IJRET: International Journal of Research in Engineering and Technology
eISSN: 23191163  pISSN: 23217308
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: AMS05C69, 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 uvpath 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
is adjacent to denoted by
*(
∈ $,
∉
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
+(
).
_______________________________________________________________________________________ Volume: 02 Issue: 10  Oct2013, Available @ http://www.ijret.org
552
IJRET: International Journal of Research in Engineering and Technology
eISSN: 23191163  pISSN: 23217308
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
is adjacent to some vertices
) − . Then every vertex D ∈
vertex in <
. If there
− { } is an end
>. Further, if D is adjacent to a vertex
_______________________________________________________________________________________ Volume: 02 Issue: 10  Oct2013, 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: 23191163  pISSN: 23217308
−
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
, a contradiction.
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
adjacent. Then 1 and (
)−
and
= ( ={ }∪
since deg( ) = @ −
set of
, a contradiction.
⊆ (
where
of
are not
) − { } . Further, since
not adjacent,
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 viceversa. 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 , '(
(
)+
) ≤ @ + ( ).
) ≤ @.
_______________________________________________________________________________________ Volume: 02 Issue: 10  Oct2013, Available @ http://www.ijret.org
554
IJRET: International Journal of Research in Engineering and Technology Proof: For Z ≤ 5, the result follows immediately. Let Z ≥ 6, ={ ,
suppose
7}
,…,
eISSN: 23191163  pISSN: 23217308
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.
_______________________________________________________________________________________ Volume: 02 Issue: 10  Oct2013, Available @ http://www.ijret.org
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: 23191163  pISSN: 23217308
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!]^( , )
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(
≅ F , then obviously,
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=
!"#( ).
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. Then the complementary set V(G ) − D contains the ⊆ V(G ) − D, which covers all the vertices in
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forms a minimal inverse dominating set of
and it follows that  Hence
(
 + !"# ( ) ≤ @ + R ∪ u  − 1.
) + !"# ( ) ≤ @ + ( ) − 1.
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