A note on 0 T Domination

637

A note on

T

0

Domination

Annie Sabitha Paul

1

and Raji Pilakkat

2

1_{Assistant Professor in Mathematics,,}

Government Engineering College, Kannur,

Affiliated to APJ Abdul Kalam Technological University, 670563, Kerala, India, Email: [email protected]

2 _{Professor, Department of Mathematics,}

University of Calicut, Thenhippalm – 673635, Kerala, India, Email: [email protected]

Abstract- A set

D



V

of a graph

G

(

V

,

E

)

is called a dominating set if every vertex in

G

is either in

D

or is adjacent to an element of

D

. A simple graph

G

is said to be

T

₀, if for any two distinct vertices

u

and

v

of

,

G

either one of

u

and

v

is isolated or there exists an edge

e

such that either

e

is incident with

u

but not with

v

or

e

is incident with

v

but not with

u

. If

D

of a dominating set

D

of the graph

G

is a

T

₀ graph, then it is called a

T

₀ dominating set and if

D

is both connected and

T

₀, then it is called a connected

0

T

dominating set. The minimum cardinality of all

T

₀ dominating sets and connected

T

₀ dominating sets are respectively called

T

₀ domination number and connected

T

₀ domination number and are denoted respectively

by

(

)

0

G

T



and

(

)

0

G

cT



. In this paper

T

₀ domination number and connected

T

₀ domination number are introduced and some results on these new parameters are established.

Keywords- Domination number,

T

₀ domination number, connected

T

₀ domination number

1 INTRODUCTION

Graphs

G



(

V

(

G

),

E

(

G

))

discussed in this paper are finite, simple and undirected. Any undefined term in this paper may be found in [1,4]. The degree [1] of a vertex

v

in graph

G

is denoted by

d

_G

(

v

)

(or

)

(

v

d

if no specification of the graph

G

is needed), which is the number of edges incident with

v

. The maximum degree of

G

is denoted by



(

G

)

. The

complement

G

of graph

G

[5] has

V

(

G

)



V

(

G

)

and

uv



E

(

G

)

if and only if

uv

is not in

E

(

G

)

. For a graph

G

, the number of vertices is called the order [5] of

G

and is denoted by

o

(

G

)

. An empty graph [1] is a graph with no edges. An isolated vertex [4] is one whose degree is zero. A vertex in a graph is called a pendant vertex [6] if its degree is one. Any vertex adjacent to a pendant vertex is called a support vertex. A simple graph in which each pair of distinct vertices is joined by an edge is called a complete

graph [1]. A complete graph on

n

vertices is denoted by

K

_n. A bipartite graph

G

is one whose vertex set can be partitioned into two subsets

X

and

Y

so that each edge has its ends in

X

and

Y

respectively. Such a partition

(

X

,

Y

)

is called a bipartition of

G

. A complete bipartite graph [1] is a simple bipartite graph with bipartition

(

X

,

Y

)

in which every vertex of

X

is joined to every vertex of

Y

.

The complete bipartite graph with

X



m

and

n

Y



is denoted by

K

_m,n. The graph

H

is said to be an induced sub graph [2] of the graph

G

if

)

(

)

(

H

V

G

V



and two vertices in

H

are adjacent ifand only if they are adjacent in

G

. A tree [1] is a connected acyclic graph. A cut edge [1] of a graph

G

is an edge such that whose removal makes the graph disconnected. The open neighborhood [5] of

v

in

)

(

G

V

consists of those vertices adjacent to

v

in

G

638 [5] of

v

is

N

 

v



N

(

v

)



 

v

. A matching [1] in

a graph is a set of pair wise nonadjacent edges. Let

)

,

(

V

E

G



be a graph. A set

D



V

is called a dominating set [5] if every vertex in

G

is either in

D

or is adjacent to an element of

D

. The minimum cardinality of all dominating sets in

G

is called the domination number and is denoted by



(

G

)

. Different types of dominating sets have been studied by imposing conditions on the dominating sets. A detailed survey can be found in [5]. A dominating set

D

is called an independent dominating set [3] if

D

is the empty graph. A dominating set

D

is

called a connected dominating set [7] if

D

is connected. The corresponding minimum cardinality of independent dominating set and connected dominating set are respectively called independent domination number and connected domination number and are denoted respectively by

i

(

G

)

and



_c

(

G

)

.

In [8], V Seena and Raji Pilakkat defined the

0

T

Graph as follows. A simple graph G is said to be

0

T

, if for any two distinct vertices

u

and

v

of

G

, one of the following hold

1. At least one of

u

and

v

is isolated.

2. There exists an edge

e

such that either

e

is incident with

u

but not with

v

or

e

is incident with

v

but not with

u

.

In this paper, a new domination parameter,

T

₀ domination number is introduced and some of its properties are studied. A

T

₀ dominating set is a dominating set

D



V

such that

D

is

T

0. Also it

is proved that every independent dominating set in a graph is

T

₀ dominating. So that every graph has a

T

₀ dominating set. Hence the property of

T

₀ domination is applicable to all simple graphs.

2

T

0

Domination

0

T

domination is defined as follows.

Definition 2.1. Let

G

be any finite undirected simple graph with vertex set

V

. A dominating set

S



V

is said to be

T

0 dominating if

D

is a

T

0 graph. The

minimum cardinality of all such

T

₀ dominating sets is called

T

₀ domination number and is denoted by

)

(

0

G

T



. Such a

T

₀ dominating set with cardinality

)

(

0

G

T



is called a

0 T



-set.

Seena V and Raji P [8] proved that a graph G is

0

T

if and only if

K

₂is not a component of

G

. A characterization property of a

T

₀ dominating set follows directly from this result.

Theorem 2.1. Let

G



(

V

,

E

)

be any graph. A dominating set

S



V

is a

T

₀ dominating set if and only if no component of

S

is

K

₂.

Theorem 2.2. For any graph

G

, every independent dominating set is

T

₀ dominating.

Proof. Let

I



V

be an independent dominating set of a graph

G



(

V

,

E

)

. Since

K

2is not a

component of

I

,

I

is a

T

₀ graph.

Corollary 2.3. For any graph

G

,

(

)

(

)

0

G

G



T





Remark 2.4. The converse of Theorem 2.2 need not be true. For example the set of all darkened vertices shown in figure 1 is

T

₀dominating but not independent. Here

(

)

3

0

G



T



and

i

(

G

)



5

.

Figure 1- G

Theorem 2.5. For any positive integer

k

, there exist a graphs

G

such that

i

(

G

)



_T

(

G

)



k

0



Proof. Consider the path

P

₃. Let

G

be the graph obtained from

P

₃ by attaching exactly

j

pendant edges to each vertex of

P

₃,where

j



2

. Then

3

)

(

0

G



T



and

i

(

G

)



3



(

j



1

)

when

j



2

,

Therefore

(

)

(

)

1

0







G

j

G

i



_T . Since

j



2

,

3...

2,

1,

=

,

)

(

)

(

0

G

k

k

G

639 Theorem 2.6 characterizes graphs

(

)

1

0

G



Proof. (1) is obvious.

(2) Suppose that

(

)

2

D

is independent dominating. Also since

)

independent dominating, contradicting i(G) = 2. Hence

2

640 with two vertices is not

T

₀,

S

is an independent

dominating set, a contradiction.

Remark 2.10. The converse of Theorem 2.9 need not be true. For example in figure.1,

(

)

3

Theorem 2.11. For complete bipartite graphs

K

_m,n

,



other vertex from

V

then these two vertices dominate

n 2, which contain no

K

₃as an induced subgraph, then

2

G

and all vertices which are non adjacent to both

u

and

v

are dominated by both

u

and

v

in G. So

G

. Then

D

consists of disconnected edges. That is,

D

has

K

₂as a component, a contradiction to

D

is minimum cardinality of all connected

T

₀

dominating connected

T

₀dominating set with cardinality

641 Proof. Since



c

(

G

)



2

, the graph induced by any

c



-set is not

K

₂and hence the



_c-set is connected

0

T

dominating. Also since

(

)

(

)

0

G

G

cT

c







, it

follows that

(

)

(

)

0

G

c

G

cT







.

Theorem 3.3. Let

a

and

b

be two positive integers with

a



2

and

b



2

a



2

. Then there is a graph G

on

b

vertices with



(

G

)





c

(

G

)





cT₀

(

G

)



a

and

i

(

G

)



a



1

.

Proof. Consider the path

P



(

u

₁

,

u

₂

,...,

u

_a

)

on a vertices. Let

b



2

a



r

,

r



2

.

Let

G

be the graph obtained from

P

by attaching two or more pendant edges at

u

₁ and

u

₂and one pendant edge at each

3

,

i



u

_i . Let

v

_i

,

i



3

be the pendant vertices attached to

u

_i

,

i



3

. Clearly

D

=



u

₁

,

u

₂

,...,

u

_a



is a



-set which is also a connected

T

₀ dominating set. Hence



(

G

)





c

(

G

)





cT₀

(

G

)



a

. Any

independent dominating set of

G

of minimum cardinality will be one among the following.



u

i

,

v

3

,

v

4

,...,

v

a





N

(

u

j

)

where

u

i is the vertex

of maximum degree among

u

₁ and

u

₂ and

N

(

u

_j

)

is the open neighborhood of

u

₁ or

u

₂with minimum cardinality such that

i



j

or



u

₁

,

u

₃

,

v

₄

,

u

₅

,

v

₆

...





N

(

u

2

)

if

d

(

u

₁

)



d

(

u

₂

)

and

N

(

u

₂

)



N

(

u

₁

)

or



u

₂

,

v

₃

,

u

₄

,

v

₅

,

u

₆

...





N

(

u

₁

)

if

d

(

u

2

)



d

(

u

1

)

and

N

(

u

1

)



N

(

u

2

)

.

In all these cases the cardinality of the i-set is (a − 1) + min



d

(

u

1

),

d

(

u

2

)



,

where min



d

(

u

1

),

d

(

u

2

)



≥ 2.

Therefore

i

(

G

)



a



1

.

Theorem 3.4. Let

T

be any tree of order

n

,

n



4

. If

T

is not isomorphic to

K

_1,n1 then

(

)

3

0

T



cT



.

Proof. Since

T

is not isomorphic to

K

_1,n1 (a star graph),



(

T

)



n



2

. Consider the following cases.

Case (i)

T

is not a path.

Since

T

is not a path, it has at least three pendant vertices say

v

₁

,

v

₂

and

v

₃. Therefore,

)

(

_i

T

v

d

=

n



2

,

i



1

,

2

,

3

. Also since

T

is not a star, the support vertex of at least one of the

v

_iwill be different from support vertices of the other two. Therefore



v

₁

,

v

₂

,

v

₃



forms a dominating set of

T

. In

T

,



v

₁

,

v

₂

,

v

₃



, the graph induced by



v

1

,

v

2

,

v

3



is either

K

3 or

P

3. Hence



v

1

,

v

2

,

v

3



is a connected

T

0 dominating set of

T

. So that

(

)

3

0

T



cT



. Since

T

has no isolated

vertices,

(

)

0

T

cT



cannot be one.

Also since there are no connected

T

₀ dominating sets of cardinality two,

(

)

3

0

T



cT



. Hence it follows

that

(

)

3

0

T



cT



.

Case (ii)

T

is a path

P

_n on

n

vertices,

n



4

. Let

v

₁ and

v

₂ be the pendant vertices and

v

₃be any one of the support vertices.

Then

d

_T

(

v

_i

)

=

n



2

,

i



1

,

2

and

d

T

(

v

3

)

=

n



3

.

In

T

, the subgraph induced by



v

₁

,

v

₂

,

v

₃



is

P

₃

and it forms a connected

T

₀ dominating set of

T

. Hence by the same reasoning as in case (i), it follows

that

(

)

3

0

T



cT



.

Corollary 3.5. Let

T

be a tree of order > 1, then

T

has a connected

T

₀ dominating set if and only if

T

is not a star.

Proof. If

T

is a star on n > 1 vertices, then

T

is disconnected. Hence

T

cannot have a connected

T

₀ dominating set. Conversely, let

T

be a tree, which is not a star. Then

n



4

. Therefore by Theorem 3.4,

T

has a connected

T

₀ dominating set.

Proposition 3.6. Let G be a bi-star B(m, n) on

p

vertices, then _cT

(

G

)

p

max

 

m

,

n

0







Theorem 3.7. [7] For any tree

T

of order

p

, the connected domination number of

T



p



e

, where

642 Theorem 3.8. Let

T

be any tree. Then the



_c-set

and

0 T



-set are the same if and only if

T

is not a bi-star

Proof. Let

T

be a tree on

p

vertices. Assume that the

c



-set and

0 T



-set are the same. Suppose if possible,

T

is a bi-star.

Then by Theorem 3.7,



c

(

T

)



p



e



2

, where,

e

is the number of pendant vertices. So that the graph induced by the



_c-set is

K

₂, which is not

T

₀. By Proposition 3.6 and Theorem 3.7, if

T

is a bi-star,

then

(

)

(

).

0

T

T

cT

c







We conclude with a conjecture.

Conjecture 3.8. There are no simple graphs

G

for

which

(

)

(

)

0

G

G

T







REFERENCES

[1] Bondy J. A, Murty U.S.R. (2008): Graph Theory ; Springer

[2] . G. Chartrand, P. Zhang (2009) Chromatic Graph Theory ; CRC Press, USA

[3] E.J. Cockayne and S.T. Hedetniemi (1974) Independence graphs. Congr. Numer..X : 471-491

[4] F. Harary, Graph Theory (1969) Addison Wesley

[5] W.Haynes, Stephen Hedetniemi, Peter Slater (1998) Fundamentals of Domination in Graphs ; Marcel Dekker Inc

[6] K.R Parthasararhy (1994) Basic Graph theory ; McGraw-Hill Pub., USA

[7] E. Sampathkumar (1979) The connected domination number of a graph ; J. Math. Phy. Sci. volume 13, 607-613