View of Reserved Domination Number of some Graphs

Reserved Domination Number of some Graphs

Dr. G. Rajasekar

1

_{, G. Rajasekar}

2

1_{Associate Professor, PG and Research Department of Mathematics, Jawahar Science College, Neyveli}

2_{Research Scholar, PG and Research Department of Mathematics, Jawahar Science College, Neyveli} 1_{grsmaths@gmail.com.,} 2_{r.g.raja3007@gmail.com}

Article History: Received: 10 January 2021; Revised: 12 February 2021; Accepted: 27 March 2021; Published

online: 10 May 2021

Abstract: In this paper the definitions of Reserved domination number and 2-reserved domination number are introduced as for the graph

G

=

(

V E

,

)

a subset

S

of

V

is called a Reserved Dominating Set

(

RDS

)

of

G

_{if (i)}

R

be any nonempty proper subset of

S

; (ii) Every vertex in

V

−

S

_{is adjacent to a vertex in}

S

. The dominating set

S

is called a minimal reserved dominating set if no proper subset of

S

containing

R

is a dominating set. The set

R

is called Reserved set. The minimum cardinality of a reserved dominating set of

G

is called the reserved domination number of

G

and is denoted by

( )k

( )

R

−



G

where

k

is the number of reserved vertices. Using these definitions the 2-reserved domination number for Path graph

P

_n, Cycle graph

C

_n, Wheel graph

W

_n, Star graph

S

_n, Fan graph

F

_1,n, Complete graph

K

_n and Complete Bipartite graph

K

_{m n,} are found.

Keywords: Dominating set, Reserved dominating set, 2-Reserved dominating set.

1. Introduction (Times New Roman 10 Bold)

Domination in graphs has wide applications to several fields such as mobile Tower installation, School Bus Routing, Computer Communication Networks, Radio Stations, Locating Radar Stations Problem, Nuclear Power Plants Problem, Modeling Biological Networks, Modeling Social Networks, Facility Location Problems, Coding Theory and Multiple Domination Problems with hierarchical overlay networks. Domination arises in facility location problems, where the number of facilities (e.g., Mobile towers, bus stop, primary health center, hospitals, schools, post office, hospitals, fire stations) is fixed and one attempts to minimize the distance that a person needs to travel to get to the nearest facility. A similar problem occurs when the maximum distance to a facility is fixed and government or any service provider attempts to minimize the number of facilities necessary so that everyone is serviced. Concepts from domination also appear in problems involving finding sets of representatives, in monitoring communication or electrical networks, and in land surveying.

Let

G

=

(

V E

,

)

be a graph. A subset

S

of

V

is called a dominating set of

G

if every vertex in

V

−

S

is adjacent to a vertex in

S

. A dominating set

S

is called a minimal dominating set if no proper subset of

S

is a dominating set. The minimum cardinality of a dominating set of

G

is called the domination number of

G

and is denoted by



( )

G

. The maximum cardinality of a minimal dominating set of

G

is called the upper domination number of

G

and is denoted by



( )

G

.

Consider the situation of installing minimum number of Mobile phone towers so that it will be utilized by all the people living in towns or villages. Here each town or village is considered as separate entity called as vertex. These towns and villages are connected by roads (In this case Ariel distance) called edges. The situation of installing minimum number of Mobile phone Towers is domination problem and this minimum number is domination number.

In real life situation it is not always in practice of installing the Towers only at the public utility pattern. If the installing authorities are very much interested in installing the tower, nearer to their residence or nearer to the residence of VIPs or to their favorites’ residence then the concerned person will reserve some of the installation points without any concern about the nearer or proximity of the other towns or villages. Such type installation points are a called reserved installation points. This situation motives the development of the reserved domination points. These reserved points are automatically included in the domination set.

2.Resaserved domination number Definition: Dominating Set

Let

G

=

(

V E

,

)

be a graph. A subset

S

of

V

is called a dominating set of

G

if every vertex in

V S

\

is adjacent to a vertex in

S

. A dominating set

S

is called a minimal dominating set if no proper subset of

S

is a dominating set. The minimum cardinality of a dominating set of

G

is called the domination number of

G

and is denoted by



( )

G

. The maximum cardinality of a minimal dominating set of

G

is called the upper domination number of

G

and is denoted by



( )

G

.

Definition: Reserved Domination set

Let

G

=

(

V E

,

)

be a graph. A subset

S

of

V

is called a Reserved Dominating Set

(

RDS

)

of

G

_if (i)

R

be any nonempty proper subset of

S

.

(ii) Every vertex in

V

−

S

is adjacent to a vertex in

S

.

The dominating set

S

is called a minimal reserved dominating set if no proper subset of

S

containing

R

is a dominating set. The set

R

is called Reserved set.

The minimum cardinality of a reserved dominating set of

G

is called the reserved domination number of

G

and is denoted by

R

_{( )k}

−



( )

G

where

k

is the number of reserved vertex.

Remark:

Then

k =

2

, we get the 2-reserved domination number of

G

and is denoted by

R

_{( )2}

−



( )

G

. In this paper let us find the 2-reserved domination number of various graphs.

Definition: Graph Connected by a Bridge

Let

G

₁,

G

₂ be any two graphs and

G

be a graph attained by connecting

G

₁ and

G

₂ by a bridge

e

=

v v

_{1 2}

where

v

₁



V G

( )

₁ ,

v

₂



V G

( )

₂ .

3. Some Preliminaries results Domination Number:

(i) Lollipop Graph:

( )

_,

2

3

m n

n

L



_{= }



+



_





.

(ii) Tadpole Graph:

( )

_,

2

1

3

3

m n

m

n

T



=



_

+

 

_{ }

+

+



_



 



.

(iii) Pan Graph:

( )

_,1

3

n

n

P



_{=  }

 

 

.

(iv) Barbell Graph:



( )

B

_n

=

2

.

Definition: Lollipop Graph

The lollipop graph is the graph obtained by joining a complete graph

K

_m with

m 

3

to a path graph

P

_n with a bridge. It is denoted by

L

_{m n,} .

( )

(

)

(

)

1 , 1

1

1

, if

3

1

, if

,

2,3, 4,...,

,

3

1

3

2

, if

,

1, 2,3,...,

3

3

k m n k

n

u

n

u k

m

R

L

n

k

k

v k

n













₊



−



₌

 

_

_



_





_{ }

 +

=

=

−

_{= }

_

_{ }

 







_

₋

_

_

₋

₊

_

+

+

=

=



_

_

_

_













Proof:

Fig 1: A Lollipop Graph

L

_{m n,} .

For convenience let us consider the lollipop graph

L

_{m n,} into two entities. One is complete graph

K

_m

with vertices



u u u

₁

,

₂

,

₃

,...,

u

m



and another one is the path graph

P

n with vertices



v v v

1

,

2

, ,...,

3

v

n



.

Case (i): Suppose

u

₁ is the reserved vertex.

Then

u

₁ must be in the dominating set and

u

₁ dominates the vertices



u u u

₂

,

₃

,

₄

,...,

u

_m

  

v

₁ . Now it is enough to find the dominating set for the remaining vertices



v v

₂

, ,...,

₃

v

_n



.

The

L

_{m n,}

 

V

₁ with

V

₁

=



v v

₂

, ,...,

₃

v

_n



is nothing but

P

_n−1. Hence

R

_{( )1}

−



(

L

_{m n,}

,

u

₁

)

= +

1



(

P

_n−1

)

1

1

3

n −





= + 

_



_

.

Case (ii): Suppose

u k

_k

,

=

2,3, 4,...,

m

is the reserved vertex.

k

u

Dominates the vertices

u u

₁

,

₂

,...,

u

_k−1

,

u

_k+1

,...,

u

_m and doesn’t dominate the vertices

v v v

₁

,

₂

, ,...,

₃

v

_n. Since the

L

_{m n,}

 

V

₂ with

V

₂

=



v v v

₁

,

₂

, ,...,

₃

v

_n



is

P

_n,

( )1

(

m n,

,

)

1

( )

n

R

−



L



= +



P

1

3

n

 

= +  

_{ }

, where



=

u k

_k

,

=

2,3,...,

m

.

Case (iii): Suppose

v k

k

,

=

1, 2,3,...,

n

is the reserved vertex.

To dominate the set



u u u

₁

,

₂

,

₃

,...,

u

_m



it is enough to choose anyone of the vertex from that set. But if we choose

u

₁ alone then it would dominate the vertex



u u

₂

,

₃

,...,

u

_m



as well as

 

v

₁ . So

u

₁ must be in the required dominating set. Since

v

_k is the reserved vertex, it dominates

v

_k−1 and

v

_k+1.

Now the vertices which are not dominated while considering the set



v u

_k

,

₁



as a subset of the dominating set are

U

=



v v

₂

, ,...,

₃

v

_k−2

,

v

_k+2

,...,

v

_n



.

 

_{( )} , 3 1 m n k n k

L

U

=

P

₋

P

_{− +} .





(

L

_{m n,}

 

U

)

=



(

P

_k−3

)

+



(

P

_{n− +(k 1)}

)

. Hence

R

_{( )1}

−



(

L

_{m n,}

,



)

=



u v

₁

,

_k



+



(

P

_k−3

)

+



(

P

_{n− +(k 1)}

)

2

3

(

1

)

3

3

n

k

k

−



− +







= +

_

_

_{+ }

_



_{ }

_

, where



=

v k

k

,

=

1, 2,3,...,

n

.

Definition: Tadpole Graph:

The

(

m n

,

)

Tadpole graph, also called dragon graph denoted by

T

_{m n,} is the graph obtained by joining a cycle graph

C

_m to a path graph

P

_n with a bridge.

( )

(

)

(

)

(

)

(

)

(

)

, 1 1, 2, 3,..., 1 , if 3 3 2, 3, 5,..., 1, for , 0 mod 3 1 1 , if 1, 4, 7,..., 5, 2 3 3 1, 4, 7,..., 1 , if 3 3 , k k k k m n u k m m n v k n n m n m n v k n n u k m n R T











 =  −    ₊  _{= }      _{= −}     _  _  − +    ₊  _{= = −} −           = −   ₊  ₌         − =

(

)

(

)

(

)

(

)

(

)

(

)

(

)

5, 2 3, 6, 9,..., 4, 1

for 0 mod 3 , 1 mod 3

2, 3, 5,..., 1, , if 3 3 1, 2, 4,..., 2, 1, 2, 3,..., , if 3 3 1, 2, 3,..., k k k k k m m v k n n m n u k m m m n v k n n u k m m n v k





 − −  _  _{= − −}   _{ }  = −  _     _{+ = }      _{= −}     _{ } =    _{+ =}     ₌    

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

for 0 mod 3 , 2 mod 3 1, 2, 3,...,

, if for 1 mod 3 , 0 mod 3

3 3 1, 2, 3,..., 1, 2, 3,..., 1 , if 3 3 k k k k m n n u k m m n m n v k n u k m m n v k





 _{ }   =     _{+ =  }      ₌     _ = −   ₊  ₌        

(

)

(

)

(

)

(

)

(

)

1, 3, 4,..., 1, for , 1 mod 3 , if 2, 5,8,..., 5, 2 3 3 1, 2, 4,..., 2, 1 , if 3 3 2, 3, 5,..., 2, 1 1 , if 3 3 k k k n n m n m n v k n n u k m m m n v k n n m n





  _  _{= −}   _{ }     _{+ = = −} −           = −  −    ₊  =      _{= −}     _ − +   ₊         

(

)

(

)

(

)

(

)

(

)

(

)

(

)

for 1 mod 3 , 2 mod 3

3, 6, 9,..., 4, 1 1, 4, 7,..., 4, 1 1, 2, 3,..., 1 , if 3 3 2, 3, 5,..., 1, 1 1 , if 1, 4, 7,..., 5, 2 3 3 k k k k k m n u k m m v k n n u k m m n v k n n m n v k n n







      = − −  _  =  _ = − −    =  −    ₊  _{= }      _{= −}     _   − +   ₊  _{= = − −}        

(

)

(

)

(

)

(

)

(

)

(

)

for 2 mod 3 , 0 mod 3

1, 3, 4,..., 2, 1 1 , if 3 3 3, 6, 9,..., 4, 1 for 2 mo 2, 5,8,..., 3, , if 3 3 1, 2, 4,..., 2, k k k k m n u k m m m n v k n n m u k m m m n v k n n





      = − −  −    ₊  _{= }      _{= − −}       _  = −  _     _{+ = }      _{= −}     _{ }

(

)

(

)

(

)

(

)

(

)

d 3 , 1 mod 3 1, 2, 3,..., , if for , 2 mod 3 3 3 1, 2, 3,..., k k n u k m m n m n v k n



                                                  _     _{     =} _{   }+ =  =      _  Proof:

For convenience let us consider the tadpole graph

T

_{m n,} into two entities. One is cycle graph

C

_m with vertices



u u u

₁

,

₂

,

₃

,...,

u

_m



and another one is the path graph

P

_n with vertices



v v v

₁

,

₂

, ,...,

₃

v

_n



.

Fig 2: A Tadpole Graph

T

_{m n,} . Case (i):

m 

0 mod 3

(

)

, then

m

=

3

k

where

k =

1, 2, 3,...

.

The possible minimum dominating sets of

C

_m are





1 1

,

4

,

7

,...,

m 5

,

m 2

S

=

u u u

u

−

u

− ,

S

2

=



u u u

2

,

5

,

8

,...,

u

m−4

,

u

m−1



and

S

3

=



u u u

3

,

6

,

9

,...,

u

m−3

,

u

m



.

Sub case (i):

n 

0 mod 3

(

)

, then

m

=

3

k

where

k =

1, 2, 3,...

.

a) Let any vertex of the set

S

₁ be a reserved vertex say

u

_k. Then

u

_k must be in the required dominating set, if

u

_k



S

₁ then

S

₁ must be in the required dominating set, also dominated in the vertex

v

₁. Now it is enough to find the dominating set



v v v

₂

, ,

_{3 4}

,...,

v

_n



. i.e.,

P

_n−1.

Hence

R

_{( )1}

−



(

T

_{m n,}

,



)

=

S

₁



(

P

_n−₁

)

1

3

3

m

n −

  



=

_{  }

+

_

  



, where



=

S

1.

b) Let any vertex of the sets

S

₂and

S

₃ be a reserved vertex say

u

_k. Then

u

_k must be in the required dominating set, if

u

_k



S

₂ then

S

₂ must be in the required dominating set. If

u

_k



S

₃ then

S

₃ must be in the required dominating set. Now it is enough to find the dominating set



v v v

₁

,

₂

, ,...,

₃

v

_n



. i.e.,

P

_n.

Hence

R

_{( )1}

−



(

T

m n_,

,



)

=

S

₂



( )

P

n

3

3

m

n

   

=

_{   }

+

   

1

3

3

m

n −

  



=

_{  }

+

_

  



, where



=

S

2.

1

3

3

n

n



  

₌

− 





  

_{  }



_







R

_{( )1}

−



(

T

_{m n,}

,



)

=

S

₃



( )

P

_n

3

3

m

n

   

=

_{   }

+

   

1

3

3

m

n −

  



=

_{  }

+

_

  



, where



=

S

3.

1

3

3

n

n



  

₌

− 





  

_{  }



_







c) Let any one of the vertex

v v v

₁

,

₄

,

₇

,...,

v

_n−5

,

v

_n−2 be a reserved vertex say

v

_k. Then

v

_k must be in the required dominating set, also dominated in the vertex

u

₁. Consider the induced sub graph of the set



u v v

1

, ,

1 2

,...,

v

n



=

P

n+1. Now it is enough to find the dominating set



u u u

2

,

3

,

4

,...,

u

m



. i.e.,

P

m−1. Hence

R

_{( )1}

−



(

T

m n_,

,



)

=



(

P

m−₁

) (

+



P

n+₁

)

1

1

3

3

m

−

n

+



 



=

_

_{ }

+

_



 



, where



=

v v v

1

,

4

,

7

,...,

v

n−5

,

v

n−2.

d) Let any one of the vertex

v v v

₂

, , ,...,

_{5 8}

v

_n−4

,

v

_n−1 be a reserved vertex say

v

_k. Then

v

_k must be in the required dominating set. Now it is enough to find the dominating set



u u u

₁

,

₂

,

₃

,...,

u

_m



. i.e.,

C

_m.

Hence

R

_{( )1}

−



(

T

m n_,

,



)

=



( ) ( )

C

m

+



P

n

3

3

m

n

   

=

_{   }

+

   

1

3

3

m

n −

  



=

_{  }

+

_

  



,

1

3

3

n

n



  

₌

− 





  

_{  }



_







, where



=

v v v

₂

, , ,...,

_{5 8}

v

_n−4

,

v

_n−1.

e) Let any one of the vertex

v v v

₃

, , ,...,

_{6 9}

v

_n−3

,

v

_n be a reserved vertex say

v

_k. Then

v

_k must be in the required dominating set and also

u

₁ dominating in the vertex

v

₁. Now it is enough to find the dominating set



u u u

1

,

2

,

3

,...,

u

m



. i.e.,

C

m. Hence

R

_{( )1}

−



(

T

_{m n,}

,



)

=



( ) (

C

_m

+



P

_n−1

)

1

3

3

m

n −

  



=

_{  }

+

_

  



, where



=

v v v

3

, , ,...,

6 9

v

n−3

,

v

n.

Combining the results of (a), (b), (c), (d) and (e),

( )

(

)

(

)

(

)

(

)

, 1

1, 2,3,...,

1

, where

3

3

2,3,5,...,

1,

,

1

1

, where

1, 4, 7,...,

5,

2

3

3

k k m n k

u

k

m

m

n

v

k

n

n

R

T

m

n

v

k

n

n











_{  }

₋

_



_

=

+

=



_{  }

_

_

=

−

  



_



−

_{= }





−

 

₊

+



₌

₌

₋

₋



_

 

_{ }



_



.

Sub case (ii):

n 

1 mod 3

(

)

, then

m

=

3

k

+

1

where

k =

1, 2, 3,...

.

a) Let any vertex of the set

S

₁ be a reserved vertex say

u

_k. Then

u

_k must be in the required dominating set, if

u

_k



S

₁ then

S

₁ must be in the required dominating set, also dominated in the vertex

v

₁. Now it is enough to find the dominating set



v v v

₂

, ,

_{3 4}

,...,

v

_n



. i.e.,

P

_n−1.

Hence

R

_{( )1}

−



(

T

m n_,

,



)

=

S

₁



(

P

n−₁

)

1

3

3

m

n −

  



=

_{  }

+

_

  



, where



=

S

1.

b) Let any vertex of the sets

S

₂and

S

₃ be a reserved vertex say

u

_k. Then

u

_k must be in the required dominating set, if

u

_k



S

₂ then

S

₂ must be in the required dominating set. If

u

_k



S

₃ then

S

₃ must be in the required dominating set. Now it is enough to find the dominating set



v v v

₁

,

₂

, ,...,

₃

v

_n



. i.e.,

P

_n.

Hence

R

_{( )1}

−



(

L

_{m n,}

,



)

=

S

₂



( )

P

_n

3

3

m

n

   

=

_{   }

+

   

, where



=

S

2.

R

_{( )1}

−



(

T

_{m n,}

,



)

=

S

₃



( )

P

_n

3

3

m

n

   

=

_{   }

+

   

, where



=

S

3.

c) Let any one of the vertex

v v v

₁

,

₄

,

₇

,...,

v

_n−3

,

v

_n be a reserved vertex say

v

_k. Then

v

_k must be in the required dominating set, also dominated in the vertex

u

₁. Consider the induced sub graph of the set



u v v

1

, ,

1 2

,...,

v

n



=

P

n+1. Now it is enough to find the dominating set



u u u

2

,

3

,

4

,...,

u

m



. i.e.,

P

m−1. Hence

R

_{( )1}

−



(

T

_{m n,}

,



)

=



(

P

_m−1

) (

+



P

_n+1

)

1

1

3

3

m

−

n

+



 



=

_

_{ }

+

_



 



3

3

m

n

   

=

_{   }

+

   

,

1

1

,

3

3

3

3

m

n

n

n





−

   

₌

+

  

₌







_

   

_{   }

  

_{  }







where



=

v v v

₁

,

₄

,

₇

,...,

v

n−₃

,

v

n.

d) Let any one of the vertex

v v v

₂

, , ,...,

_{5 8}

v

n−₂

,

v

n be a reserved vertex say

v

k. Then

v

k must be in the

required dominating set. Now it is enough to find the dominating set



u u u

₁

,

₂

,

₃

,...,

u

_m



. i.e.,

C

_m. Hence

R

_{( )1}

−



(

T

m n_,

,



)

=



( ) ( )

C

m

+



P

n

3

3

m

n

   

=

_{   }

+

   

, where



=

v v v

2

, , ,...,

5 8

v

n−2

,

v

n.

e) Let any one of the vertex

v v v

₃

, , ,...,

_{6 9}

v

_n−4

,

v

_n−1 be a reserved vertex say

v

_k. Then

v

_k must be in the required dominating set and also

u

₁ dominating in the vertex

v

₁. Now it is enough to find the dominating set



u u u

1

,

2

,

3

,...,

u

m



. i.e.,

C

m. Hence

R

_{( )1}

−



(

T

_{m n,}

,



)

=



( ) (

C

_m

+



P

_n−1

)

1

3

3

m

n −

  



=

_{  }

+

_

  



, where



=

v v v

3

, , ,...,

6 9

v

n−4

,

v

n−1. Combining the results of (a), (b), (c), (d) and (e),

( )

(

)

(

)

(

)

(

)

(

)

, 1

1, 4, 7,...,

5,

2

1

, where

3

3

3, 6, 9,...,

4,

1

,

2, 3, 5,...,

1,

, where

3

3

1, 2, 4,...,

2,

k k m n k k

u

k

m

m

m

n

v

k

n

n

R

T

u

k

m

m

m

n

v

k

n

n











_{  }

₋

_



_

=

−

−

+

=



_{  }

_

_

=

−

−

  





_



−

_{= }

=

−





   

₊

_{= }



   

_{   }

₌

₋





.

Sub case (iii):

n 

2 mod 3

(

)

, then

m

=

3

k

+

2

where

k =

1, 2, 3,...

.

a) Let any vertex of the set

S

₁ be a reserved vertex say

u

_k. Then

u

_k must be in the required dominating set, if

u

_k



S

₁ then

S

₁ must be in the required dominating set, also dominated in the vertex

v

₁. Now it is enough to find the dominating set



v v v

₂

, ,

_{3 4}

,...,

v

_n



. i.e.,

P

_n−1.

Hence

R

_{( )1}

−



(

T

m n_,

,



)

=

S

₁



(

P

n−₁

)

1

3

3

m

n −

  



=

_{  }

+

_

  



3

3

m

n

   

=

_{   }

+

   

,where



=

S

1.

1

3

3

n

n





−

  

₌







_

  

_{  }







b) Let any vertex of the sets

S

₂and

S

₃ be a reserved vertex say

u

_k. Then

u

_k must be in the required dominating set, if

u

_k



S

₂ then

S

₂ must be in the required dominating set. If

u

_k



S

₃ then

S

₃ must be in the required dominating set. Now it is enough to find the dominating set



v v v

₁

,

₂

, ,...,

₃

v

_n



. i.e.,

P

_n.

Hence

R

_{( )1}

−



(

L

_{m n,}

,



)

=

S

₂



( )

P

_n

3

3

m

n

   

=

_{   }

+

   

, where



=

S

2.

R

_{( )1}

−



(

T

_{m n,}

,



)

=

S

₃



( )

P

_n

3

3

m

n

   

=

_{   }

+

   

, where



=

S

3.

c) Let any one of the vertex

v v v

₁

,

₄

,

₇

,...,

v

_n−4

,

v

_n−1 be a reserved vertex say

v

_k. Then

v

_k must be in the required dominating set, also dominated in the vertex

u

₁. Consider the induced sub graph of the set



u v v

1

, ,

1 2

,...,

v

n



=

P

n+1. Now it is enough to find the dominating set



u u u

2

,

3

,

4

,...,

u

m



. i.e.,

P

m−1. Hence

R

_{( )1}

−



(

T

m n_,

,



)

=



(

P

m−₁

) (

+



P

n+₁

)

1

1

3

3

m

−

n

+



 



=

_

_{ }

+

_



 



3

3

m

n

   

=

_{   }

+

   

,

1

1

,

3

3

3

3

m

n

n

n





−

   

₌

+

  

₌







_

   

_{   }

  

_{  }







where



=

v v v

₁

,

₄

,

₇

,...,

v

_n−4

,

v

_n−1.

d) Let any one of the vertex

v v v

₂

, , ,...,

_{5 8}

v

_n−3

,

v

_n be a reserved vertex say

v

_k. Then

v

_k must be in the required dominating set. Now it is enough to find the dominating set



u u u

₁

,

₂

,

₃

,...,

u

_m



. i.e.,

C

_m.

Hence

R

_{( )1}

−



(

T

m n_,

,



)

=



( ) ( )

C

m

+



P

n

3

3

m

n

   

=

_{   }

+

   

, where



=

v v v

2

, , ,...,

5 8

v

n−3

,

v

n.

e) Let any one of the vertex

v v v

₃

, , ,...,

_{6 9}

v

_n−2

,

v

_n be a reserved vertex say

v

_k. Then

v

_k must be in the required dominating set and also

u

₁ dominating in the vertex

v

₁. Now it is enough to find the dominating set



u u u

1

,

2

,

3

,...,

u

m



. i.e.,

C

m. Hence

R

_{( )1}

−



(

T

_{m n,}

,



)

=



( ) (

C

_m

+



P

_n−1

)

1

3

3

m

n −

  



=

_{  }

+

_

  



,

3

3

m

n

   

=

_{   }

+

   

,

1

3

3

n

n





−

  

₌





_

_

_{  }

_{  }







where



=

v v v

₂

, , ,...,

_{5 8}

v

_n−3

,

v

_n. Combining the results of (a), (b), (c), (d) and (e),

( )

(

)

(

)

(

)

, 1

1, 2, 3,...,

,

, where

3

3

1, 2, 3,...,

k m n k

u

k

m

m

n

R

T

v

k

n





   





=



−

=

_{   }

+

_{= }

=

   

_

.

Case (ii):

m 

1 mod 3

(

)

, then

m

=

3

k

+

1

where

k =

1, 2, 3,...

. i)

n 

0 mod 3

(

)

, then

m

=

3

k

, where

k =

1, 2, 3,...

.

We can also get the same result for

m



0 mod 3 ;

(

)

n



2 mod 3

(

)

. ( )

(

)

(

)

(

)

, 1

1, 2, 3,...,

,

, where

3

3

1, 2, 3,...,

k m n k

u

k

m

m

n

R

T

v

k

n





   





=



−

=

_{   }

+

_{= }

=

   

_

ii)

n 

1 mod 3

(

)

, then

m

=

3

k

+

1

, where

k =

1, 2, 3,...

. We can also get the same result for

m



0 mod 3 ;

(

)

n



1 mod 3

(

)

.

( )

(

)

(

)

(

)

(

)

, 1

1, 2,3,...,

1

, where

3

3

1,3, 4,...,

1,

,

, where

2,5, 6,...,

5,

2

3

3

k k m n k

u

k

m

m

n

v

k

n

n

R

T

m

n

v

k

n

n











_{  }

₋

_



_

=

+

=



_{  }

_

_

=

−

  



_



−

_{= }

   

₊

₌

₌

₋

₋

   

_{   }



.

iii)

n 

2 mod 3

(

)

, then

m

=

3

k

+

2

, where

k =

1, 2, 3,...

. We can also get the same result for

m



0 mod 3 ;

(

)

n



0 mod 3

(

)

.

( )

(

)

(

)

(

)

(

)

(

)

, 1

1, 2, 4,...,

2,

1

, where

3

3

2, 3, 5,...,

2,

,

2, 6, 9,...,

4,

1

1

1

, where

3

3

1, 4, 7,...,

4,

1

k k m n k k

u

k

m

m

m

n

v

k

n

n

R

T

u

k

m

m

m

n

v

k

n

n











_{  }

₋

_



_

=

−

+

=



_{  }

_



=

−

  





_



−

_{= }

=

−

−







−

 

₊

+



_{= }





_

 

_{ }



_

₌

₋

₋





Case (iii):

m 

2 mod 3

(

)

, then

m

=

3

k

+

2

where

k =

1, 2, 3,...

. i)

n 

0 mod 3

(

)

, then

m

=

3

k

, where

k =

1, 2, 3,...

.

We can also get the same result for

m



0 mod 3 ;

(

)

n



0 mod 3

(

)

.

( )

(

)

(

)

(

)

(

)

, 1

1, 2,3,...,

1

, where

3

3

2,3,5,...,

1,

,

1

1

, where

1, 4, 7,...,

5,

2

3

3

k k m n k

u

k

m

m

n

v

k

n

n

R

T

m

n

v

k

n

n











_{  }

₋

_



_

=

+

=



_{  }

_



=

−

  



_



−

_{= }





−

 

₊

+



₌

₌

₋

₋



_

 

_{ }



_



.

ii)

n 

1 mod 3

(

)

, then

m

=

3

k

+

1

, where

k =

1, 2, 3,...

. We can also get the same result for

m



0 mod 3 ;

(

)

n



1 mod 3

(

)

.

( )

(

)

(

)

(

)

(

)

(

)

, 1

1, 3, 4,...,

2,

1

1

, where

3

3

3, 6, 9,...,

4,

1

,

2, 5,8,...,

3,

, where

3

3

1, 2, 4,...,

2,

k k m n k k

u

k

m

m

m

n

v

k

n

n

R

T

u

k

m

m

m

n

v

k

n

n











_{  }

₋

_



_

=

−

−

+

=



_{  }

_

_

=

−

−

  





_



−

_{= }

=

−





   

₊

_{= }



   

_{   }

₌

₋





.

iii)

n 

2 mod 3

(

)

, then

m

=

3

k

+

2

, where

k =

1, 2, 3,...

. We can also get the same result for

m



0 mod 3 ;

(

)

n



2 mod 3

(

)

.

( )

(

)

(

)

(

)

, 1

1, 2, 3,...,

,

, where

3

3

1, 2, 3,...,

k m n k

u

k

m

m

n

R

T

v

k

n





   





=



−

=

_{   }

+

_{= }

=

   

_

.

The pan graph is the graph obtained by joining a cycle graph

C

_n to a singleton graph

K

₁ with a bridge. It is denoted by

P

_n,1.

Theorem: For the Pan graph

P

_n,1,

n 

3

the reserved domination number is

( )

(

)

(

)

₍

₎

,1 1 1

2

1

1

, if

1, 2, 3,...,

3

3

,

1

1

, if

3

k n

n

k

k

u

k

n

R

P

n

v











₊



−



₊



−

+



₌

₌















_{ }

_

−

_{= }

−





 +

_

_

=



_

_



Proof:

Fig 3: A Pan Graph

P

_n,1.

For convenience let us consider the pan graph

P

_n,1 into two entities. One is cycle graph

C

_n with vertices



u u u

1

,

2

,

3

,...,

u

n



and another one is the singleton graph

K

1 with vertex

 

v

1 . Case (i): Suppose

u k

_k

,

=

1, 2,3,...,

n

is the reserved vertex.

In order to dominate the vertex

v

₁, the vertex

u

₁ must chose in the dominating set.

So the minimal reserved dominating set with the vertex

u

_k

(

k

=

1, 2,3,...,

n

)

of the pan graph

P

_n,1 is the minimal

2

-reserved dominating set with the vertices



u v

₁

,

_k



of the cycle graph

C

_n. Hence

( )

(

)

(

)

( )

(





)

,1 1 1 2

where

,

1, 2,3,.

..,

,

,

,

k n n k

u k

R

P

R

C

u u

n











−

_{ =}

−





=

=

2

1 3

(

1

) (

3

)

3

3

n

k

k

− −



+ − +







= +

_

_

_{+ }

_



_{ }

_

2

1

1

2

3

3

k

−

n k

− −









= +

_

_

− +

_

_









1

1

(

2

)

3

3

n

k

k

−



− +







= +

_

_

_{+ }

_



_{ }

_

.

Case (ii): Suppose

v

₁ is the reserved vertex.

Then

v

₁ must be in the dominating set and

v

₁ dominates the vertex

u

₁. Now it is enough to find the dominating set for the remaining vertices



u u

₂

,

₃

,...,

u

_n



.

The

P

_n,1

 

V

with

V

=



u u

₂

,

₃

,...,

u

_n



is nothing but

P

_n−1. Hence

R

_{( )1}

−



(

P

_n,1

,

v

₁

)

= +

1



(

P

_n−1

)

1

1

3

n −





= + 

_



_

.

Definition: Barbell Graph

The barbell graph is the simple graph obtained by joining two copies of complete graph

K

_n by a bridge (where

3

n 

). It is denoted by

B

_n.

Remark: For the barbell graph

B

_n,

n 

3

the reserved domination number is

R

₍₁₎

−



(

B

_n

,



)

=

2

, if

(

)

(

)

1, 2, 3,...,

1, 2, 3,...,

k k

u

k

n

v k

n



= 



=

=



.

5. 2-reserved domination number:

Definition: Path graph:

The path graph

P

_n is a tree with two nodes of vertex degree

1

, and the other

n −

2

nodes of vertex degree

2

.

Result: For the path graph

P

₂, the

2

-reserved domination number is

R

_{( )2}

−



( )

P

₂

=

2

.

Result: For the path graph

P n 

_n

,

3

the

2

-reserved domination number is

( )2

(





)

(

)

1

2

3

,

,

2

, if

3

3

3

n k l

n

l

k

l

k

R

−



P

v v

= +



_

−

_{ }

 

+

− −



_

+



_

− +



_

k



l



 

_{ }

_

(

k l

,

=

1, 2,3,...,

n

)

.

Definition: Cycle graph:

A cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with n vertices is called

n

C

. The number of vertices in

C

_n equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it.

Result: For the cycle graph

C

_n,

n 

3

the

2

-reserved domination number is

( )2

(





)

(

) (

)

3

3

,

,

2

, if

3

3

n k l

n k

l

l

k

R

−



C

v v

= +

_



− −



_

+



_

+

− +



_

k



l



_{ }



(

k l

,

=

1, 2,3,...,

n

)

. Definition: Wheel Graph

A wheel graph is a graph

W

_n formed by connecting a single universal vertex to all vertices of a cycle. To denote a wheel graph with

n +

1

vertices

(

n 

3

)

, which is formed by connecting a single vertex to all vertices of a cycle of length

n

.

Result: For the wheel graph

W

_n

,

n 

5

the

2

-reserved domination number is

( )

(

)









1 2

2 if

,

,

3 if

,

.

l n k l

u v

R

W

v v









=



−

_{= }

=



Definition: Star Graph

A star graph

S

_n is the complete bipartite graph

K

_1,n, a tree with one internal node and

n

leaves (but no internal nodes and

k +

1

leaves when

k 

1

).

Result: For the star graph

S

_n

=

K

_1,n,

n 

3

the 2-reserved domination number is

( )

(

)









1 2

2 if

,

,

3 if

,

.

l n k l

u v

R

S

v v









=



−

_{= }

=

