Communications on Applied Nonlinear Analysis 

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Complementary Tree Domination Number of Corona Product of 

Complete graph with Some Graphs 

*S. Jayalakshmi 1 and P. Vidhya2 
1Research Scholar (Part Time), School of Mathematics 

Madurai Kamaraj University, Madurai - 625021, Tamilnadu, India 

E-mail: jayark83@gmail.com , jayalakshmi.s1@sdnbvc.edu.in 
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: 12-08-2024 

Revised: 15-09-2024 

Accepted: 25-10-2024 

Abstract:  A set D of a graph  G  =  (V, E)  is  a  dominating  set,  if  every vertex in V − D 

is adjacent to some vertex in D. The domination number γ(G) of G is the minimum 

cardinality of a dominating set. A dominating set D is called a complementary tree 

dominating set if the induced subgraph <V  − D> is a tree.  The minimum cardinality of a 

complementary tree dominating set is called the complementary tree domination number of 

G and is denoted by γctd(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. In this paper, we discussed complementary 

tree domination number of corona product of complete graph with some graphs. 

AMS Subject Classification: 05C69. 

Keywords: Dominating set, Complementary tree domination  number.A subset D of the 

vertex set V(G) of a graph G is said to be a dominating set if every vertex not in D is 

adjacent to at least one vertex in D. A dominating set D is said to be an eccentric dominating 

set if for every , there exists at least one eccentric point of v in D. An eccentric dominating 

set D of G is a non split eccentric dominating set if the induced sub graph < V- D > is 

connected. The minimum of the cardinalities of the non split eccentric dominating sets of G 

is called the non split eccentric domination number of G. This paper evaluates the non split 

eccentric domination number of Corona product and join of some standard graphs.  

Keywords: Domination, Eccentric Domination, Non Split Eccentric Domination, Corona 

product, Join. 

 

1 Introduction 

 

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

with p vertices and q edges. Roberto Frucht and Frank Harary [1] introduced the binary 

product of two graphs named Corona in 1970. 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 concept of 

domination in graphs was introduced by Ore [4]. 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 



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2 

denoted by γ(G). The complementary tree domination number of a graph was introduced 

by  S. Muthammai, M. Bhanumathi and          P. Vidhya [3] 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 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). Sergio canoy Jr and Carmelito E.Go [5] have 

obtained the domination number of corona graphs. Any undefined term in this paper 

may be found in Harary[2] 

For notation convenience 
vG2  be a copy of G2 corresponding to the vertex ( ).1GVv

Also 
jiu be the vertex of G which are adjacent to the vertex ( ).1GVvi   

 In this paper we discussed complementary tree domination number of corona product 

of complete graph and their bounds are determined. 

 

2 Prior Results 

 

Observation 2.1. [3] 

 

(i) For any path Pn  with  n  vertices,  γctd(Pn) = n − 2 ,  n ≥ 4 . 

(ii) For any cycle Cn  with  n  vertices,  γctd(Cn) = n − 2 ,  n ≥ 3 . 

(iii) For any complete graph Kn with n vertices, γctd(Kn) = n − 2 , n ≥ 3 . 

(iv) For any star K1,n , γctd(K1,n) = n , n ≥ 2 . 

(v)For any complete bipartite graph Km,n with m , n ≥ 2 , γctd(Km,n) = min{m, n} . 

(vi) γctd(Cn ◦ K1) = n + 1 , n ≥ 3 , where  Cn ◦ K1  is the corona of  Cn  and  K1 . 

(vii) For any wheel Wn  with  n  vertices,  γctd(Wn) = 2 ,  n ≥ 4 . 

Preposition 2.2 . [3] 

If ( ) ,2− pGctd then  pendant vertices are the members of every ctd-set. 

Example 2.3. 

 

 

                             1v                      4v  

          

 

 

                                                                           1u                      2u  

                             2v                   3v                                 

         G1         G2 

 

   

 

 

 



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                                     12u  

                                                               41u                        

                 11u  

                                                                                   42u  

 

                                  1v                                 4v  

 

 

 

                                   2v                               3v  

                                                                              32u  

                  21u  

       22u    

31u  

                                           21 GG   

                                        Figure 1: 

For the graph  G1 ◦ G2 given in Figure 1.  

In the following, a necessary and sufficient condition for a ctd-set of a    corona 

product of graphs G1 ◦ G2 is found. 

 

Theorem 2.4 

 Let 1G  and 2G  be connected  graphs  then ( )21 GGVD   is a ctd-set in 21 GG   if 

and only if one of the following conditions  holds. 

(i)  For each ( ) ( ) DGVGVv v  21 ,  is a dominating   in 
vG2  and  ( )

( )


GVu

uGVD


 2  . 

(ii)  ( ) DGV 1  is a complementary tree dominating in 1G  and ( ) DGV v 2  whenever  

( ) DGVv  1  and ( ) DGV v 2  is dominating in  
vG2  whenever  ( ) .1 DGVv −  

Proof. Suppose ( ) .1 = DGV   

Let ( )1GVv  and ( ) .2 DGVx v −  Hence ( ) .21 DGGVx −  Since D is a ctd-set of  

21 GG  .There exists Dy  such that ( ) .1,
21

=yxd GG   Since ( )vGVx 2  and  ( ) 1,
21

=yxd GG   
either ( )vGVy 2  or vy = . If vy =  then ( ) ,1 DGVy   a contradiction. Therefore 

( ) DGVy v  2 is a dominating set of 
vG2 .Since ( ) ,1 =DGV  ( )

( )


1

2

GVu

uGVD


 . Hence (i) 

holds.  

Suppose ( )  DGV 1  and ( ) ( )11 GVDGV   .  



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Let ( ) DGVx − 1  it follows ( ) DGGVx − 21  . Let D is a dominating set of 21 GG   
there exists Dy  such that ( ) .1,

21
=yxd GG 

 If ( )1GVy  then ( ) DGVy  1  then 

( ) .1, =yxdG  
Hence ( ) DGV 1  is a dominating set in .1G Now we have to prove ( ) DGV 1  

is a complementary tree dominating  set in 1G  it is enough to prove ( ) − DGV 1  is a tree. Let 

( ) ., 1 DGVyx −  Since − DGGV )( 21   is a tree T. Therefore )(, 1GVyx   there exist 

( )xGVu 2  and v ( )yGV 2 . Since T is connected and acyclic. There exists a unique path 

between u  and v .Hence x and y are the vertices transverse from u  and v . Hence 
( ) DGV 1  

is a ctd-set in  G1.  

 Suppose  DGVv  )( 1    and ( ) − DGV v

2  .  

Let   DGVu v − )( 2 . Since )()( 11 GVDGV   there  exists .)( 1 DGVw −  Since 

− DGGV )( 21   is a tree. There exists a unique path between u - w with vertices from 

.)( 21 DGGV −  However any u-w path must contain a vertex v which is impossible. Hence 

.)( 2 =− DGV v

 Hence .)( 2 DGV v   
 

 Suppose DGVv − )( 1 .  

Let  DGVx v − )( 2 . This implies that .)( 21 − DGGVx  Since D is a ctd-set in 

21 GG   there exists Dy  such that ( ) .1,
21

=yxd GG   Consequently vy =  or ).( 2

vGVy  

Since Dy  and  .)( 1 DGVv − Hence vy   then it follows  )( 2

vGVy now 

( ) 1,
21

=yxd GG   implies ( ) .1,
2

=yxd vG
 Hence DGV v )( 2  is a dominating set in

vG2 .Therefore  (iii) 

holds.  

Conversely, Suppose (i) holds . Let .)( 21 DGGVx −   
Suppose )( 1GVx .Since DGV x )( 2  is  dominating set in 

xG2  and .)( 2 DGV x

 
Let  .)( 2 DGVu x  Then 1),(

21
=yxd GG  .  

Suppose )( 1GVx then there exist  )( 1GVy  such that ( ).2

yGVx since DGV y )( 2  
is a dominating set in 

yG2  and DGVx y − )( 2  there exist DGVt y  )( 2  such that  

1),(
21

=txd GG   then D is a dominating set in 21 GG  . Since

( )
( )

( ) .12

1

=


DGVandGVD
GVu

u
 

          Consequently, )()()( 1

)(

221 GVDGVDGGV
GVv

v −=−


 .Let 

qpDGGVqp − ,)(, 21  .  If DGVqp v − )(, 2 for some )( 1GVv  then there is a path with 

vertices qvp ,,  in .)( 21 DGGV −   
If )(, 1GVqp  .Then there is a tree which contains a p-q path in DGGV −)( 21 

.Since G1 is connected. 

If ( )1GVp  and ( ) DGVq v − 2  for some ( )1GVv   then  DGGV −)( 21   contains a 

tree with vertices p  and v . Suppose vp   since G1 is connected then DGGV −)( 21   

contains a tree with vertices qvp ,, .Suppose ( ) DGVp v − 2  and ( )wGVq 2  for some 



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).(, 1GVwv  Since G1 is connected then there exist a tree with vertices qwvp ,,,  in 

DGGV −)( 21  .Hence DGGV −)( 21   is connected and acyclic. Therefore DGGV −)( 21   is 

a tree . Hence D is a complementary tree dominating in 21 GG  . 

Suppose (ii) holds. Let  .)( 21 DGGVx −  Suppose  .)().()( 11 DGVxeiGVx −

Since DGV )( 1  is a dominating in G1, there exist DGVy  )( 1  such that ( ) 1, =txdG  for  

some )( 1GVt  is follows that ( ) .1,
21

=yxd GG   Suppose ( )vGVx 2  for some )( 1GVv

.1),().()().(
212 =− vxdeiDGVxei GG

v

 If Dv  which is a contradiction to 

( ) .2 DGVx v − Hence  Dv is ( ) DGVv − 1 . In this case ( ) DGV v 2  is dominating in 
vG2

).( ei  there exist ( ) DGVw w  2  such that .1),(
2

=wxd vG
It follows that ( ) .1,

21
=wxd GG   

Hence D is a dominating set in 21 GG  . 

 Let  ( )21, GGVyx   suppose ( )1, GVyx  ).( ei ( ) DGVyx −, . Since DGV )(  is a 

complementary tree dominating set in G1 and − DGV )( 1  is a tree. From this 

− DGGV )( 21   is a tree which contains a path x-y in .)( 1 DGV −   

Suppose ( )1GVx  and ( )vGVy 2  for some ( ) ( ) DGVxeiGVv − 11 ).(  and 

( ) .2 DGVy v −  If vx =  there exists path between x-y. Suppose  vx   if  ( ) DGVv  1  then 

( ) .2 DGV v  This contradicts the fact that ( ) .02 − DGV v
Then ( ) ,1 DGVv −  consequently,

( ) DGV v −2  is a dominating set in 
vG2  ).( ei  ( ) .2 DGV v

 Suppose ( ) .2 DGVx v  Since 

( ) DGVv − 1  and ( ) − DGV 1  contains a path with vertices .vx − Thus ( ) − DGGV 21   

contains a tree with vertices .vx −   
Suppose ( ) yxDGVyx v − ,, 2  for some ( ).GVv If ,Dv then ( ) DGV v 2 .This 

contradicts the fact that ( ) .2 − DGV v
Thus Dv  (i.e) .)( 1 DGVv − Now there exists a 

path with vertices yvx ,, in ( ) .21 DGGV −  

Suppose ( )vGVx 2  and ( )wGVy 2  for some ( ) .,, wvGVwv  Then ( ) DGVx v − 2  

and ( ) DGVy w − 2  if Dv or  Dw then ( ) DGV v 2  and ( ) .2 DGV w   This contradicts the 

facts that ( ) − DGV v

2  and ( ) − DGV w

2 . Thus Dwv ,  that is

( ) ( ) ., 211 DGGVDGVwv −−   Since ( ) − DGV 1  is a tree.  There is a tree with 

support vertices v and w in ( ) DGGV −21  .Hence ( ) − DGGV 21   is a tree. Hence D is a 

complementary tree dominating set in 21 GG  . 

 

Corollary 2.5. 

 

Let 1G  and 2G  be any connected graph . Then ( ) ( ) ( ).12 2121 GGGGctd  −
 

G1  is not a tree. 
 

Proof.  Let .1 nG = Suppose there exist D such that D satisfies (ii) of theorem [ 4.1.2]  

 



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  ( )

( ) 22

2

2

2).(

2

Since

GGei

G
G







  

 

        
( ) ( ) ( ).12 2121 GGGGctd  −

 
 

Corollary 2.6. 

 

 Let G1 be a tree and G2 be any connected graph respectively. Then  

( ) ( ).2121 GGGGctd  =
 

Proof.  For each ( ).1GVv  Let 
v

G2
 be a copy of G2 corresponding to vertex .v  Further, for 

each ( ).1GVv  Let vD be a minimum dominating set in .2

vG By theorem [4.1.2] 
( )


1GVv

vDD


=  

is a complementary tree  dominating set in .21 GG   Thus  

                          

( )

( )

( )

( )21

21

1

1

GG

D

D

DGG

GVv

v

GVv

v

ctd





=

=

=












 

Therefore, ( ) ( ).2121 GGGGctd  
 

Here , we  consider G1  and G2 be any connected graph of order n and  m respectively. Then 

the vertex set {𝑢𝑖𝑗/1 ≤ 𝑖 ≤ 𝑛, 1 ≤ 𝑗 ≤ 𝑚} is the ith copy of G2 is adjacent to the ith vertex of 

G1 and let D is a minimum ctd-set of 21 GG   . Hence ( ) − DGGV 21  is a tree. 

 

3 Complementary Tree Domination Number of Corona Product of complete graph 

with Some  Graphs 

 

       In this section, for 4n complementary tree domination number of ,1GKn    

where G1 is any connected graph with 3m vertices are obtained. 

      Here, we consider D1 is a ctd-set of Kn, hence 21 −= nD  and D2 be the  set 

whose elements are the vertices of G1 which are adjacent to each vertex of 

( )

( ) ( ) ( )

( ) ( )

( ) ( )21

2

22

2

12

12

222

.2)2(

GG

Gn

GGnD

GGnD









−

−=

+−

+−



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D1.Hence mnD )2(2 −= .Clearly, DDD  21 ,where D is a minimum ctd-set of  

1GKn   and ( )( ).2121 −+= nmDD  

Proposition 3.1. ( ) .221 −= nKKnctd   

Proof.   Let 𝐺 = nK ∘ 𝐾1 .  

Let 𝑉(𝐾𝑛) = {𝑣1, 𝑣2, . . . . . . . 𝑣𝑛}  and  vertex  𝑢𝑖 be the ith copy of 𝐾1 attached to the vertex 

𝑣𝑖 .Then 𝑉(𝐺) = {𝑣𝑖 , 𝑢𝑖/1 ≤ 𝑖 ≤ 𝑛}.Here  𝑢1, 𝑢2. . . . . . . 𝑢𝑛 are the pendant vertices of 𝐺. We 

have  pendant vertices are members of ctd-set of G [3]. Hence, D = {vi : 1 ≤ i ≤ n − 2}∪{ui : 

1 ≤ i ≤ n }=              n − 2 + n = 2 n − 2.  is a minimum ctd-set of  𝐺. Hence, |𝐷|   =

 𝛾𝑐𝑡𝑑(𝐺) = 2𝑛 − 2. 

Proposition 3.2. ( ) .432 −= nKKnctd   

Proof.  Let V (Kn)  =  {v1, v2, . . . , vn}  and V (K2)  =  {u1, u2}.   Then 

V (Kn ◦ K2) = {vi/1 ≤ i ≤ n} ∪{uij/1 ≤ i ≤ n, 1 ≤ j ≤ 2} .We have, γctd (Kn) = n − 2 

[3].  By choosing (n − 2) vertices of Kn say v1, v2, . . . , vn−2 which form a minimum 

ctd-set in Kn and  vertices which are  adjacent to {v1, v2, . . . , vn−2 }  are {uij/1 ≤ i ≤ n − 

2, 1 ≤ j ≤ 2}. Therefore, 

D = {vi : 1 ≤ i ≤ n − 2} ∪ {uij : 1 ≤ i ≤ n-2, 1 ≤ j ≤ 2}∪ 1,1,1 , nn uu −  

= n − 2 + 2n – 4+2 

= 3n − 4 

 

Proposition 3.3 . For m ≥ 2, γctd(Kn ◦ mK ) = mn + n − 2. 

Proof.  Take G = Kn ◦ mK . Let V (Kn) = {v1, v2, . . . , vn} and {u1, u2, . . . , um} be the vertex set 

of the ith copy of mK  is adjacent to the vertex vi. Then V (G) = {vi/1 ≤ i ≤ n} ∪ {uij/1 ≤ i ≤ n, 

1 ≤ j ≤ m} where uij’s is the ith copy of mK  is adjacent to the vertex vi in Kn. Let D be a 

minimum ctd-set of G. Since pendant vertices are members of every ctd-set G. By choosing 

pendant vertices, it dominates all the vertices of Kn but ⟨V(G) − D⟩ forms a cycle. So we are 

choosing   2

1

−

=

n

iiv   and all the pendant vertices in a graph G. Therefore, 

D = {vi : 1 ≤ i ≤ n − 2} ∪ {uij : 1 ≤ i ≤ n, 1 ≤ j ≤ m} 



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= n − 2 + nm 

= mn + n − 2. 

 

Proposition 3.4.  For m ≥ 3, γctd(Kn ◦ K 1,m−1) = n + (m+1)(n − 2). 

 

Proof. Take  G = Kn ◦ K1,m−1. Let V (Kn) = {vi /1 ≤ i ≤ n} and  V (K1,m−1) = { w, u1, u2, . . . ,         

um−1}. Then V (G) = {vi/1 ≤ i ≤ n}∪{wi uij , 1 ≤ i ≤ n, 1 ≤ j ≤ m − 1}. By choosing a vertex               

{c i /1 ≤ i ≤ n} which dominates all the vertices of Kn ◦K1,m−1. But ⟨V (G) −D⟩ mn PK   

which contains a cycle. Let D = D1 ∪ D2 ∪{c1, c2, . . . , cn} is a minimum ctd-set of G. 

|D| = n + (m+1)(n − 2). 

Proposition 3.5. For, m ≥ 3, ( ) ( )( ) .
2

221 







+−+=

m
nmPK mnctd   

 Proof.  Let V (Kn)  =  {v1, v2, . . . , vn}  and V (Pn)  =  {u1, u2, . . . , um}.   Then 

V (Kn ◦ Pm) = {vi/1 ≤ i ≤ n} ∪{uij/1 ≤ i ≤ n, 1 ≤ j ≤ m} .We have, γctd(Pm) = m − 2, 

γctd(Kn) = n − 2 [3].  By choosing (n − 2) vertices of Kn say v1, v2, . . . , vn−2 which 

form a minimum ctd-set in Kn and  vertices which are  adjacent to {v1, v2, . . . , vn−2 }  

are {uij/1 ≤ i ≤ n − 2, 1 ≤ j ≤ m}. By Proposition 2.3  

Case i : m is even 

   mnnnmnnnn uuuuuuuDDD ,3,1,,15,13,11,121 ,.....,,.....,, = −−−−
 is a minimum 

ctd-set of mn PK   and ( )
2

,
2

mmmn SDPKV − 

 

 

 

 

Case ii : m is odd 

   1,3,1,1,15,13,11,121 ,.....,,.....,, −−−−−− = mnnnmnnnn uuuuuuuDDD  is a minimum ctd-set of 

mn PK   and ( )
















−

22

mmmn SDPKV   

( )( ) )1(.
2

221

22
21









+−+=









+







++=

m
nm

mm
DDD



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







+








++=

22
21

mm
DDD

                                                                                                 (2) 

From 1 & 2 

=D ( ) ( )( ) .
2

221 







+−+=

m
nmPK mnctd   

Proposition 3.6.  For m ≥ 3, ( ) ( )( ) .
2

221 







+−+=

m
nmCK mnctd 

 

Proof. Let V (Kn) = {v1, v2, . . . , vn} and the set {u1, u2, . . . , um} be  the vertices of 

Cm.  Then V (Kn ◦ Cm) ={vi/1 ≤ i ≤ n} ∪{uij/1 ≤ i ≤ n, 1 ≤ j ≤ m} By preposition 

2.4 

Case i : m is even 

   1,3,1,1,15,13,11,121 ,.....,,.....,, −−−−−− = mnnnmnnnn uuuuuuuDDD  is a minimum ctd-set of 

mn CK   and ( )
2

,
2

mmmn SDCKV − 

 

 

Case ii : m is odd 

   mnnnmnnnn uuuuuuuDDD ,3,1,,15,13,11,121 ,.....,,.....,, = −−−−
 is a minimum ctd-set of 

mn CK   and ( )
2

1

2

1 ++− mmmn SDCKV   









++=

2
221

m
DDD

                                                                                                            (4) 

From 3 & 4 

=D ( ) ( )( ) .
2

221 







+−+=

m
nmCK mnctd   

( )( ) )3(.
2

221

22
21









+−+=









+








++=

m
nm

mm
DDD



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Proposition 3.7. For m ≥ 4, γctd(Kn ◦Wm) =  (m+1)(n − 2) +2 






 −

2

1m
 

Proof. Let V (Kn) = {vi /1 ≤ i ≤ n} and V (Wm) = {c, uj /1 ≤ j ≤ m − 1} and V (Kn ◦Wm) =                

{v1, v2, . . . , vn}∪{ci uij /1 ≤ i ≤ n, 1 ≤ j ≤ m−1} .Let  

D = D1∪ D2∪{ nn CC ,1− }∪ {ctd-set of  1−mn CK  }and ( )








−

2

mmn SDWKV   

|D| = (n − 2) +m(n − 2) +2 






 −

2

1m
 

Therefore, 

|D| =(m+1)(n − 2) +2 






 −

2

1m
. 

 

Proposition 3.8.  For m ≥ 4, γctd(Kn ◦ Km) = n(m + 1) − 4. 

 

Proof.  Take G = Kn ◦ Km. Let V (Kn) = {v1, v2, . . . , vn} and V (Km) = {u1, u2, . . . , um}. Then           

V (G) = {vi/1 ≤ i ≤ n} ∪ {uij/1 ≤ i ≤ n, 1 ≤ j ≤ m} .We have, γctd(Kn) = n − 2 [3]. 

       Suppose by choosing (n−2) vertices of Kn which dominates all the vertices of i

mK
 
1 ≤ i ≤ 

n − 2 but it does not dominates n and n − 1 copy of Km which contradict the ctd-set. 

        Suppose by choosing (m − 2) vertices of Km which dominates all the vertices of Kn but  

⟨V(G)−D⟩ contains a cycle which is contradict to ctd-set. Let D = D1∪D2∪ 

 11,,1/ −−= mjnniuij
 is a minimum ctd-set of G. 

= n − 2 + m(n − 2) + m − 1 + m − 1 

Hence, |D| = n(m + 1) − 4. 

 

Proposition 3.9. For m1,m2 ≥ 2,γctd(Kn ◦ K m1,m2) = (n − 2)(m1+m2+ 1) + 2 min(m1,m2). 

 

Proof. Take G = Kn ◦ K m1,m2 . Let V (Kn) = {vi/1 ≤ i ≤ n},V (K m1,m2) = {uj/1 ≤ j ≤ m1} ∪ {wj/1 

≤ j ≤ m2}. Then V (G) = {vi/ 1 ≤ i ≤ n}∪{uij/1 ≤ i ≤ n; 1 ≤ j ≤ m1}∪{wij/1 ≤ i ≤ n; 1 ≤ j ≤ m2}. 

Let 

D be the minimum ctd-set of Kn ◦K m1,m2 .  

Case i. m1 < m2 



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By choosing {uij/1 ≤ i ≤ n, 1 ≤ j ≤ m1} which dominates all the vertices of G and ⟨V (G) − D⟩ 

  

2mn KK  forms a cycle. Let D = D1 ∪ D2  ∪ {un−1,1, un−1,2, . . . , un−1,m1} ∪ {un,1, un,2, . . . , 

un,m1} is a minimum ctd-set of G. Hence, 

|D| = n − 2 + m1(n − 2) + m2(n − 2) + 2m1 

= (n − 2)(m1 + m2 + 1) + 2m1                                                                                                 (5) 

Case ii. m2 < m1 

By choosing a vertex D = D1 ∪ D2 ∪ {wn−1,1, wn−1,2, . . . , wn−1,m2} ∪ {wn,1, wn,2, . . . , wn,m2} 

which dominates all the vertices of V (G) and ⟨V(G)−D⟩ is a tree. Here D is a ctd-set of G. 

Hence, 

|D| = n − 2 + m1(n − 2) + m2(n − 2) + 2m2 

= (n − 2)(n1 + n2 + 1) + 2m2                                                                                                (6) 

From (5) and (6) 

|D| = (n − 2)(m1 + m2 + 1) + 2 min(m1,m2). 

Proposition 3.10.   For, m ≥ 2, γctd(Kn ◦ Km) ≥ n(m + 1) − 4. 

 

Proof. Take G = Kn ◦Km. Let V (Kn) = {v1, v2, . . . , vn} and {u1, u2, . . . , um}be the vertex set 

of the ith copy of Km is adjacent to the vertex vi in Kn.Then V (G) = {vi/1 ≤ i ≤ n} ∪ {uij/1 ≤ i ≤ 

n, 1 ≤ j ≤ m} . Let D be a minimum ctd-set of G. By choosing {uij/1 ≤ i ≤ n, 1 ≤ j ≤ 






 +

2

1m
} 

dominates all the vertices in a graph G but ⟨V (G) − D⟩ forms a cycle so we are choosing (n − 

2) vertices from {vi} then their corresponding {uij} vertices and for the remaining n − 1 and n 

vertices we choose 






 +

2

1m
 vertices from {uij}.Therefore, |D| ≥ n(m + 1) − 4. 

 

4 Bounds  for  Complementary Tree Domination Number of Corona Product of 

Complete Graph with Some Graphs 

 

Theorem 4.1. 

 

For any connected graph G, with m ≥ 2 vertices then                                                                         

3n − 4 ≤ γctd(Kn ◦ G) ≤ (m + 1)(n − 2) +  2(m − β0). 

Proof. Let V (Kn) = {v1, v2, . . . , vn}. Let T be any induced subgraph of Kn having maximum 

number of edges such that T  K2 is a tree. Then |T| = 2. Let S be a maximum independent 



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set of G such that |S| = β0.Let D1 be the set of vertices of S in copies of G which are adjacent 

to the 

vertices of T. Then |D1| = 2β0. Let D = V (Kn ◦ G) − (V (K2) ∪ D1). Then V (Kn ◦ G) − D =        

V (K2) ∪ D1 and each vertex in V (K2) is adjacent to m−β0 vertices in a copy of G. Also, each 

vertex in D1 is adjacent to atleast m−β0 vertex in a copy of G. Therefore D is a dominating set 

of Kn ◦G and ⟨V (Kn ◦ G) − D⟩ is the tree obtained from T by attaching m − β0 pendant edges 

at each vertex of K2. Therefore D is a ctd-set of Kn ◦ G,  

γctd(Kn ◦ G) ≤ |D| = |V (Kn ◦ G) − (V (K2) ∪ D1)| 

= mn + n − (2 + 2β0) 

= m(n − 2) + (n − 2) + 2(m − β0) 

= (n − 2)(m + 1) + 2(m − β0) 

The lower bound equality holds if G  K2. 

References 

[1] R.Frucht and F.Harary, “On the corona of two graphs”, Aequationes Math., Vol. 

4,pp. 322–325, 1970. 

[2] F.Harary,“Graph Theory”, Addison Wesley, Reading Mass,1972. 

[3] S.Muthammai, M.Bhanumathi and P.Vidhya, “Complementary tree domination 

number of a graph”, International Mathematical Forum, Vol. 6, No. 26,pp. 1273–

1282, 2011. 

[4] O.Ore, “Theory of Graphs”, Amer. Math. Soc. Colloq. Publ., Vol .  38,1962. 

[5] Sergio CanoyJr and Carmelito E.Go,“ Domination in the corona and join of 

graphs”, International Mathematical Forum,Vol. 6, No. 13,2011. 

[6] P. Vidhya and S. Jayalakshmi, “Complementary Tree Domination of Corona Product of 

Cycle Cn with Some Standard Graphs”, Design Engineering, Vol. 9,2021. 

[7]    S. Muthammai and P. Vidhya, “More Results on Complementary Tree Domination         

Number of Graphs”, International Journal of Mathematics And its Applications, Vol.          

4, No. 1-D, pp. 17-20, 2016.