Domination in Semitotal-Point Graph

Domination in Semitotal-Point Graph

B. BASAVANAGOUD*, S.M.HOSAMANI and S.H.MALGHAN Department of Mathematics,

Karnatak University,Dharwad-580 003, Karnataka, India e-mail*: [email protected]

[email protected]

*Corresponding author.

ABSTRACTS

A set D of vertices in a graph G is a dominating set of G if every vertex not in D is adjacent to atleast one vertex in D. The domination number



(G)



of G is the cardinality of a minimum dominating set of G. The semitotal-point graph T₂ (G) of G is the graph whose vertex set is

) ( )

( G E G

V 

, where two vertices are adjacent if and only if (i) they are adjacent vertices of G or (ii) one is a vertex of G and the other is an edge of G, incident with it. In this paper, some results on domination number of semitotal-point graph are obtained.

2000 Mathematics Subject Classification: 05C69.

Keywords: semitotal-point graph, domination number, connected domination number, total domination number.

INTRODUCTION

The graphs considered here are finite, undirected without loops or multiple edges. Any undefined term in this paper may be found in Harary³. Let

G = ( V , E )

be a graph with

V = p

and

E = q

. In a graph G if deg(v) = 1, then v is called an end vertex or a pendant vertex of G. Each

vertex

v  V

dominates every vertex in itss closed neighbourhood

N [v ]

. A set

S  V

is a dominating set if each vertex in V is dominated by atleast one vertex of S. The dominating number

 (G )

is the smallest cardinality of a dominating set.

A dominating set D of a graph G is a split dominating set if the induced subgraph

V  D

is disconnected. The

split domination number



_s

(G )

of G is the minimum cardinality of a split dominating set.

A dominating set D of a graph G is a connected dominating set if the induced sub graph

D

is co nne cte d. The connected domination number



_c

(G )

of G is th e minimu m ca rdin ality of a connected dominating set.

A dominating set D is a total dominating set if the induced subgraph

D

has no isolated vertex. The total domination number



_t

(G )

of a graph G is the minimum cardinality of a total dominating set.

A set D of vertices of a graph G is a maximal dominating set of G if D is a dominating set and

V  D

is not a dominating set. The maximal domination number



_m

(G )

of G is the minimum cardinality of a maximal dominating set.

A dominating set D of G is a cototal dominating set if the induced subgraph

D

V 

has no isolated vertices. The cototal domination number



_cot

(G )

of G is the minimum cardinility of a cototal dominating set.

A dominating set D of vertices in a graph G is a dominating clique if the induced subgraph

D

is a complete graph. The clique domination number

)

cl

(G



of G is the minimum cardinality of a dominating clique.

A dominating set D of a connected graph G is a path nonsplit dominating set if the induced subgraph

V  D

is a path

in G. The path nonsplit domination number

)

pns

(G



is the minimum cardinality of a path nonsplit dominating set.

The maximum number of classes of a domatic partition of G is called the domatic number of G and is denoted by d(G). The vertex independence number

)

0

( G



is the maximum cardinality among the independent set of vertices of G.

E.Sampatku mar and S. B.

Chikkodimath⁶ introduced the new graph valued function, namely the semitotal-point graph of a graph.

The semitotal-point graph T₂(G) of a graph G is the graph 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 of G and the other is an edge of G, incident with it.

In Figure 1, a graph and its semitotal- point graph are shown.

1

2 3 4

1

2 3 4

G : T₂(G) :

a

b

c d

a

b

c d

Figure 1

PRELIMINARY RESULTS

We need the following Theorems for our further results.

Theorem A[4]. For any graph G with a pendant vertex

 ( G ) = 

_s

( G ) .

Theorem B[1]. If G is a connected graph with

p  3

vertices, then

.

3 ) 2

( p

t

G 



Theorem C[7]. If T is a tree with

p  3

vertices, then



_c

( T ) = p  e .

where e denotes the number of pendant vertices in T.

The orem D[3 ]. For a ny n ontrivial con nected

( p , q )

graph G

.

=

=

_{1 1}

0

0

  

  p 

Theorem E[8]. For any connected

( p , q )

graph G

( ) .

)

( d G

G p

p  





 



 

Theorem F[2]. If G is a graph without iso late d vertice s of orde r p, th en

) 2

( p

 G 

_.

Next we give the domination number of semitotal-point graphs of some standard class of graphs.

Proposition 1. For any path P_P with

p  2

vertices,

.

= 2 )) (

(

₂







  P  p T

_p



Proposition 2. For any cycle C_P with

p  3

vertices,

.

= 2 )) (

(

₂







  C  p T

_p



Proposition 3. For any wheel W_P with

4



p

vertices,

1 .

= 2 )) (

(

₂

 





  W  p T

_p



Proposition 4. For any star

K

_{1, p 1} with

 2

p

vertices,

 ( T

₂

( K

_1,p1

)) = 1 .

Proposition 5. For any K_P with

p  2

vertices,

 ( T

₂

( K

_p

)) = p  1 .

Proposition 6. If

G = K

_p^{, then}

G  T

₂

( G )

.

MAIN RESULTS

Theorem 3.1. For any

( p , q )

^{graph G,}

))

( ( )

( G  T

₂

G

 

. The equality holds for

G = K

₂ or

K

_p^.

Proof. We consider the following cases.

Case 1. Let G be a graph without isolated vertices.Then by Theorem F,

) 2

( p

 G 

_. The refo re by def inition o f

T

₂

( G )

^,

)) 2 (

(

₂

p q

G

T 

 

_{. Hence}

)) ( ( )

( G  T

₂

G

 

^.

Case 2. Let G be a graph with isolated vertices of the type

G = H  nK

₁^{, where}

) , (

= p q

H  

is any connected graph.Then

1

2

( G ) = H nK

T  

. By Case 1,

) ( )

( H   H 



. Therefore

p n

G  

 2 ) (



and

p q n

G

T    

 2

) ) ( (

₂



. Hence the

result. Equality can be easily verified for

= K

2

G

or

K

p^.

Theorem 3. For any nontrivial tree T,

)

( )) (

( T

₂

T  T

 

^.

Proof. Let T be any nontrivial tree. Let

}

, , {

=

_{1 2}

1

v v v

r

D 

be a subset of vertex set such tha t

degv

_i

 2

f or

1  i  r

^{a nd}

} , , {

=

_{1 2}

2

u u u

s

D 

where

degu

_i

= 1

f or

s

i 



1

be the subsets of

V (T )

^where

2

=

1

)

( T D D

V 

^{. Clearly}

D

₁is a minimum dominating set o f T. Therefo re

r D T ) =

( 

₁



. Let

D

₁

 = { v

₁

 , v

₂

 ,  v

_r

 }

^and

}

, , {

=

_{1 2}

2

u u u

s

D     

be the corresponding vertice s of D₁ and D₂ in

T

₂

( G )

respectively. Let

u

₁

, u

₂

,  u

_n

 E ( T )

and

u

n

u

u

₁

 , 

₂

,  

be the corresponding edge vertices in

T

₂

( G )

. For each edge

u

_i for

n i 



1

there exists a vertex

v

_j

 D

₁



. Clearly

 ( T )  D

₁

 D '

₁

  ( T

₂

( T ))

. Hence the result.

Corollary 1. In a tree if every nonpendant vertex is adjacent to at least one pendant vertex, then

 ( T

₂

( T )) =  ( T )

Theorem 3.3 For any

( p , q )

^{graph G,}

p G T ( ))  (

₂



. The equality holds for

K

p

G =

^.

Proof. We consider the following cases.

Case 1. Suppose G is a connected graph.

The n

 ( G )  p  1

. There fore by Proposition 5,

 ( T

₂

( G ))  p  1

^.

Case 2. Let G be a disconnected graph with n components that is

G = H₁



H₂







H_n.

The n

T

₂

( G ) = H

₁

  H 

₂

 





H’_n. W ee consider the following subcases.

Subcase 2.1. Suppose each component

H

i;

1  i  n

is a complete graph of order

2 

, then by Ca se 1 ,

2 ) ' (

₁

p

¹

H  



,

2 ) ' (

₂

p

²

H  



_,,

2 ) '

(

_n

p

ⁿ

H  



_{whe re}

) (

= '

₁

V H

₁

p 

,

p '

₂

= V ( H

₂

 )

,

p '

_n

= V ( H

_n

 )

. Then

 ( T

₂

( G )) = p  n

, where n is the number of components in G. Hence

1

<

)) (

( T

₂

G p 



^.

Subcase 2.2. Suppose the components

H

i;

1  i  n

, are either complete or other than complete graphs then by Subcase 2.1,

 ( T

₂

( G )) < p  n < p  1

.The equality is easy to verify for

G = K

_p^.

Corollary 2. For a ny con nected

)

,

( p q

^{graph G,}

 ( T

₂

( G ))  p  1

^.

Proposition 7. For a ny g raph G ,

0 0 2

( ))

(  

 T G  

.

Proof. Follows from Theorem D and Theorem 3.3.

Theorem 3.4 For any

( p , q )

graph G,

0 0 2

( )) ) (

(

1      



 T G

G p

.

Further the equality of lower bound is attained if

G = K

_p^.

Proof. We first consider the lower bound.

Let D be a dominating set of G. Each vertex dominates at most itself and other vertice s o f

 (G )

^{. Hen ce}

)) ( ) (

(

1 T

²

G

G p  





^.

If

G = K

_p, then by The ore m 3.3,

p

K T (

p

)) = (

₂



. Hence the lower bound is attained.

Th e up per boun d fo llows from the Proposition 7.

Theorem 3.5 For any

( p , q )

graph G,

p

G T G

p  

₀

( )   (

₂

( ) 

.

Further the equality of lower bound is attained if

G = P

_por

C

_p.

Proof. We first consider the lower bound.

It is obvious that fo r any graph G ,

))

( ( )

(

₂

0

G T G

p    

. For equality,we consider the following cases:

Case 1. If

G = P

_p for

p  2

vertices then

 



 



= 2 )

0

( P

_p

p



. Therefore

p  

₀

( G ) =

2 .

2 = 





 



 



 

  p p

p

By Pro position 1,

 



 



= 2 )) (

(

₂

p

P T

_p



. Hence the equality..

Case 2. If

G = C

_p

p  3

vertices then

2 .

= )

0

( 





  C

_p

 p



Therefore

p  

₀

( G ) = ^.

= 2

2 





 



 



 

  p p

p

By Proposition 2,







 



= 2 )) (

(

₂

p

C T

_p



The upper bound follows from Theorem 3.3.

Corolla ry 3 . Fo r any grap h G ,

)).

( ( )

(

₂

0

G  T G

 

The orem 3 .6 For a ny g raph G ,

)

( ))

(

( T

₂

G  diam G



^.

Proof. Let A = {e₁, e₂,e_t } be the set of edges which constitutes the longest path between any two vertices of G such that

) (

= diam G

A

. We consider the following cases:

Case 1. If G contains only one edge then

3 2

( G ) = C

T

. Therefore

 ( T

₂

( G )) = 1 = A

. Case 2. Let

u

₁

, u

₂

,  



, u

_m

 E ( G )

and

u

u

₁

 , 

₂

, 



, u

_mbe the corresponding edge vertices of

T

₂

( G )

. Let

v

₁

, v

₂

, 

,

, v

_n

 V ( G )

and

v

₁

 , v 

₂

,  



, v

_nbe the corresponding

vertices of

T

₂

( G )

. Since all the edge vertices are incident with

V ( T

₂

( G ))  u

_m



, therefore

A D u G T V G

T ( ))  ( ( )) 

_m

 = 

(

_{2 2}



.

Hence

 ( T

₂

( G ))  diam ( G )

. Theorem 3.7 For any tree T,

) ( )) (

( T

₂

T 

_s

T

 

.

Proof. By Theorem 3.2 and Theorem A we get the required result.

Theorem 3.8 For any connected graph G of order atleast three,

 ( T

₂

( G ))  

_t

( G )

. Proof. By Corollary 2 and Theorem we get the required result.

Theorem 3.9 For any connected graph G,

)

( )) (

( T

₂

G 

_m

G

 

where



_m

(G )

is the maximal domination number of G.

Proof. Let G be any connected graph.

Suppose

 ( T

₂

( G )) > 

_m

( G )

. Then each

u

i,

1  i  m

of a corresponding edge vertex of G in

T

₂

( G )

will be adjacent to at least two vertices of

V ( T

₂

( G )  v

_n

 )

in

)

2

( G

T

, where

v

_n is the corresponding vertex set of G in

T

₂

( G )

. Therefore by taking alternate vertices

v

_j,

1  j  n

of

)

2

( G

T

, the set

v

_j

, j = 1,2,...., n

will be a dominating set of

T

₂

( G )

. Since each

u

_i,

m i 



1

is an independent vertex in

)

2

( G

T

, then the set

u

_i

, i = 1,2,...., n

is a maxiaml independent set. Since every

maximal independent set is a minimal dominating set, the refo re

D G T V u

D  =

_m

= (

₂

( )) 

is also a dominating set o f

T

₂

( G )

^{, Th us}

) (

<

)) (

( T

₂

G 

_m

G



, a contradiction.

Theorem 3.10 For any tree T of order

 2

p

vertices,



_c

( T )   ( T

₂

( T ))

.

Proof. Let T be a tree of order

p  2

^{. Let}

}

, , , {

= v

₁

v

₂

v

_r

D 

^{be such th at}

2 ) ( v

_i



deg

for

1  i  r

^{. Then in}

T

₂

( G )

degree of each

u

_i and each

v

_r will become twice. Therefore, each

v

_i in

T

₂

( G )

will be a minimal dominating set. Therefore

e p T

T ( )) =  (

₂



where e is the number of pendant vertices in T. Hence by Theorem C,



_c

( T )   ( T

₂

( T )

.

Theorem 3.11 For any

( p , q )

^{graph G,}

1

= )) ( ( T

₂

G



cot if and only if

G = K

_1,p1. Proof. If

G = K

_1,p1, then

deg v

_i

 = 2 degv

_i where

v

_i is the corresponding vertex of

v

i of G in

T

₂

( G )

. Each component of

)

2

( G

T

will be a triangle and each triangle will have one common vertex which is adjacent to all other vertices of

T

₂

( G )

^. Therefore



_cot

( T

₂

( G )) = 1

.

Conversely, suppose



_cot

( T

₂

( G )) = 1

^and

1

 K

1, _p

G

, then there exists a vertex of

1

= )) (

(

₂



 T G p

. Let u be a vertex of degree

p  1

, then all other vertices of

)

2

( G

T

are adjacent to u. But in

T

₂

( G )

, this is the only case when G is a star, oth erwise

 ( T

₂

( G )) < p  1

^{. Th us}

2 )) ( ( T

₂

G 



, a contradiction.

Theorem 3.12 For any

( p , q )

^{graph G,}

1

= ) )(

( T

₂

G



cl if and only if

G = K

_1,p1. Proof. Proof is similar to the Theorem 3.11.

The orem 3.1 3 Fo r any grap h G ,

)

( )) (

( T

₂

G 

_cot

G

 

.

Proof. Since



_t

( G )  

_cot

( G )

, therefore by Theorem 3.8, we get the required result.

Theorem 3.14 For any

( p , q )

^{graph G,}

p

G

m

( T

₂

( )) 



.

Further equality holds if and only if

K

p

G =

^.

Proof . For any graph G, it is easy to

observe that



_m

( T

₂

( G ))  p

. Suppose

K

p

G =

^{, then}



_m

( G ) = p = 

_m

( T

₂

( G ))

. Con versely, let



_m

( T

₂

( G )) = p

a nd

K

p

G 

. If G contains an isolated vertex then



_m

( T

₂

( G )) < p

. So consider a graph G without isolated vertices. Then there exists at least one vertex with

 ( G )  2

which will be present in both dominating set and maximal dominating set of

T

₂

( G )

^. Therefore

 ( T

₂

( G )) < p

, a contradiction.

Hence

G = K

_p^.

Proposition 7. For any path

P

_p with

p  2

vertices,



_pns

( T

₂

( G )) = q

where q is the number of edges in

P

_p.

Proof. Let G be a path with

p  2

^vertices.

Let

u 

_m;

1  m  q

be the corresponding edge vertices in

T

₂

( G )

^and

v

_n

1  n  p

is the corresponding vertices of G in

)

2

( G T

^{, then}



_pns

( T

₂

( G ) = V ( T

₂

( G )  u

_m

 ) u

m

= = q

.

The orem 3.1 5 Fo r any grap h G ,

))

( ( )

( G T

₂

G

d  

where d(G ) is the domatic number of G.

Proof. For any graph G,

d ( G )   ( G )  1

^. Let u be a vertex of minimum degree in G, then degree of the corresponding vertex

u

i in

T

₂

( G )

will be twice. Also by Theorem 3.3,

 ( T

₂

( G ))  p

. Therefore

)) ( ( )

( G T

₂

G

d  

^{. If}

G = K

p^{, then}

)) ( (

=

= )

( K p T

₂

G

d

p



.Hence the equality

holds.

Theorem 3.16

For any graph G,

( ( ))

)

( T

²

G

G p

p 

  





 





^.

Proof. Follows from Theorem E and Theorem 3.15.

In the next Th eore m we pro ve the

Nordhaus-Gaddum type results.

Theorem 3.17 For any connected

( p , q )

graph G

(i)

 ( T

₂

( G ))   ( T

₂

( G ))  2 p  1

(ii)

 ( T

₂

( G )).  ( T

₂

( G ))  p ( p  1)

^. Proof. B y u sing Propo sition 5 a nd Proposition 6 we get the required result.

ACKNOWLEDGEMENT

This research was supported by the University Grants Commission, New Delhi,India-No.F.4-3/2006(BSR)/7-101/

2007(BSR) dated: September 2009.

This research was supported by the University Grants Commission, New Delhi,India-No.F.FIP/ Plan/KAKA047TF02 dated:October 8, 2009.

REFERENCES

1. E. J. Cockayne, R. M. Dawes and S.

T. Hedetniemi, Total domination in graphs, Networks , 10, 211-219 (1980).

2. E. J. Cockayne and S. T. Hedetniemi,

Towards a theory of domination in Graphs, Networks, 7, 247-261(1977).

3. F. Harary, Graph Theory, Addison- Wesley, Reading, Mass, (1969).

4. V. R. Kulli and B.Janakiram,The split domination number of a graph, Graph Theory Notes of New York, New York Academy of Sciences, 32, 16-19 (1997).

5. V. R. Kulli and B. Janakiram, The maximal domination number of a graph, Graph Theory Notes of New York, New York Acad emy of Sciences, XXXIII, 11-13 (1997).

6. E. Sampathkumar and S. B. Chikko- dimath,The Semi-Total Graphs of a Graph-I, Journal of the Karnatak University-Science, XVIII, 274-280 (1973).

7. E.Sampathkumar and H.B.Walikar, The Connected domination number of a graph, J.Math.Phys.Sci., 13, 607- 613 (1979).

8. B.Zelinka, Domatic numbers and degrees of vertices of a Graph, Math.Slovaca, 33, 145-147 (1983).