IJRET: International Journal of Research in Engineering and Technology

eISSN: 2319-1163 | pISSN: 2321-7308

BOUNDS ON INVERSE DOMINATION IN SQUARES OF GRAPHS M. H. Muddebihal1, Srinivasa G2 1 2

Professor, Department of Mathematics, Gulbarga University, Karnataka, India, mhmuddebihal@yahoo.co.in Assistant Professor, Department of Mathematics, B. N. M. I. T , Karnataka, India, gsgraphtheory@yahoo.com

Abstract Let

be a minimum dominating set of a square graph

called an inverse dominating set with respect to domination number of

(

)−

contains another dominating set

, then

of

is

. The minimum cardinality of vertices in such a set is called an inverse (

and is denoted by

. If

). In this paper, many bounds on

(

) were obtained in terms of

elements of . Also its relationship with other domination parameters was obtained.

Key words: Square graph, dominating set, inverse dominating set, Inverse domination number. Subject Classification Number: AMS-05C69, 05C70 --------------------------------------------------------------------***---------------------------------------------------------------------1. INTRODUCTION

A set \$ ⊆ ( ) is said to be a dominating set of

In this paper, we follow the notations of [1]. We consider

vertex in ( − \$) is adjacent to some vertex in \$. The

only finite undirected graphs without loops or multiple edges.

minimum cardinality of vertices in such a set is called the

In general, we use <

domination number of

the set of vertices

> to denote the subgraph induced by ( ) and

and

[ ] denote the open and

and is denoted by ( ). Further, if

the subgraph < \$ > is independent, then \$ is called an independent dominating set of . The independent omination

closed neighborhoods of a vertex .

&(

number of , denoted by The minimum (maximum) degree among the vertices of

is

denoted by ( ) ∆( ) . A vertex of degree one is called an ( )(

end vertex. The term

, if every

( )) denotes the minimum

number of vertices(edges) cover of . Further,

( )( ( ))

) is the minimum cardinality of

an independent dominating set of . A dominating set \$ of

is said to be a connected dominating

set, if the subgraph < \$ > is connected in . The minimum cardinality of vertices in such a set is called the connected

represents the vertex(edge) independence number of .

domination number of

and is denoted by

'(

).

A vertex with degree one is called an end vertex. The distance between two vertices shortest uv-path in two vertices in

and

is the length of the

. The maximum distance between any

is called the diameter of

and is denoted

by !"#( ).

A dominating set \$ is called total dominating set, if for every vertex

∈ , there exists a vertex

*(

∈ \$,

such that

. The total domination number of

,

) is the minimum cardinality of total

dominating set of . The square of a graph as in

denoted by

and the two vertices

only if they are joined in

and

, has the same vertices are joined in

if and

by a path of length one or two.

The concept of squares of graphs was introduced in [2].

Further, a dominating set \$ is called an end dominating set of , if \$ contains all the end vertices in

. The minimum

cardinality of vertices in such a set is called the end domination number of

and is denoted by

+(

).

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IJRET: International Journal of Research in Engineering and Technology

eISSN: 2319-1163 | pISSN: 2321-7308

Domination related parameters are now well studied in graph

Analogously, let

be a minimum dominating set of a square

theory (see [3] and [9]).

graph

(

set Let \$ be a minimum dominating set of

, if the compliment

− \$ of \$ contains a dominating set \$ , then \$ is called an ′

with respect to \$. The minimum

inverse dominating set of

cardinality of vertices in such a set is called an inverse domination number of

and is denoted by

( ). Inverse

. If of

, then

respect to

)−

contains another dominating

is called an inverse dominating set with

. The minimum cardinality of vertices in such a

set is called an inverse domination number of (

denoted by

and is

). In this paper, many bounds on

(

)

were obtained in terms of elements of . Also its relationship with other domination parameters was obtained.

domination was introduced by V. R. Kulli and S. C. Sigarkanti [10].

2. RESULTS Theorem 2.1: For any connected graph

⊆ (

A set

) is said to be a dominating set of

every vertex not in

is adjacent to a vertex in and is denoted by (

)≤ ( )+

*(

)+

).

. The

minimum cardinality of vertices in such a set is called the domination number of

*(

, if

(

,

)(see [4]).

Proof: Let \$ = { ,

,…,

4}

⊆ ( ) be the minimum set

of vertices which covers all the vertices in G. Suppose the subgraph < \$ > has no isolated vertices, then \$ itself is a

A set

of

is said to be connected dominating set of

every vertex not in

is connected. The minimum

cardinality of a connected dominating set of connected domination number of

A dominating set

of

for every vertex

∈ (

of

is called

(see [5]).

*(

total dominating set of

of

), let

be the minimal

. Suppose

in

vertices of by

.(

is said to be restrained dominating

*|

)−

⊆ \$ is a minimal =

which covers all the vertices

forms a minimal inverse dominating set

≤ |\$ | + deg ( ).

*(

⊆ \$

. Then there exists a vertex set

⊆ (

. Clearly,

>}

∈ ( ), such that deg( ) = ( ), it follows that, |

, there exists atleast one vertex

Therefore,

(

)+

|∪

*(

)≤

) + ( ).

is adjacent to a vertex in -+ (

) is the minimum . Further, a

Theorem 2.2: Let respectively. If

(

and )=

be, -set and (

-set of

), then each vertex in

is

of maximum degree.

is said to be double dominating set, if ∈ (

) is dominated by at least two

. The double domination number of

, denoted

) is the minimum cardinality of a double

dominating set of

>}

,…,

(see [6]).

cardinality of a restrained dominating set of

for every vertex

,…,

of

*

. Since for any graph

, denoted by

of

( & ) ∈ { & }.

) is the minimum cardinality of a

and to a vertex in ( − ). The restrained domination

dominating set

* -set

{ ,

set, if for every vertex not in

number of

⊆ \$ be the set of

={ ,

set of . Since ( ) = (

), there exists a vertex

,

7}

Clearly, \$ = \$ ∪ 6 ∪ { & } forms a minimal total dominating

dominating set of

,…,

1 by adding vertices { & } ⊆ ( ) − \$ and

| A dominating set

. Otherwise, let 6 = { ,

is said to be total dominating set, if

is adjacent to . The total domination number

denoted by

set of

-

vertices with deg( & ) = , 1 ≤ ! ≤ <. Now make deg ( & ) =

is adjacent to at least one vertex in

and the set induced by

, such that

if

*

(see [7] and [8]).

Proof: For @ = 2, the result is obvious. Let @ ≥ 3. Since, ( )= (

) such that

={ ,

let

exists a vertex in

(

,…, ∈

7}

doesnot contain any end vertex,

be a dominating set of

such that

) − . Then every vertex D ∈

vertex in <

. If there

− { } is an end

>. Further, if D is adjacent to a vertex

_______________________________________________________________________________________ Volume: 02 Issue: 10 | Oct-2013, Available @ http://www.ijret.org

553

IJRET: International Journal of Research in Engineering and Technology ∈ (

) − . Then

− { , D} ∪ { } is a dominating

=

, a contradiction. Hence each vertex in

set of

maximum degree in

is of

eISSN: 2319-1163 | pISSN: 2321-7308

Suppose

is not independent, then there exists at least ∈

one vertex

− . Clearly, |

(

,

Theorem 2.3: For any connected graph

) = 1 if

|

is independent. Then in this case,

| = | − | in

Hence, (

has at least two vertices of degree (@ − 1).

. Clearly, it follows that | | ∪ |

)+

(

Theorem 2.5: For any connected (@, K) -graph

cases.

vertices,

@ − 1. Then in this case −

=

= { } is a -set of

− { }. Further, if

deg( ) ≤ @ − 2 in D∉

( ) in

of deg( ) =

has exactly one vertex = { }E ( ) in

. Clearly, (

)− ,

=

∪ {D} forms an inverse

)+

&(

,…,

(

)+

&(

4}

be the minimum set of vertices

\$={ ,

Suppose >}

,…,

) ≰ @ − 1.

there

Consider

∈ R, ( ) ∩ ( ) ∈

,

such that for every two vertices ( ) − R.

with @ ≥ 3

) ≤ @ − 1.

@ ≤ 2,

For

@ ≥ 3, let R = { ,

. Then there exists at least one vertex

such that

exists

a

vertex

⊆ R which covers all the vertices in

and if the subgraph < \$ > is totally disconnected. Then S

dominating set of

Proof:

(

| = @.

) = @.

Proof: To prove this result, we consider the following two

Case 1. Suppose

|=

|{ − } − { }| and hence, | | ∪ | ′| ≤ @, a contradiction.

.

Conversely, if

and only if

( )⊆

such that

forms the minimal independent dominating set of . Now in Case 2. Suppose

has at least two vertices

deg( ) = @ − 1 = deg ( ) such that = { } dominates

)−

and

= ( ={ }∪

since deg( ) = @ −

-set of

⊆ (

where

of

are not

) − { } . Further, since

and

)−

and

are

where { } ⊆ ( obtain |

)−

Proposition

(

.

HG

=

, the

)+

(

={ ,

={ ,

,…,

7}

inverse dominating set of |

F

,G

=

)+

(

(

the complementary set such that

(

)=

(

(

)−

&(

7}

,…, ⊆ (

. Then

contains another set

). Clearly,

forms an inverse

and it follows that | )+

=

| ∪ |\$| ≤ @ − 1.

) ≤ @ − 1.

Theorem 2.6: If every non end vertex of a tree is adjacent to

Proof: Let R = { , ⊆ R is a

vertex without

) ≤ @.

(W ) ≤ X

,…,

4}

-set of

G 4

Y + 1.

be the set of all end vertices (R). Suppose,

and R ∈

⊆ (

. Then

R forms a minimal inverse dominating set of

)−

. Since every

tree W contains at least one non end vertex, it follows that ⊆ ( )−

) be the -set of forms a minimal

. Since | | ≤ L@/4O and

| ≤ L@/3O, it follows that, | | ∪ | (

⊆ \$, which forms minimal -set of

in W such that vertex |R| = #

Theorem 2.4: For any connected (@, K)-graph

Proof : Suppose

, there exists a vertex set

at least one end vertex, then ,

FGI, GJ = 1.

isolates, then (

UV

Therefore,

) = 1.

FG =

2.1:

,…,

= { }E ( ),

and vice-versa. In any case, we

| = 1. Therefore

T ,

dominating set of

are adjacent to all the vertices in

) and distance between two vertices

is at most two in

forms a

Conversely, suppose deg( ) = @ − 1 = deg( ), such that and

( )= (

, since

| ≤ @. Therefore,

G 4

|

|≤X

Y + 1. Therefore,

(W ) ≤ X

G 4

Y + 1.

Theorem 2.7: For any connected (@, K)-graph , '(

(

)+

) ≤ @ + ( ).

) ≤ @.

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554

IJRET: International Journal of Research in Engineering and Technology Proof: For Z ≤ 5, the result follows immediately. Let Z ≥ 6, ={ ,

suppose

7}

,…,

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Proof: Let R = { ,

, deg( & ) ≥ 2, 1 ≤ ! ≤ < be a

[R] in

and let

7}

,…,

be the set of all end vertices in

. Suppose d ⊆

is a dominating

>. Then R ∪ d forms a minimal end

minimal dominating set of . Now we construct a connected

set of the subgraph <

dominating set

dominating set of

. In

diam( ) ≤ 3, then

= { , } is a minimal

'

from

two components of

by adding in every step at most

forms a connected component in

Thus we get a connected dominating set , ( )= (

− 1 steps. Now in minimal { ,

,…,

all ,

after at most

). Suppose

⊆ (

)−

'(

'|

immediately.

=

{ ,

(

(

)+

>}

⊆ (

)≤X

2.8:

For

any

'(

)≤ (

connected

)+

(@, K) -graph

,

( ). Equality holds for

(

)+

)−

,…,

>}

be the maximum set of

vertices such that < 6 > is totally disconnected and |6| = ( ). Now in

={ ,

, let

7}

) be the

⊆ (

( )= (

minimum set of vertices such that

). Clearly,

. Suppose the subgraph <

forms a minimal -set of is connected, then

,…,

>

itself is a connected dominating set of

. Otherwise, construct the connected dominating set by adding at most two vertices ,

from

vertices of

(

such that

')

(

=

'

between the

) and <

'

{ ,

,…,

7}

. Clearly,

⊆ (

=

follows (

)+

Suppose

which covers all the vertices in

forms an inverse dominating set of . Since !"#( , ) ≥ 2, ∀ ,

respect to -set of it

)−

'(

that

|

)≤ (

|∪| )+

'|

with

∈ (

| = 1. Therefore,

(

)+

),

≤ | | ∪ |6|. Therefore,

( ).

≅ FG . Then in this case, |6| = | | = | '(

)= (

Theorem 2.9: For any connected (@, K)-graph ,

)+

'|

=

( ). (

− R is

)≤

a

=

minimal

inverse of

G {h∪i}

|≤X

Y + 3. Therefore,

For

any

connected

(@, K)-graph

)≤@−

,

( ) + 1. Equality holds for

FG , Zj . ,…,

&}

be the minimal set of vertices such that |k| =

which covers all the edges in

,obtain d, the induced subgraph of

in { ,

,…,

>}

) − R and

diam ( , ) ≥ 2, for all

l⊆

∈ l and

(

)− .

∈ d. Let R = .

is adjacent to a vertex in Clearly,

, such

∈ R. Then

where l ⊆ l, covers all the vertices in vertex not in

>

be the set of all such end vertices in

= (

Suppose

( ). Now

, i.e.,d = <

such that d contains at least one end vertex

> is

connected. Further, there exists a vertex set

, and the result follows

Y + 3.

Proof: Let k = { ,

Proof: Let 6 = { ,

-set of

with respect to the dominating set

G ab (c)

-+ (

) . Suppose

diam( ) ≥ 4, then

Suppose

Theorem 2.10:

FG .

|

(

)+

) ≤ @ + ( ).

Theorem

of

. Clearly, it follows that |

. Hence it

≤ @ ∪ | |. Therefore,

,…,

-set

dominating set of

. Clearly,

forms a minimal inverse dominating set of |∪|

with respect to

be

!]^( , ) ≥ 2, for

, such that

, which covers all the vertices in

follows that |

. Then there exists a vertex set

-set of >}

'

.

, ( )= (

forms a

that =R∪l ,

such that every and to a vertex in minimal restrained

dominating set of

. Let

= {D , D , … , D7 } ⊆ (

forms a minimal

-set of

with respect to dominating

. Clearly, it follows that |

set of

Therefore,

(

)+

For equality, suppose |

-+ (

Suppose G

(

)+

| ∪ | | ≤ @ − |k| + 1.

)≤@−

( ) + 1.

≅ FG . Then in this case, | | = 1 =

| and |k| = @ − 1. Therefore, |

and hence

)

-+ (

| ∪ | | = @ − |k| + 1

)=@−

( ) + 1.

≅ Zj . Then in this case, | | = 2 = |

|k| = . Clearly, it follows that,

(

)+

-+ (

| and )=@−

( ) + 1.

G ab (c)

X

Y + 3.

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555

IJRET: International Journal of Research in Engineering and Technology

Theorem (

,

2.11: )+

For

.(

any

(@, K)-graph

connected

) ≤ 2@ − K + 1.

Equality

holds

for

eISSN: 2319-1163 | pISSN: 2321-7308

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(

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.

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, -set

. Then the complementary set V(G ) − D contains the ⊆ V(G ) − D, which covers all the vertices in

vertex set . Clearly,

forms a minimal inverse dominating set of

and it follows that | Hence

(

| + !"# ( ) ≤ @ + |R ∪ u | − 1.

) + !"# ( ) ≤ @ + ( ) − 1.

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