘2
′ , … , 𝑢𝑘𝑝

′  are the corresponding edge vertices of 

(𝑢, 𝑢𝑘1), (𝑢𝑘1, 𝑢𝑘2), … , (𝑢, 𝑢𝑘,𝑝−1). Let 𝐷𝑘 = {𝑢𝑘1, 𝑢𝑘3
′ , … , 𝑢𝑘𝑝

′ }, 𝑘 =  1, 2, … , 𝑡  𝐷 = ⋃ 𝐷𝑘
𝑡
𝑘=1 ⊆

𝑉(𝑇2(𝐺)). Then ⟨𝑉(𝑇2(𝐺))  −  𝐷 ⟩ is a tree. Hence D is a minimum ctd-set of 𝑇2(𝐺). Therefore, 

|𝐷| = γ𝑐𝑡𝑑(𝑇2(𝐺)) = (𝑝 − 1)𝑡.                                                                  □ 



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Proposition 3.3. 

Let G be a unicyclic graph by attaching a path of length (𝑛 ≥ 1) to the 𝑡(≤ 𝑝) consecutive 

vertices of 𝐶𝑝(𝑝 ≥ 3). Then γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑛𝑡 + 𝑝 − 1. 

 

Proof. 

In the cycle 𝐶𝑝(𝑝 ≥ 3) say 𝑣1, 𝑣2, … , 𝑣𝑝. Consider a path of length 𝑃𝑝−1 in 𝐶𝑝 say 𝑣1, 𝑣2, … , 𝑣𝑝−1 

and attach a path 𝑃𝑛
′(𝑛 ≥ 1) say 𝑣𝑖 , 𝑢2, … , 𝑢𝑛, 𝑖 =  1, 2, … , 𝑡 to the 𝑡(≤ 𝑝) consecutive vertices of 

𝐶𝑝. In 𝑇2(𝐺), the set of all edge vertices of path 𝑃𝑛
′ of t consecutive vertices of 𝐶𝑝, edge vertices of a 

path 𝑃𝑝−1 and a vertex 𝑣𝑝 forms a minimum ctd-set of 𝑇2(𝐺). Therefore, γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑛𝑡 + 𝑝 − 1.                                                                   

                                                                                                                                                    □ 

  

Corollary 3.4. 

G be a unicyclic graph by attaching one pendant vertex to exactly one vertex of 𝐶𝑝 then 

γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝. 

 

Corollary 3.5. 

G be a unicyclic graph by attaching one pendant vertex to 𝑝 − 1 vertices of 𝐶𝑝. Then 

γ𝑐𝑡𝑑(𝑇2(𝐺)) = 2𝑝 − 2. 

 

Corollary 3.6. 

G be a unicyclic graph by attaching a path of length n to exactly one vertex of 𝐶𝑝 

thenγ𝑐𝑡𝑑(𝑇2(𝐺)) =  𝑝 + 𝑛 + 1. 
 

4. Bounds and Some Exact Values for the Complementary Tree Domination Number of Semi 

Total Point Graph of Graphs 

Theorem 4.1. 

γ𝑐𝑡𝑑(𝑇2(𝐺)) = 2 if and only if 𝐺 ≅ 𝐾1,2 or 𝐶3. 

 

Proof. 

Let D be a γ𝑐𝑡𝑑-set of 𝑇2(𝐺) such that |𝐷| = 2.   

Let 𝐷 = {𝑢1, 𝑢2} where 𝑢1, 𝑢2 ∈ 𝑉(𝑇2(𝐺)). 

 

Case 1. 

𝑢1 and 𝑢2 are vertices in G. Then, D is also a γ𝑐𝑡𝑑-set of G. By Theorem 3.1, it can be seen that 

 𝐺 ≅ 𝐾1,2. 

 

Case 2. 

𝑢1, 𝑢2 ∈ 𝑉(𝑇2(𝐺)) − 𝑉(𝐺). Let 𝑢1 = 𝑢1
′  and 𝑢2 = 𝑢2

′  where (𝑢, 𝑢1) and (𝑢, 𝑢2) ∈ 𝐸(𝐺). Since 

𝑢1
′  and 𝑢2

′  are edge vertices in 𝑇2(𝐺) and ⟨𝑉(𝑇2(𝐺))  −  𝐷⟩ is connected and acyclic. Hence it can be 

seen that  𝐺 ≅ 𝐾1,2. 

 

Case 3. 

Let 𝑢1 ∈ 𝑉(𝐺) and 𝑢2 = 𝑢2
′ ∈ 𝑉(𝑇2(𝐺)) − 𝑉(𝐺) 

 

 



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Sub case 3.1.    𝑢2 = 𝑢2
′ . 

That is 𝐷 = {𝑢1, 𝑢2
′ } is a γ𝑐𝑡𝑑-set of 𝑇2(𝐺).  Hence 𝐺 ≅ 𝐶3. 

 

Sub case 3.2.  𝑢2 ≠ 𝑢2
′ . 

Let 𝑢2 = 𝑣′ for some (𝑢1, 𝑣) ∈ 𝐸(𝐺) and 𝑣′ ≠ 𝑢2
′ . Then 𝐷 = {𝑢, 𝑣′} is a γ𝑐𝑡𝑑-set of 𝑇2(𝐺). 𝑢2

′ ≠
𝑣′ implies that ⟨𝑢2

′ , 𝑣, 𝑥 ⟩ form a cycle in ⟨𝑉(𝑇2(𝐺))  −  𝐷 ⟩ forms either cycle or disconnected where 

𝑥, 𝑣 ∈ 𝑉(𝐺). 

 

Conversely, if 𝐺 ≅ 𝐾1,2 or 𝐶3. Then γ𝑐𝑡𝑑(𝑇2(𝐺)) = 2.                                                                □ 

 

Theorem 4.2. 

γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝 + 𝑞 − 2 if and only if 𝐺 ≅ 𝐾2. 

 

Proof. 

Assume γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝 + 𝑞 − 2. Let D be a minimum ctd-set of 𝑇2(𝐺) having 𝑝 + 𝑞 − 2 

vertices. Since ⟨𝑉(𝑇2(𝐺)) − 𝐷⟩ ≅ 𝐾2. Let 𝑉(𝑇2(𝐺)) − 𝐷 = {𝑢, 𝑣} either  

(i) 𝑢, 𝑣 ∈ 𝑉(𝐺) or 

 

(ii) 𝑢 ∈ 𝑉(𝐺) and 𝑣 = 𝑢′ ∈ 𝑉(𝑇2(𝐺)) − 𝑉(𝐺) where 𝑢′ ∈ (𝑢, 𝑣). 

 

Case 1. 

Let 𝑢′ ∈ 𝐷 where 𝑢′ be the edge vertex of (𝑢, 𝑣) in 𝑇2(𝐺). Therefore, no vertex of 𝑉(𝐺) is an 

element of D hence 𝑢′ ∈ 𝐷. Therefore 𝐺 ≅ 𝐾2. 

 

Case 2. 

Let 𝑢 ∈ 𝑉(𝐺), 𝑢′ ∈ 𝑉(𝑇2(𝐺)) − 𝑉(𝐺). Since ⟨𝑉(𝑇2(𝐺)) − 𝐷⟩ ≅ 𝐾2. 𝑢′ is adjacent to u in   

⟨𝑉(𝑇2(𝐺))  −  𝐷 ⟩.  That is v is adjacent to a vertex u in G. Therefore, 𝐺 ≅ 𝐾2. 

 

Conversely, if 𝐺 ≅ 𝐾2 then γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝 + 𝑞 − 2.                                                                 □ 

 

Remark 4.3. 

If 𝑝 ≥ 3 then γ𝑐𝑡𝑑(𝑇2(𝐺)) ≤ 𝑝 + 𝑞 − 3. Equality holds if 𝐺 ≅ 𝑃3.                         

 

Theorem 4.4. 

Let 𝐺(𝑝, 𝑞) be a complete graph with 4 ≤ 𝑝 ≤ 8 then γ𝑐𝑡𝑑(𝑇2(𝐺)) ≤ ⌈
𝑝+𝑞

2
⌉. 

 

Proof. 

We prove induction on p. Let 𝑝 =  4, 𝑒 = (𝑢1, 𝑢2) ∈ 𝐸(𝐺) where 𝑢1, 𝑢2 ∈ 𝑉(𝐺). Let 𝐷 =
{𝑢1, 𝑢2, 𝑢12

′ , 𝑢34
′ } ⊆ 𝑉(𝑇2(𝐺)) is a ctd-set of 𝑇2(𝐺) where 𝑢12

′ , 𝑢34
′ ∈ 𝑉(𝑇2(𝐺)) − 𝑉(𝐺) and 𝐷′ =

𝑉(𝑇2(𝐺)) − 𝐷 = {𝑢3, 𝑢4, 𝑢13
′ , 𝑢23

′ , 𝑢24
′ , 𝑢41

′ } since 𝑝 ≥ 4 and δ(𝐺) ≥ 2 each vertex in ⟨𝑉(𝑇2(𝐺))  −

 𝐷′ ⟩ is adjacent to atleast one vertex in D and ⟨𝑉(𝑇2(𝐺)) − 𝐷⟩ ≅ 𝑆𝑚,𝑚,𝑚, 𝑚 ≥ 2 in 𝑇2(𝐺). 

Therefore, |𝐷| ≤ ⌈
𝑝+𝑞

2
⌉. Hence γ𝑐𝑡𝑑(𝑇2(𝐺)) ≤ ⌈

𝑝+𝑞

2
⌉. Equality holds if 𝐺 ≅ 𝐾8.                 □ 

 

 

 



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Theorem 4.5. 

Let G be a connected graph such that δ(𝐺) ≥ 2 then γ𝑐𝑡𝑑(𝑇2(𝐺)) ≤ 𝑞 − Δ(𝐺) + 1. 

 

Proof. 

Let v be a vertex of maximum degree in G. Let 𝑆 = {𝑣𝑖
′: (𝑣, 𝑣𝑖) ∈ 𝐸(𝐺), 𝑖 = 1, … , Δ(𝐺)}.  Any set 

𝐷 ⊆ 𝑉(𝑇2(𝐺)) such that V(𝑇2(𝐺)) − D = {⋃ 𝑣𝑖
𝑝−1
𝑖=1 } ∪ 𝑆. Since δ(𝐺) ≥ 2, 𝑑𝑒𝑔(𝑣𝑖) ≥ 2, 𝑖 =

1,2, … , 𝑝 − 1 and hence 𝑣𝑖 is adjacent to the vertices of G other than v, Then 𝑣𝑖
′ is adjacent to the 

vertex 𝑣𝑖, 𝑖 =  1, 2, … , 𝑝 − 1 where 𝑣𝑖 is a vertex in 𝑉(𝑇2(𝐺)) − 𝐷.  Also, v is adjacent to atleast one 

vertex of G and hence in S. Therefore, D is a dominating set of 𝑉(𝑇2(𝐺)).  Moreover ⟨𝑉(𝑇2(𝐺)) −

𝐷⟩ ≅ 𝑇 ∘ 𝐾1 and hence D is a ctd-set of 𝑇2(𝐺). Therefore, 

γ𝑐𝑡𝑑(𝑇2(𝐺)) ≤ |𝑉(𝑇2(𝐺)) − (𝑉(𝐺) − 𝑣) − 𝑆| 

 = 𝑝 + 𝑞 − 𝑝 + 1 − Δ(𝐺) 

 ≤ 𝑞 − Δ(𝐺) + 1 

Equality holds if 𝐺 ≅ 𝐾3, 𝐾4&𝐾1 + 𝑃𝑛−1.                                                                   □ 

 

Theorem 4.6. 

Let 𝑇2(𝐺1(𝑝1, 𝑞1)) and 𝑇2(𝐺2(𝑝2, 𝑞2)) be two connected graphs of order atleast two. Let T be an 

induced sub-graph of 𝑇(𝐺1) having maximum number of vertices such that T is a tree. If β0 is the 

independence number of 𝑇2(𝐺2) and vertices corresponding to the edge joining from copies of 𝑇(𝐺2) 

to 𝑇(𝐺1) then  

γ𝑐𝑡𝑑(𝑇2(𝐺1 ∘ 𝐺2)) ≤ 2𝑝1𝑝2 + 𝑝1(1 + 𝑞2) + 𝑞1 − (𝑡 − 2)β0 − 𝑡. 

 

Proof. 

Let T be an induced sub-graph of 𝑇(𝐺1) having maximum number of vertices such that T is a tree 

and |𝑇| = 𝑡.   

 

Let S be a maximum independent set of 𝑇(𝐺2) and vertices corresponding to the edge joining to 

the vertices of each copies of 𝑇(𝐺2) to 𝑇(𝐺1) such that |𝑆| = β0 and 𝐷′ be the set of vertices in S in 

copies of 𝑇(𝐺2) which are adjacent to the vertices of T then |𝐷′| = (𝑡 − 2)β0 if δ(𝐺1) ≥ 2. 

 

Let 𝐷 =  (𝑉(𝑇2(𝐺1 ∘ 𝐺2))) − (𝑉(𝑇)  ∪ 𝐷′) then 𝑉(𝑇2(𝐺1 ∘ 𝐺2)) − 𝐷 = 𝑉(𝑇) ∪ 𝐷′ and (𝑡 − 2) 

vertices of 𝑉(𝑇) are adjacent to (𝑝2 − β0) vertices in a copy of 𝑇2(𝐺2).  Also each vertex in 𝐷′ is 

adjacent to atleast one (𝑝2 − β0) vertices in a copy of 𝑇2(𝐺2). 

 

Therefore, D is a dominating set of 𝑇2(𝐺1 ∘ 𝐺2) and  

⟨𝑉(𝑇2(𝐺1 ∘ 𝐺2))  −  𝐷 ⟩ is a tree. 

γ𝑐𝑡𝑑(𝑇2(𝐺1 ∘ 𝐺2)) ≤ |𝐷| 

≤ |𝑉(𝑇2(𝐺1 ∘ 𝐺2)) − (𝑉(𝑇) ∪ 𝐷′)| 

= 2𝑝1𝑝2 + 𝑝1(1 + 𝑞2) + 𝑞1 − 𝑡(1 + β0) − 2β0.                                                                                 □  

 

5. Relation Between ctd(G) and ctd(T2(G)) 

Observation 5.1. 

For any connected graph 𝐺(𝑝, 𝑞), 𝑝 ≥ 2 then γ𝑐𝑡𝑑(𝐺) ≤ γ𝑐𝑡𝑑(𝑇2(𝐺)). Equality holds if 𝐺 ≅

𝐾1,𝑝−1. 

 



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Theorem 5.2. 

For any connected graph 𝐺(𝑝, 𝑞), 𝑝 ≥ 2 with δ(𝐺) = 1 then γ𝑐𝑡𝑑(𝑇2(𝐺)) ≤ 𝑝 − 1 + γ𝑐𝑡𝑑(𝐺). 

 

Proof. 

Let D be a minimum ctd-set of G and hence |𝐷| = γ𝑐𝑡𝑑(𝐺). Therefore, ⟨𝑉(𝐺)  −  𝐷 ⟩ is a tree. 

Now, the set 𝐷′ = 𝐷 ∪ (𝑉(𝑇2(𝐺)) − 𝑉(𝐺)) is a minimum ctd-set of 𝑇2(𝐺). Hence, γ𝑐𝑡𝑑(𝑇2(𝐺)) ≤

|𝐷′| = γ𝑐𝑡𝑑(𝐺) + 𝑝 − 1.                                                                                          □ 

 

Theorem 5.3. 

Given two integers a and b with 2 ≤ 𝑎 ≤ 𝑏 there exists a graph with 𝑎 + 𝑏 + 1 vertices such that 

γ(𝑇2(𝐺)) = 𝑎 + 1 and γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑎 + 𝑏. Also γ𝑐𝑡𝑑(𝑇2(𝐺)) ≤ γ(𝑇2(𝐺)) + 𝑏. 

 
 

Figure 1. 

 

Proof. 

In the cycle 𝐶𝑎+2(𝑎 ≥ 2) by {𝑣1, 𝑣2, … , 𝑣𝑎+2} of length 𝑎 + 2. Consider a path of length a. In this 

path attach (𝑏 − 𝑎) pendant edges at exactly one vertex and attach one pendant edge at each of the 

remaining (𝑎 − 1) vertices. Let the graph thus obtained be denoted by G and G has 𝑎 + 𝑏 + 1 

vertices.  

 

In 𝑇2(𝐺), edges of cycle 𝐶𝑎+2 and edges of pendant vertex of G and the vertices of G are the 

vertex set of 𝑇2(𝐺). Therefore,|𝑉(𝑇2(𝐺))| = 2𝑎 + 2𝑏 + 2. 

 

The set {𝑣1, 𝑣2, … , 𝑣𝑎+1} forms a minimum dominating set of 𝑇2(𝐺) and the set consisting of edge 

vertex of path 𝑃𝑎+1 a vertex 𝐶𝑎+2 and all the edge vertex of pendant vertices of G forms a minimum 

ctd-set of 𝑇2(𝐺). Therefore, γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑎 + 𝑏. 

If a = b then, the equality holds.                                                                           □ 

 

Theorem 5.4. 

If γ𝑐𝑡𝑑(𝐺) = 1 then γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝 − 1 where 𝑝 ≥ 2 is the number of vertices in G. 

 

Proof. 

Assume γ𝑐𝑡𝑑(𝐺) = 1, then 𝐺 ≅ 𝐾1 + 𝑇 where T is a tree with atleast two vertices. Let 𝑉(𝐾1) = 𝑣 

and 𝑉(𝑇) = {𝑣1, 𝑣2, … , 𝑣𝑝−1} then 𝑉(𝐺) = {𝑣, 𝑣1, 𝑣2, … 𝑣𝑝−1} and 𝑉(𝑇2(𝐺)) = 𝑉(𝐺) ∪ 𝑣𝑖
′ ∪ 𝑣𝑖,𝑖+1

′  

where 𝑣𝑖
′ ∈ (𝑣, 𝑣𝑖) and 𝑣𝑖,𝑖+1

′ ∈ (𝑣𝑖 , 𝑣𝑖+1) where 𝑖 =  1, 2, … , 𝑝 − 1.  Let 𝐷 = {𝑣𝑖,𝑖+1
′ /𝑖 =

1,2, … , 𝑝 − 2} ∪ {𝑣}. Then 𝐷 ⊆ 𝑉(𝑇2(𝐺)) and ⟨𝑉(𝑇2(𝐺))  −  𝐷 ⟩ is a tree. Therefore D is a ctd-set of 

𝑇2(𝐺) and hence, γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝 − 1.                                                                              □ 



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Theorem 5.5. 

Let G be a connected graph, if γ𝑐𝑡𝑑(𝐺) = 2 then 

𝛾𝑐𝑡𝑑(𝑇2(𝐺)) = {
𝑝 𝑖𝑓 𝐺 ≅ 𝐺1&𝐺2

𝑝 − 1 𝑖𝑓 𝐺 ≅ 𝐺3
. 

(i) 𝐺1 is the graph obtained from 𝐾1 + 𝑇 with one pendant edge attached at the vertex of 𝐾1, where 

𝑇 is any tree with 𝑝 − 2 vertices. 

 

(ii) 𝐺2 is the graph obtained from a tree T where (|𝑇|  =  𝑝 − 2) by joining each of the vertices of 

the tree to the vertices of 𝐾2 such that deg𝐺(𝑣) ≥ 2 for all 𝑣 ∈ 𝑉(𝐾2). 

 

(iii) 𝐺3 is the graph obtained from a tree by joining each of the vertices of the tree to the vertices of 

2𝐾1 such that deg𝐺(𝑣) ≥ 1 for all 𝑣 ∈ 𝑉(2𝐾1) and |𝑉(𝐺)| = 𝑝. 

 

Proof. 

Assume γ𝑐𝑡𝑑(𝐺) = 2.  

 

Let G be a connected graph with 𝑝 ≥ 4 and 𝑆1 = {𝑢1, 𝑢2} is a minimum ctd-set of G then 

⟨𝑉(𝐺)  −  𝑆1⟩ is a tree 𝑇. Hence |𝑇| = |𝑉(𝐺) − 𝑆1| = 𝑝 − 2.   

 

Now construct 𝑇2(𝐺), the vertices of complementary dominating set of G (tree T) and its edge 

vertex forms a cycle 𝐶3. 

Let𝑆2 = {𝑣𝑖𝑗
′ /𝑖 ≠ 𝑗, 𝑖 = 1,2, … , 𝑝 − 3, 𝑗 = 1,2, … , 𝑝 − 2} ⊆ 𝑉(𝑇2(𝐺))where(𝑣𝑖 , 𝑣𝑗) ∈ 𝐸(𝑉(𝐺) − 𝑆1). 

 

Case 1. 

𝑢1 and 𝑢2 are connected then 𝐺 ≅ 𝐺1 or 𝐺2. Let 𝐷 = 𝑆1 ∪ 𝑆2 ∪ {𝑢12
′ } is a minimum ctd-set of 

𝑇2(𝐺1) or 𝑇2(𝐺2).  Hence |𝐷| = |𝑆1| + |𝑆2| + 1 = 𝑝.Therefore, γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝 if 𝐺 ≅ 𝐺1 or 𝐺2. 

 

Case 2. 

𝑢1 and 𝑢2 are not connected. Then 𝐺 ≅ 𝐺3.  Let 𝐷 = 𝑆1 ∪ 𝑆2 is a minimum ctd-set of 𝑇2(𝐺3). 

Hence |𝐷| = |𝑆1| + |𝑆2| = 𝑝 − 1.  Therefore, γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝 − 1 if 𝐺 ≅ 𝐺3.                           □ 

 

Theorem 5.6. 

Let G be a connected graph with p vertices (𝑝 ≥ 3), 𝑉(𝑇2(𝐺)) = 𝑉(𝐺) ∪ 𝑉′(𝐺) then, 𝑉′(𝐺) =

𝑉(𝑇2(𝐺)) − 𝑉(𝐺) is a ctd-set of 𝑇2(𝐺) if and only if G is a tree. 

 

Proof. 

Assume 𝑉′(𝐺) is a ctd-set of 𝑇2(𝐺). Then, each vertex in 𝑉(𝑇2(𝐺)) − 𝑉(𝐺) is adjacent to atleast 

one vertex in 𝑉′(𝐺) and ⟨𝑉(𝑇2(𝐺))  −  𝑉′(𝐺) ⟩ is a tree. That is ⟨𝑉(𝐺)⟩ is a tree.  

 

Conversely, Assume G is a tree.Let𝐷 =  𝑉′(𝐺) that is D contains all the edge vertices of G. Since 

G is connected each vertex v in 𝑉(𝑇2(𝐺))  −  𝐷 =  𝑉(𝐺) forms a cycle 𝐶3 in 𝑇2(𝐺) and 𝑉(𝑇2(𝐺)) −

𝐷 = 𝑉(𝐺) is a tree. Hence, D is a ctd-set of G.       □ 

 

Theorem 5.7. 

For any connected (𝑝, 𝑞) graph G, 

γ𝑐𝑡𝑑(𝑇2(𝐺)) + Δ(𝐺) = 2𝑝 − 2 or 𝑝 + 𝑞 − 1 if and only if 𝐺 ≅ 𝐾1,𝑝−1 (𝑝 ≥ 4). 



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Proof. 

When 𝐺 ≅ 𝐾1,𝑝−1, γ𝑐𝑡𝑑(𝑇2(𝐺)) + Δ(𝐺) = 𝑝 + 𝑞 − 1. 

 

Conversely, γ𝑐𝑡𝑑(𝑇2(𝐺)) + Δ(𝐺) = 2𝑝 − 2 is possible if γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝 − 1 and Δ(𝐺) = 𝑝 − 1 is 

possible only if G is a star on p vertices.                                                                                              □ 

 

Theorem 5.8. 

For any connected (𝑝, 𝑞) graph G, γ𝑐𝑡𝑑(𝑇2(𝐺)) + Δ(𝐺) = 2𝑝 − 𝑛 (𝑝 ≥ 4) where 𝑛 = 𝑑𝑖𝑎𝑚(𝐺), 

𝑛 ≥ 2 if 𝐺 ≅ broom graph. 

 

Proof. 

For the graphs given in the theorem Δ(𝐺) = 𝑝 − 𝑛, γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝 then γ𝑐𝑡𝑑(𝑇2(𝐺)) + Δ(𝐺) =

2𝑝 − 𝑛. 

 

Conversely, γ𝑐𝑡𝑑(𝑇2(𝐺)) + Δ(𝐺) = 2𝑝 − 𝑛 only possible if 

(i) γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝 − 1 and Δ(𝐺) = 𝑝 − (𝑛 − 1) in this case γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝 − 1 if and only 

if G is a tree on p vertices. But for a star Δ(𝐺) = 𝑝 − 1 and 𝑑𝑖𝑎𝑚(𝐺) = 2. 

 

(ii) If G is a broom graph on p vertices with path of length 2, Δ(𝐺) = 𝑝 − 2 and  𝑑𝑖𝑎𝑚(𝐺) = 3. 

γ𝑐𝑡𝑑(𝑇2(𝐺)) + Δ(𝐺) = 𝑝 − 1 + 𝑝 − 2 =  2𝑝 − 3 

Therefore G is a broom graph of length 𝑛 − 1.i.e., 𝑛 = 𝑑𝑖𝑎𝑚(𝐺)(𝑛 ≥ 2).               □ 

 

Theorem 5.9. 

Let 𝑡1 and 𝑡2 be two trees with order 𝑝1 ≥ 2 and 𝑝2 ≥ 2 respectively. Then 

γ𝑐𝑡𝑑(𝑇2(𝑡1 ∘ 𝑡2)) ≤ 2γ𝑐𝑡𝑑(𝑡1 ∘ 𝑡2) + 𝑝1(1 + 𝑝2) − 1. 

 

Proof. 

We have γ𝑐𝑡𝑑(𝐺) ≤ 𝑝1(𝑝2 − 1)[4]. Let D be the minimum ctd-set of 𝑡1 ∘ 𝑡2. Hence |𝐷| ≤
𝑝1(𝑝2 − 1). 

 

Let 𝐷′ be the number of edge vertices of 𝑇2(𝑡1 ∘ 𝑡2). Then 𝐷 ∪ 𝐷′ ⊆ 𝑉(𝑇2(𝐺)) is a minimum ctd-

set of 𝑇2(𝐺). Therefore, 

 

γ𝑐𝑡𝑑(𝑇2(𝐺)) ≤ |𝐷 ∪ 𝐷′| 

 ≤ 2𝑝1(𝑝2 − 1) + 𝑝1(1 + 𝑝2) − 1 

 ≤ 2γ𝑐𝑡𝑑(𝐺) + 𝑝1(1 + 𝑝2) − 1.                                                                                       □  

 

Theorem 5.10. 

Let G be a (𝑝, 𝑞), 𝑝 ≥ 5, graph such that both G and 𝐺 are connected then 

(i) 8 ≤ γ𝑐𝑡𝑑(𝑇2(𝐺)) + γ𝑐𝑡𝑑 (𝑇2(𝐺)) ≤ 2(𝑝 + 𝑞 − 4) 

(ii) 4 ≤ γ𝑐𝑡𝑑(𝑇2(𝐺)). γ𝑐𝑡𝑑 (𝑇2(𝐺)) ≤ (𝑝 + 𝑞 − 4)2 

 

 

 



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Proof. 

By Theorem 3.4 γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝 + 𝑞 − 2 if and only if G is a graph 𝐾2. But in this case 𝐺 is 

disconnected. γ𝑐𝑡𝑑(𝑇2(𝐺)) = 𝑝 + 𝑞 − 3 if and only if G is the graph 𝑃3&𝐶4. But in this case 𝐺 is 

disconnected. Therefore, γ𝑐𝑡𝑑(𝑇2(𝐺)) ≤ 𝑝 + 𝑞 − 4. Hence,γ𝑐𝑡𝑑(𝑇2(𝐺)) + γ𝑐𝑡𝑑 (𝑇2(𝐺)) ≤

2(𝑝 + 𝑞 − 4). 

For lower bound γ𝑐𝑡𝑑(𝑇2(𝐺)) = 4 if and only if 𝐺 ≅ 𝑃4 and 𝐶5. In this case 𝐺 is connected. 

Hence γ𝑐𝑡𝑑(𝐺) + γ𝑐𝑡𝑑(𝐺) ≥ 4.  (ii) follows similarly.                                                                □ 

 

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