i(G)-Graph - G(i) Of Some Special Graphs

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. IV (Jan - Feb. 2015), PP 70-84
www.iosrjournals.org
DOI: 10.9790/5728-11147084 www.iosrjournals.org 70 |Page
)(Gi -Graph - )(iG Of Some Special Graphs
V. Anusuya,1 R. Kala2
Department of Mathematics S. T. Hindu college Nagercoil 629 002 Tamil Nadu, India
Department of Mathematics Manonmaniam Sundaranar University Tirunelveli 627 012 Tamil Nadu, India.
Abstract: The )(Gi -graph is defined as a graph whose vertex set correspond 1 to 1 with the )(Gi -sets of G .
Two )(Gi - sets say 1S and 2S are adjacent in )(Gi if there exists a vertex 1Sv , and a vertex 2Sw
such that v is adjacent to w and }{}{= 21 vwSS  or equivalently }{}{= 12 wvSS  . In this paper
we obtain )(Gi -graph of some special graphs.
I. Introduction
By a graph we mean a finite, undirected, connected graph without loops and multiple edges. For graph
theoretical terms we refer Harary [12] and for terms related to domination we refer Haynes et al. [14].
A set VS  is said to be a dominating set in G if every vertex in SV  is adjacent to some vertex
in S . The domination number of G is the minimum cardinality taken over all dominating sets of G and is
denoted by )(G . A subset S of the vertex set in a graph G is said to be independent if no two vertices in
S are adjacent in G . The maximum number of vertices in an independent set of G is called the independence
number of G and is denoted by )(0 G . Any vertex which is adjacent to a pendent vertex is called a support. A
vertex whose degree is not equal to one is called a non-pendent vertex and a vertex whose degree is 1p is
called a universal vertex. Let u and v be (not necessarily distinct) vertices of a graph G . A vu  walk of
G is a finite, alternating sequence vueeeuu nn =,,,,,= 210  of vertices and edges beginning with vertex
u and ending with vertex v such that niuue iii ,1,2,3,=,,= 1  . The number n is called the length of
the walk. A walk in which all the vertices are distinct is called a path. A closed walk ),,,,( 210 nuuuu  in
which nuuuu ,,,, 210  are distinct is called a cycle. A path on p vertices is denoted by pP and a cycle on
p vertices is denoted by pC .
Gerd H.Frickle et. al [11] introduced  -graph.The  -graph of a graph G denoted by
))(),((=)(  EVG is the graph whose vertex set corresponds 1to1  with the  -sets. Two  -sets
say 1S and 2S are adjacent in )(E if there exist a vertex 1Sv and a vertex 2Sw such that v is
adjacent to w and }{}{= 21 uwSS  or equivalently }{}{= 12 wuSS  . Elizabeth et.al [10] proved
that all graphs of order 5n have connected  -graphs and determined all graphs G on six vertices for
which )(G is connected. We impose an additional condition namely independency on  -sets and study
)(Gi -graphs denoted by )(iG . The )(Gi -graph is defined as a graph whose vertex set correspond 1 to 1 with
the )(Gi -sets of G . Two )(Gi - sets say 1S and 2S are adjacent in )(Gi if there exists a vertex 1Sv ,
and a vertex 2Sw such that v is adjacent to w and }{}{= 21 vwSS  or equivalently
}{}{= 12 wvSS  . In this paper we obtain )(Gi -graph of some special graphs.
II. Main Results
Definition 2.1 A set VS  is said to the independent if no two vertices in S are adjacent. The minimum
cardinality of a maximal independent dominating set is called the independent domination number and is denoted
by )(Gi . A maximal independent dominating set is called a )(Gi set.
Definition 2.2 Consider the family of all independent dominating sets of a graph G and define the graph

))(),((=)( iEiViG to be the graph whose vertices )(iV correspond 1-1 with independent dominating sets of
G and two sets 1S and 2S are adjacent ib )(iG if there exists a vertex 1Sv , and 2Sw such that (i)
v is adjacent to w and (ii) 1S = }{}{2 vwS  and }{}{= 12 wvSS  .
Proposition 2.3 If a graph G has a unique )(Gi -set then 1)( KiG  and conversely.
Corollary 2.4 11, =)( KiK n .
Proof. Since the central vertex of nK1, is the only )(Gi -set, 11, =)( KiK n .
Proposition 2.5 1)( KiKn  , whereas nn KiK )( .
Proof. Let },,,,{ 321 nvvvv  be the set of vertices of nK . Each singleton set nivS ii ,1,2,3,=},{=  is
an element of )(iV and each pair ),(1),,( njiSS ji  form an edge in )(iKn . Hence nn KiK )( .
Since the set of all vertices of nK is the only independent dominating set of 1)(, KiKK nn  .
Proposition 2.6 For nm 1 ,






n<ifm
2nandm=ifm
1=n=ifm
)(
1
2
2
,
K
K
K
iK nm
Proof. Let },,,,{= 3211 muuuuS  and },,,,{= 3212 nvvvvS  be the bipartition of nmK , .
If 1== nm , }{ 1u and }{ 1v are the )(Gi sets and clearly 2, =)( KiK nm .
If nm = and 2m , 1S and 2S are the only two independent dominating sets of nmK , and they are
non-adjacent vertices of )(, iK nm . Hence 2, =)( KiK nm for all values of m . If 1,< Snm is the only )(Gi
-set and so 1, )( KiK nm  .
Proposition 2.7 2323 )(   kk CiC .
Proof. Case(i). 1=k
Let the cycle be ),,,,,( 154321 vvvvvv .
},{=},,{=},,{=},,{=},,{= 535524423412311 vvSvvSvvSvvSvvS are the 5 )(Gi -sets of 5C and
)(5 iC is the cycle ),,,,,( 154321 SSSSSS .
Case(ii). 2=k
Let the cycle be ),,,,,,,,( 187654321 vvvvvvvvv .
},,{=},,,{=},,,{=},,,{=},,,{=},,,{=},,,{= 8637742675258524631364127411 vvvSvvvSvvvSvvvSvvvSvvvSvvvS
are the 8 )(Gi -sets of 8C and )(8 iC is the cycle ),,,,,,,,( 187654321 SSSSSSSSS .
Case(iii). Let the vertices of the cycle be ),,,,,( 123321 vvvvv k
We know that 1=)( 23  kCi k . },,,,,,{= 1323107411  kk vvvvvvS  and
},,,,,,{= 323107412 kk vvvvvvS  are two )(Gi -sets of 23 kC .
Now finding the first vertex of 1S and changing the other vertices of 1S we get
},,,,,,{= 33396313 kk vvvvvvS  . Now fixing the first two vertices of 1S and changing the other vertices of
1S we get },,,,,,{= 33396414 kk vvvvvvS  . Proceeding like this, fixing the first 1k vertices and
changing the
th
k vertex alone we get },,,,,,,{= 33353107411 kkkk vvvvvvvS   . Now consider the two

)(Gi -sets },,,,,{= 23138522  kkk vvvvvS  and },,,,,,{= 1313438523  kkkk vvvvvvS  .
As before, fixing the first vertex and changing from the
rdnd
,32 vertices upto
th
k of 3kS we get
},,,,,,,{= 132353107424  kkkk vvvvvvvS  ,
},,,,,,,,{= 132353107425  kkkk vvvvvvvS 

}.,,,,,,,{= 1323431185222  kkkk vvvvvvvS 
Now consider },,,,,{= 23396332  kkk vvvvvS  . As before, fixing the first vertex and changing from the
thrdnd
,4,32 vertices of 32 kS we get
},,,,,,{= 23131185342  kkk vvvvvvS 
},,,,,,{= 23131185352  kkk vvvvvvS 

}.,,,,,,,{= 2313331296323  kkkk vvvvvvvS 
Now 23321 ,,,, kSSSS  are )(Gi -sets of 23 kC . Here 1S is adjacent 2S and 4kS . kSSSS ,,, 432
are adjacent to preceeding and succeeding vertices. 1kS is adjacent to 2S and kS . 2kS is adjacent to 3kS
and 42 kS . 3kS is adjacent to 2kS and 22 kS . 4kS is adjacent to 5kS and 1S .
12765 ,,,,  kkkk SSSS  are adjacent to the preceeding and suceeding vertices. 22 kS is adjacent to 12 kS
and 3kS . 32 kS is adjacent to 3S and 23 kS . 42 kS is adjacent to 2kS and 52 kS .
13726252 ,,,,  kkkk SSSS  are adjacent to the preceeding and suceeding vertices. 23 kS is adjacent to
13 kS and 32 kS .
Thus we get a cycle
),,,,,,,,,,,,,,,,( 14212223242313233231121 SSSSSSSSSSSSSSSSSS kkkkkkkkkkkkkk  
which is isomorphic to 23 kC .
Proposition 2.8 For 33 )(2, KiCk k 
Proof. Since each kC3 for 3k has 3 disjoint )(Gi -sets, 33 )( KiC k  .
Proposition 2.9 13 )( KiPk 
Proof. Since paths kP3 of order k3 have a unique )(Gi -set, .)( 13 KiPk 
Proposition 2.10 223 )(   kk PiP
Proof. Let 23321 ,,,, kvvvv  be the vertices of 23 kP . We have 1=)( 23  kPi k .
},,,,,,{=},,,,,{= 1313438522238521  kkkk vvvvvvSvvvvS  are two )(Gi -sets of 23 kP . Now
fixing the first vertex and varying from the
ththrdnd
k,,4,32 vertices we get the following )(Gi -sets.
},,,,,{= 13107423 kvvvvvS 
},,,,,{= 13107424 kvvvvvS 

},,,,,,{= 1313118521  kkk vvvvvvS 
Also },,,,,{= 13107412  kk vvvvvS  is an )(Gi -set of .23 kP
Thus there are 2k )(Gi -sets of 23 kP . It is obvious that 1S is adjacent to 2S alone and 2kS is adjacent
to 3S alone. 1kS is adjcent to 2S and kS . kSSSS ,,,, 543  are adjacent to the preceeding and succeeding
vertices. Thus we get a path of length 2kP . Hence 223 )(   kk PiP .

Definition 2.11 Grid graph is the cartesian product of 2 paths.
The cartesian product of 2 paths mP and nP is denoted by nm PP W or nm PP  .
Proposition 2.12 For 2122 ))((2, KiPPk k  W .
Proof. )(122 iPP kW for 2k has only two disjoint )(Gi -sets. Therefore 2122 ))(( KiPP k W .
The structure of )(Gi -graphs of paths and cycles of order 13 k can be determined. Assume that the vertices
in each of these graphs have been labelled 1,31,2,3, k . For 13= kPG or
1},3{1,4,7,=,= 13  kSCG k  is a )(Gi -set of size 1k . In each case, 1 and 13 k have one
external private neighbour while the other numbers of S have two non adjacent external private neighbours. So
{2}{1}1 S and }{31}{3 kkS  are )(Gi -sets. Further if
'
S is an )(Gi -set for 13= kPG or
13= kCG and vertex i has exactly one external private neighbour, 1= ij or 1= kj , then
}{}{='
jiS  is an )(Gi -set. Let us refer to the process of changing from a )(Gi -set
'
S to the  -set
}{}{'
jiS  as a swap. We see that each swap defines an edge in )(iG .
Definition 2.13 We define a step grid )(kSG to be the induced subgraph of the kk  grid graph kk PP W
that is defined as follows:
))(),((=)( KEKVkSG where 2},,1:),{(=)(  kjikjijiKV and
}=1,=1,=,=:),(),,{(=)( ''''
jjiijjiijijiKE  .
Theorem 2.14 If 13= kPG or 13= kCG then )(iG is connected.
Proof. Each independent dominating set X of 13 kP is some number of swaps of sets of type 1
1}){}{(  iiX or sets of type 2 1}){}{(  iiX from S . Alternatively we can perform swaps from
S to X . Thus each vertex in )(13 iPk can be associated with an ordered pair ),( ji where i is the number
of swaps of type 2 needed to convert S to X . Thus vertex 1 and 13 k in 13 kP can be swapped with at
most one external private neighbour. However each vertex can be swapped at most once in either direction. Thus
the conditions on the ordered pair ),( ji are 2=,,11 jikjki  . If 1= iq and 1= jr , we
have 11,11  krkq and 21)(  krq .
Thus every )(Gi -set of 13= kPG or 13= kCG is some number of swaps from the )(Gi -set
1},3{1,4,7,= kS  . Hence )(iG is connected for these graphs.
Theorem 2.15 )(13 iPk is isomorphic to a step grid of order k with 2 pendent edges where the pendent
vertices correspond to the )(Gi -sets },,,,{ 631 kvvvv  and },,,,,{ 1313852  kk vvvvv  .
Proof. We know that 1=)( 13  kPi k . Consider the )(Gi -set },,,,{= 137411 kvvvvS  of 13 kP . Fixing
the first vertex of 1S and changing from the
ththrdnd
k,,,4,32  vertex of 1S we get,
},,,,,{= 396312 kvvvvvS 
},,,,,{= 396413 kvvvvvS 
},,,,,{= 397414 kvvvvvS 

}.,,,,,{= 3107411 kk vvvvvS 
Now consider },,,,,{= 13138522  kkk vvvvvS  . Now fixing the first and last vertices of 2kS and changing
the
th
k vertex
th
k 1)(  vertex alone,
th
k vertex
th
k 1)(  vertex (2 vertices),

ththrdnd
kk ,1)(,,,3,2  vertices, 1)( k vertices, we get 1)( k )(Gi -sets. They are
},,,,,,{= 1323538523  kkkk vvvvvvS 
},,,,,,{= 1323438524  kkkk vvvvvvS 

},,,,,,{= 13235374212  kkkk vvvvvvS 
Now fixing the first vertex of 2kS and changing the remaining vertices including the last vertex as before we
get )(2 GikC -sets. Let us denote these 2kC )(Gi sets by (3) . Thus the total number of )(Gi -sets of
2
23
=
2
1)(
12=12=
2
213


kkkk
kkCkPk . Of these
2
232
 kk
)(Gi sets, 1S gets deg
2, 2S gets deg 1 and the remaining 1)( k vertices of (1) get deg 3. 2kS gets deg 1,remaining 1)( k
vertices of (2) get deg 3. Of the 2kC vertices of 21)((3), Ck  get deg 4,remaining
1]=1)([ 22  kCkkC vertices get deg 2.
Thus these
2
232
 kk
vertices are connected in )(13 iPk and they form the step grid of order k with 2
pendent vertices },,,,,{ 39631 kvvvvv  and },,,,,{ 1313852  kk vvvvv  .
Theorem 2.16 For any triangle free graph G , )(iG is triangle free.
Proof. Suppose )(iG contains a traingle of 3 vertices corresponding to )(Gi -sets 21,SS and 3S . Since
),( 21 SS corresponds to an edge in )(iG , }{}{= 12 yxSS  for some )(, GVyx  such that
)(),( GEyx  . Further since ),( 32 SS corresponds to an edge in )(iG , }{}{= 23 dcSS  for some
)(, GVdc  such that )(),( GEdc  . However },{},{=}{}{= 123 dycxSdcSS  . But since
),( 32 SS corresponds to an edge in )(iG , }{}{= 23 baSS  for some )(, GVba  such that
)(),( GEba  . Since 3S is not two swap away from 1S , it must be the case ycax =,= and db = . But
this implies that ),(),,( bxyx and ),( by are edges in )(GE , a contradiction since G is traingle free. Thus
for any traingle free graph G , there is no 3K induced subgraph in )(iG .
Corollary 2.17 For any tree T , )(iT is traingle free.
Theorem 2.18 For any tree T , )(nT is nC -free for any odd 3n .
Proof. Suppose )(iT contains a cycle C of 3k vertices where k is odd. Let x be the vertex in C and
let S be the )(Gi -set corresponding to the vertex x . Let y and z be the two vertices on C of distance
2
1
=
k
m swaps away from x with corresponding )(Gi -sets 1S and 2S . That is there is a path 1P
corresponding to a series of vertex swaps say 1x for 1y , 2x for mxy ,2 for my so that
YXSS =1 where },,,,{= 321 mxxxxX  and },,,,{= 321 myyyyY  . Likewise there is a path
2P corresponding to a series of vertex swaps say 1w for 21,wz for mwz ,,2  for mz so that
ZWSS =2 where },,,,{= 321 mwwwwW  and },,,,{= 321 mzzzzZ  . However since
}{}{=)),((),( 12 baSSiTEzy  for some )(, TVba  . Thus this must be the case that the set
}{}{= jj xwWX  and }{}{= jj yzZY  . This implies that }{}{= 12 jj xySS  and

))((),( iTEyx jj  for mj 1 . Since jx was swapped for jy and kx was swapped for ky in 1P , we
also know that ))((),( iTEyx jj  and ))((),( iTEyx kk  . Now both jx and jy are in 2S . So there
exists a swap lx for iy in 2P such that ))((),( iTEyx il  . However in path lxP,1 was swapped for ly
and thus ))((),( iTEyx ll  . Similarly 2Syl  , so there exists some sx so that in path sxP ,2 was swapped
for ly . We can continue to find the alternating path 1P and 2P swaps. But since m is finite , we reach a
vertex qy which swapped with jx in 2P , thus creating a cycle in T and contradicting the fact that T is
cycle-free. Hence )(iT is free of odd cycle.
Theorem 2.19 Every tree T is the i -graph of some graph.
Proof. Let us prove the theorem by induction on the order n of a tree T . The trees 1= KT and 2= KT are
the i -graphs of 1K and 2K respectively.
Let us assume that the theorem is true for all trees T of order at most n and let
'
T be a tree of order 1n .
Let v be a leaf of
'
T with support u . vT '
is a tree of order n . By induction we know that the tree
vT '
is the i -graph of some graph say G . Let kGi =)( and },,,,{= 321 ku uuuuS  be the )(Gi - set
of G corresponding to the vertex u in vT '
.
Construct a new graph G by attaching k leaves to the vertices in uS say
''
3
'
2
'
1
'
,,,,= ku uuuuS  . Now add
a new vertex x and join it to each of the vertices in
'
uS . Finally attach a leaf y adjacent to x . Then every
)(Gi - set of the new graph G must either be of the form }{xS  for any )(Gi - set S in G or the one
new )(Gi - set }{ySu  .
}{xSu  is adjacent to }{ySu  in the graph i -graph of G. Also the vertex corresponding to the )(Gi -
set }{ySu  is adjacent only to the vertex corresponding to the )(Gi - set }{xSu  and the )(Gi - set
}{ySu  corresponding to the vertex v in
'
T . Thus the i -graph of the graph
"
G is isomorphic to the tree
'
T .
i -graph sequence: From a given graph we can construct the i -graph repeatedly that is ))(()( iiGiGG
ii

etc. We can also see that often the sequence ends with 1K . We can list some examples of the phenomenon.
(1). 11, KK
i
n 
(2). 133 KKC
ii
k 
(3). 1KK
i
n 
(4). i
ii
KPP  34
(5). 1332 KKPP
ii
W the sequence can be infinite.
(6). 14362 KPPPP
ii
W the sequence can be infinite.
(7). 12122 KKPP
ii
k W the sequence can be infinite.
Although all the i -graph sequences terminated after a small number of steps, for some graph the sequence can be
infinite.
For example

1. 
i
n
i
n
i
n KKK 
2. 
i
k
i
K
i
k CCC   232323
3. 
iii
CCPP  8833W
Definition 2.20 Let us define a new class of graph as follows. These graphs are combinations of cycles and
complete graphs. Consider ,kC the cycle on k vertices ),,,,( 321 kxxxx  . If k is odd, we replace each
edge )()(modk),( 1)( kii CExx  , kii  with a complete graph of size n . That is we add vertices
2321 ,,,, naaaa  and all possible edges corresponding to these vertices and ix and 1ix . This is repeated for
each of the original edges in kC . If k is even, we replace one vertex 1x with a complete graph nK and add
edges from kX and 2K to each vertex in the added nK . Then for each of the edge (modk),( 1ii xx ,
12  ki we make the same replacement as we did when k is odd. We call the graph formed in this
manner as kn CK  . The graph 34 CK  is given in fig 5.3.
(-3,-3.3483582)(6.4844894,4.3276935)
59.69716(1.9881554,-0.7504699)[linewidth=0.034,dimen=outer](3.3379886,2.2471151)(-0.042011347,0.46711
504)
119.15412(8.365909,-2.2677374)[linewidth=0.034,dimen=outer](6.538801,2.2125077)(3.1588013,0.43250778)
179.7155(6.4535127,-2.8186145)[linewidth=0.034,dimen=outer](4.920255,-0.5112962)(1.5402552,-2.2912962)
[dotsize=0.12,dotangle=-2.0515552](1.5710514,-0.51132834)
[dotsize=0.12,dotangle=-2.0515552](1.7256418,3.2455456)
[linewidth=0.042cm](1.7456291,3.2448297)(1.5524962,-0.470638)
[linewidth=0.042cm](0.082096644,0.38254735)(3.2345839,2.350954)
[linewidth=0.042cm](1.5701551,-2.2123866)(4.851097,-0.5687867)
[linewidth=0.042cm](1.5710514,-0.51132834)(4.8330774,-2.1891801)
[linewidth=0.042cm](3.27599,2.3894963)(3.275274,2.3695092)
[linewidth=0.042cm](3.275274,2.3695092)(6.364515,0.33761585)
[linewidth=0.042cm](4.7879763,3.2559245)(4.891787,-0.55023146)
-2.0515552(0.113585256,0.10894949)(3.0801435,-3.097705)Fig 1
Proposition 2.21 2))((  nkn kKiCK  .
Proof. We only prove the case when n is odd since the graph )( kn CK  consists of nkK subgraphs
arranged along an odd cycle kC . We choose vertices that will dominate the vertices in each nK subgraph. This
is minimally accomplished by choosing the vertices that are on the inner cycle. Each of these two vertices
dominate two adjacent nK subgraphs . Let kvvvv ,,,, 321  be the vertices of the inner cycle and
2321 ,,, iniii aaaa  be the vertices of nK drawn on the edge jivv of the cycle kC . Then
},,,,,,{= 1,125311  kk avvvvS  },,,,,{= 116422  kk avvvvS  and },,,,,{= 117533 avvvvS k are
three )(Gi sets of kn CK  with cardinality=
2
1k
. Since there are 2 vertices 2kv and 1v of S , the
vertices of nK drawn on the edge kk vv 1 is not determined by the first
2
1k
vertices of 1S . Hence any one

of the vertices of that nK except 1kv and kv should be an element of 1S . Hence 1ka and kv dominates
that nK . There are 2n choices for the last vertex of 1S . Now, varying the last vertex of 1S , these 2n
)(Gi sets including 1S and these 2n )(Gi sets are adjacent with each other and they form a 2nK . Now
fixing the first vertex of 1S and changing from the
nd
2 vertex, we get the )(Gi set
},,,,,{= 2,116414 avvvvS k . Now changing from the
th
ththrd k
2
1
,,,5,43

 vertices we get
2
1k
number of 2nK graph . Thus with 1v as the first vertex we get
2
1k
number of 2nK graphs. Similarly
using 2S we get
2
1k
number of 2nK graph. Now changing the last vertex of 3S by allowing all 2n
choices for it we get a 2nK graph. Thus the total number of 2nK graph= k
kk
=1
2
2
2
1




. Therefore
.=))(( 2nkn kKiCK 
To find the number of independent dominating sets of the comb:
Let nuuuu ,,,, 321  be the supports and nvvvv ,,,, 321  be the corresponding pendent vertices of the comb
nCb . In each )(Gi -set, let us arrange the pendents and supports individually in the ascending order of suffixes.
},,,,{ 321 nvvvv  is the only )(Gi -set with n pendent vertices. Hence the )(Gi -set with no support is 1.
The sets
},,,,,{,},,,,,,,,{},,,,,,,{},,,,,,{ 1321265421254311432 nnnnn uvvbvuvvvvvvuvvvbvuvvvv 
are the n )(Gi - sets with only one support.
To find the number of independent dominating sets with 2 supports:
The )(Gi - sets with 1u as first support are
},,,,,,{,},,,,,,{},,,,,,,,,{},,,,,,,,{ 11432514324176532216542 nnnnn uuvvvvuuvvvvuuvvvvvvuuvvvvv 
. Thus we get 2n )(Gi - sets with 1u as first support.
The )(Gi - sets with 2u as first support are
},,,,,,,,{,},,,,,,,,,{},,,,,,,,{ 21254315276431426531 nnnnn uuvvvvvvuuvvvvvvuuvvvvv  .
Thus there are 3n )(Gi - sets with 2u as first support. Proceeding like this we see that
},,,,,,,{ 213321 nnnn uuvvvvv  is the only )(Gi - set with 2nu as first support.
Hence the total number of )(Gi - sets with 2 supports are
2
2)1)((
=1234)(3)(2)(


nn
nnn  .
Here we see that 3Cb is the smallest comb having )(Gi -sets with 2 supports.
To find the number of independent dominating sets with 3 supports:
5Cb is the smallest comb containing )(Gi -sets with 3 supports and },,,,{ 53142 uuuvv is the only )(Gi -set
with 3 supports. For sake of brevity we use the following notation. We denote supports only. For example let us
denote },,,,{ 53142 uuuvv by },,{ 531 uuu .
For the comb ,nCb the )(Gi -set with first support 1u and second support 3u are },,{ 531 uuu ,
},,{ 631 uuu , },,{ 731 uuu ,..., },,{ 31 nuuu i.e, here we fix the first 2 supports and vary the third support. Thus
we get 4n )(Gi -sets. Now fixing 1u and 4u as the first 2 supports and varying the third support we get the

)(Gi -sets },,{ 641 uuu , },,{ 741 uuu , },,{ 841 uuu ,..., },,{ 41 nuuu . Thus we get 5n )(Gi -sets.
Proceeding like this we get },,{ 21 nn uuu  is the only )(Gi -set with 1u as the first support and 2nu as the
second support. Hence the number of )(Gi -sets with 1u as the first support is
2
3)4)((
=126)(5)(4)(=


nn
nnn  .
Now fixing 2u and 4u as the first 2 supports and varying the third support we get 5n )(Gi -sets. Similarly
by fixing 2u and 5u as the first 2 supports and varying the third support we get 6n )(Gi -sets. Proceeding
like this, by fixing 2u and 2nu as the first 2 supports we get only one )(Gi -set. Hence the number of )(Gi
-sets with first support 2u is
2
4)5)((
=123)(4)(5)(


nn
nnn  .
Continuing in a similar way , by fixing 4nu and 2nu as the first two supports we get only one )(Gi -set. Thus
the total number of )(Gi -sets with 3 supports is
3)4)((
2
1
=
2
12
2
5)6)((
2
4)5)((
2
3)4)((
=
5=








 kk
nnnnnn
n
k

(1)
Hence the number of )(Gi -sets of ,,,,, 98765 CbCbCbCbCb are .35,1,4,10,20, 
To find the number of independent dominating sets with 4 supports.
7Cb is the smallest comb having )(Gi -set with 4 supports.
For the comb, the number of )(Gi -sets with 531 ,, uuu as first 3 supports .7= n
The number of )(Gi -sets with 231 ,, nuuu as first 3 supports 1= .
Hence the number of )(Gi -sets with 1u and 3u as first two supports
2
6)5)((
=
 nn
.
Similarly number of )(Gi -set with 1u and 4u as the first 2 supports
2
7)6)((
=
 nn
Number of )(Gi -sets with 51, nuu as first 2 supports 3= .
Number of )(Gi -sets with 41, nuu as first 2 supports 1= .
Therefore number of )(Gi -sets with 1u as first support.
5)6)((
2
1
=
136
2
7)8)((
2
6)7)((
2
5)6)((
=
9
7=








 kk
nnnnnn
k

(2)
Similarly number of )(Gi -sets with first support 5)6)((
2
1
=
1
7=2 

kku
n
k
.
Number of )(Gi -sets with first support 5)6)((
2
1
=
2
7=3 

kku
n
k
.
Proceeding like this we get the number of )(Gi -sets with first support 7n is 3 and the number of )(Gi -sets

with first support 6n is 1. Hence the total number of )(Gi -sets with 4 supports is
1365)6)((
2
1
5)6)((
2
1
5)6)((
2
1 2
7=
1
7=7=
 

kkkkkk
n
k
n
k
n
k
. Hence the
number of )(Gi -sets of ,,, 987 CbCbCb are 1,5,15,35, .
To find the smallest comb with only one )(Gi -set with 5 supports:
9Cb is the smallest comb with only one )(Gi -set with 5 supports. For the comb ,7Cb the number of )(Gi
-sets with 7531 ,,, uuuu as first 4 supports is 8n . The number of )(Gi -sets with 8531 ,,, uuuu as first 4
supports is 9n .
The number of )(Gi -sets with 7531 ,,, uuuu as first 4 supports is 1. Thus the the number of )(Gi -sets with
first 3 vertices 431731631531 ,,,,,;,,;,, nuuuuuuuuuuuu  are
,1,
2
9)10)((
,
2
8)9)((
,
2
7)8)((

 nnnnnn
Therefore number of )(Gi -sets with 31,uu as first 2 supports= 7).8)((
2
1
9=
 kk
n
k
Similarly number of )(Gi -sets with 31,uu as first 2 supports= 7)8)((
2
1 1
9=


kk
n
k

Number of )(Gi -sets with 61, nuu as first 2 supports is 1.
Number of )(Gi -sets with first support
1.4107)8)((
2
1
7)8)((
2
1
7)8)((
2
1
=
2
9=
1
9=9=1  

kkkkkku
n
k
n
k
n
k
Similarly number of )(Gi -sets with 2u as first support
14107)8)((
2
1
7)8)((
2
1
=
2
9=
1
9=
 

kkkk
n
k
n
k

Number of )(Gi -sets with 2u as first support is 1.
Therefore total number of )(Gi -sets with 5 supports is
8)1.(9)4(10)6(7)8)((
2
4
7)8)((
2
3
7)8)((
2
2
7)8)((
2
1 2
9=
2
9=
1
9=9=
 

nnnkkkkkkkk
n
k
n
k
n
k
n
k

Thus the total number of )(Gi -sets of ,,,, 1211109 CbCbCbCb with 5 supports are 1,6,21,56,
By a similar method we can find the number of )(Gi -sets of the comb with more number of supports.
Note 2.22 Consider the sequence 20,165,35,56,84,11,4,10,20, (1)
This is the sequence of number of )(Gi -sets of the comb with 3 supports. Let
20,=10,=4,=1,= 4321 tttt . The partial sums of the sequence are
1== 11 tS
5=41== 212 ttS
15=1041== 3213  tttS
35=201041== 43214  ttttS

Thus the sequence of partial sums of the sequence (1) is 1,5,15,35, . 2).........(

The terms of this sequence represents the number of )(Gi -sets with 4 supports of the comb.
The sequence of partial sums of the sequence (2) are 1,6,21,56, . (3) .
The terms of this sequence represent the number of )(Gi -sets with 5 supports of the comb.
The sequence of partial sums of the sequence (3) are 210,1,7,28,84, . The terms of the sequence represent
the number of )(Gi -sets with 6 supports of the comb.
Thus if the number of )(Gi -sets with n supports is known, the number of )(Gi -sets with 1n supports can
be found out.
Note 2.23
1. Let us denote the partial sums of the sequence of number of )(Gi -sets with k supports by
,,,
131211 kkk SSS
Then 20,=10,=4,=1,= 3,43,33,23,1 SSSS
35,=15,=5,=1,= 4,44,34,24,1 SSSS
56,=21,=6,=1,= 3,45,35,25,1 SSSS and so on.
2. Number of )(Gi -sets of the comb nCb with 2 supports=number of )(Gi -sets of the comb 1nCb
with 2 supports 2)(3)2)((
2
1
=2)(  nnnn .
3. Number of )(Gi -sets of the comb nCb with 3 supports 4)3)((
2
1
= 53,  nnS n .
4. Number of )(Gi -sets of the comb nCb with 4 supports 74,63,=   nn SS .
5. Number of )(Gi -sets of the comb nCb with 5 supports 95,84,=   nn SS .
6. Number of )(Gi -sets of the comb nCb with 6 supports 116,105,=   nn SS . and so on.
Theorem 2.24 Let us denote the graph )(iCbn by nG . Then order of nG =order of 1nG +order of 2nG .
Proof. We know that nCb has n2 vertices and nCbi n =)( . Also the maximum number of supports in an
)(Gi -set of 



2
=
n
Cbn .
Let nuuuu ,,,, 321  be the supports and nvvvv ,,,, 321  be the pendent vertices of the comb nCb . Then
=)( nGo Number of )(Gi sets with n pendents+ Number of )(Gi -sets with 1n pendents+ Number of
)(Gi -sets with 2n pendents  Number of )(Gi -sets with 



2
n
pendents
  85,64,43,2)1)((
2
1
1= nnn SSSnnn .

)()(=
4)3)((
2
1
2)([1]
)(3)2)((
2
1
1)[(1=
)()(
)(3)2)((
2
1
1)[(1=
()(
)(3)]2)((
2
1
[
2)](3)2)((
2
1
[1]1)[(1=
2)1)((
2
1
1=)().
21
105,84,
63,95,
74,53,
116,105,95,84,
74,63,
116,105,95,84,
74,63,53,
85,64,43,



















nn
nn
nn
nn
nnnn
nn
nnnn
nnn
nnnn
GOGO
SS
SnnnS
SSnnn
SSSS
SSnnn
SSSS
SSnnS
nnnn
SSSnnnGoei





(3)
Example 2.25 When 1=n , 21 KCb  and
11=2|=)(| 1 iCb
21=3|=)(| 2 iCb
1)(12)(1=11)(21=131=5|=)(| 3 iCb
2)(11)3(1=2)(11)(31=341=8|=)(| 4 iCb
1)3(13)4(1=13)(31)(41=1651=13|=)(| 5 iCb
3)4(11)65(1=3)(14)(61)(51=41061=21|=)(| 6 iCb and so on.
Theorem 2.26 For any complete graph H , there exists a graph HG® such that HiG )( .
Proof. Let H be a complete graph with vertices nvvvv ,,,, 321  . By construction, let us prove that there
exists a graph HG® such that HiG )( . To form G , we add a star sK1, of order 1s with vertices
1321 ,, spppp  , center 2p and 3s and add an edge joining any one of the leaf of sK1, to a vertex iv
of niH ,1 . Since no vertex of G is adjacent to any other vertex, 2)( Gi . Obviously
nivpX ii },1,{= 2 is an )(Gi -set for G ,since each iv dominates all the other vertices of sK1, . Since
2p is the only vertex of sK1, which dominates all the vertices of sK1, there are no other )(Gi -sets for G .
Hence niXi ,1 are the only )(Gi -sets for G .
Each )(Gi -set differs by only one vertex as 2p appears in every )(Gi -set of G . Hence HiG )( . The
following figure shows the construction of the graph G with 5KH  so that HiG )( .
(0,-2.328125)(13.48917,3.328125) [dotsize=0.12](0.08,1.0755764) [dotsize=0.12](2.86,1.0755764)
[dotsize=0.12](2.86,-1.6644236) [dotsize=0.12](0.06,-1.6644236) [dotsize=0.12](4.66,-0.2644236)
[linewidth=0.042cm](2.88,1.0755764)(4.66,-0.24442361)
[linewidth=0.042cm](2.88,-1.6244236)(4.64,-0.2644236)
[linewidth=0.042cm](0.08,1.1155764)(4.64,-0.24442361)
[linewidth=0.042cm](0.06,-1.6244236)(4.66,-0.2644236)
[linewidth=0.042cm](2.82,1.0355763)(0.08,-1.6044236)
[linewidth=0.042cm](0.08,1.0755764)(2.92,-1.7044237)
[linewidth=0.042cm](4.7,-0.2644236)(7.36,-0.28442362) [dotsize=0.12](7.4,-0.28442362)
[linewidth=0.042cm](7.42,-0.28442362)(10.3,-0.28442362) [dotsize=0.12](10.34,-0.3044236)
[linewidth=0.042cm](10.34,-0.24442361)(9.32,1.7555764)
[linewidth=0.042cm](10.34,-0.2644236)(10.16,1.8355764)
[linewidth=0.042cm](10.36,-0.2644236)(11.04,1.7355764)
[linewidth=0.042cm](10.34,-0.2644236)(11.94,1.3355764)
[linewidth=0.042cm](10.38,-0.24442361)(12.44,0.6155764)

[linewidth=0.042cm](10.38,-0.2644236)(12.48,-0.28442362)
[linewidth=0.042cm](10.4,-0.28442362)(12.34,-1.1044236)
[linewidth=0.042,dimen=outer](12.18,-1.4044236)0.02 [linewidth=0.042,dimen=outer](11.96,-1.5244236)0.02
[linewidth=0.042,dimen=outer](10.36,-1.7444236)0.02 [dotsize=0.12](10.1,-1.6844236)
[dotsize=0.12](12.32,-1.1244236) [dotsize=0.12](12.46,-0.2644236) [dotsize=0.12](12.44,0.5955764)
[dotsize=0.12](11.94,1.3355764) [dotsize=0.12](11.06,1.7355764) [dotsize=0.12](10.16,1.8355764)
[dotsize=0.12](9.32,1.7755764) [linewidth=0.04cm](10.365889,-0.2503125)(10.085889,-1.6903125)
(7.3387012,-0.5003125) 1p 39.2(1.9594711,-6.852193)(10.589769,-0.6648356) 2p (9.138701,2.0196874)
3p (10.0987015,2.1396875) 4p (11.178701,2.0796876) 5p (12.398702,1.4796875) 6p
(12.978702,0.6596875) 7p (12.998701,-0.3003125) 8p (12.818703,-1.2403125) 9p
(10.168701,-2.1003125) 1sp [linewidth=0.032,dimen=outer](2.84,1.1355762)(0.02,-1.6844236)
Corollary 2.27 Every complete graph H of order n is the i -graph G of order mn where 3.m
Definition 2.28 A graph obtained by attaching a pendent edge to each vertex of the n -cycle is called a crown. Let
us denote it by nG and 1= KCG nn e . Hence a crown nG has n2 vertices.
Let nuuuu ,,,, 321  be the vertices of the cycle(supports) and nvvvv ,,,, 321  be the corresponding pendent
vertices. It is obvious that nGi n =)( . In the )(Gi -set of nG , let us arrange the pendent vertices and supports
in the increasing order of the suffixes. Note that maximum number of supports in any )(Gi -set of 



2
=
n
Gn .
To find the number of independent dominating sets of a crown:
As },,,,{ 321 nvvvv  is the only )(Gi -set with n pendent vertices, the number of )(Gi - set with no
support is 1.
The sets },,,,,{},,,,,{},,,,,{ 1321231132 nnnn uvvvvuvvvuvvv  are the n )(Gi -sets with only
one support. Hence the number of )(Gi -sets with one support is n .
The )(Gi -sets containing 2 supports with 1u as the first support are
},,,,,,{,},,,,,,,,,{},,,,,,,{ 114324176532316542 nnn uuvvvvuuvvvvvvuuvvvv  . Thus we get
3n )(Gi -sets with 1u as the first support.
The )(Gi -sets with 2u as the first support are
},,,,,,,{},,,,,,,,,{},,,,,,,,{ 2254315276431425431 nnnn uuvvvvvvuvvvvvvuuvvvvv  . Thus
there are 3n )(Gi -sets with 2u as the first support.
The )(Gi -sets with 3u as the first support are
},,,,,,,,{,},,,,,,,,,{},,,,,,,,,{ 316542163654215365421 nnnn uuvvvvvvuuvvvvvvuuvvvvvv  .
Thus there are 4n )(Gi -sets with 3u as the first support. Hence the number of )(Gi -sets with 2 supports
is
(1)..........
2
3
=
2
6562
=
2
2)3)((
3=
1]5)(4)(3)[(3)(=
22
nnnnn
nn
n
nnnn



 
(4)
4G is the smallest crown with )(Gi -sets containing 2 supports. Hence substituting 4,5,6,n in (1) we
get the sequence ....(2)90,,54,65,77,0,27,35,442,5,9,14,2 
ie)the terms in the sequence (2) represent the number of )(Gi -sets of the crown ,,, 654 GGG . Let

90,=77,=65,=54,=44,=35,=27,=20,=14,=9,=5,=2,= 121110987654321 tttttttttttt
Consider the sequence of partial sums of (2) .
275,=210,=156,=112,=77,=50,=30,=16,=7,=2,= 10987654321 SSSSSSSSSS
ie)the sequence of partial sums of (2) is ..(3)156,50,77,112,2,7,16,30, 
The terms of (3) represent the number of )(Gi -sets of the crown ,,, 876 GGG with 3 supports . The
sequence of partial sums of (3) is ....(4)105,182,2,9,25,55, 
The terms of this sequence represent the number of )(Gi -sets of the crown ,,, 1098 GGG with 4 supports. In
a similar manner the number of )(Gi -sets of the crown with more number of supports can be found out.
Note 2.29
1. We denote the partial sums of the sequence of number of )(Gi -sets with k -supports by
,,, ,3,2,1 kkk SSS .
Then
 294,=182,=105,=55,=25,=9,=2,=156,=112,=77,=50,=30,=16,=7,=2,=35,=27,=20,=14,=9,=5,=2,= 4,74,64,54,44,34,24,13,83,73,63,53,43,33,23,12,72,62,52,42,32,22,1 SSSSSSSSSSSSSSSSSSSSSS
2. Number of )(Gi -sets of the crown nG with 2 supports 2= 42,  nS n .
3. Number of )(Gi -sets of the crown nG with 3 supports 52,63,=   nn SS .
4. Number of )(Gi -sets of the crown nG with 4 supports 73,84,=   nn SS .
5. Number of )(Gi -sets of the crown nG with 5 supports 94,105,=   nn SS . and so on.
Theorem 2.30 Let nG be a crown of order n . Then =)(iGn order of  )(1 iGn order of )(2 iGn
Proof. Let nuuuu ,,,, 321  be the vertices of the cycle and nvvvv ,,,, 321  be the corresponding pendent
vertices. ))(( iGO n = Number of )(Gi sets with n pendents + Number of )(Gi -sets with 1n pendents +
Number of )(Gi -sets with 2n pendents ++ Number of )(Gi -sets with 



2
n
pendents. Therefore
))(())((=
])(2)[(
1]()(1)[(1=
()(
)(2)]([1]1)[(1=
1=))((
21
94,73,52,
84,63,42,
94,105,73,84,
52,63,42,
1)(3,95,74,53,32,
iGOiGO
SSSn
SSSn
SSSS
SSnSn
SSSSSniGO
nn
nnn
nnn
nnnn
nnn
nnnnnnn
















Example 2.31 When 3=n , 31=4|=)(| iGn
When 4=n , 2)(13)(1=21)(31=241=7|=)(| iGn
When 5=n , 3)(12)4(1=3)(21)(41=551=11|=)(| iGn
When 6=n , 2)4(15)5(1=24)(51)(51=2961=18|=)(| iGn
When 7=n ,
5)5(112)96(1=5)(25)(91)(61=71471=29|=)(| iGn
When 8=n ,
2)96(17)147(1=29)(76)(141)(71=2162081=47|=)(| iGn

When 9=n ,
7)17(12)16208(1=7)(214)(167)(201)(81=9302791=76|=)(| iGn
and so on.
References
[1]. Acharya B.D, Walikar, H.B and sampath Kumar, E, Recent developments in the theory of domination in graphs, Mehta Research
Institute, Allahabad, MRI. Lecture Notes in Math ,1 (1979).
[2]. Akers, S.B Harel, D. and Krishnamurthy, B. The star graph- An attractive alternate to the n-cube -Proc. Intl conf on parallel
processing, (1987), 393-400.
[3]. Alexandre Pinlou, Daniel Goncalves, Michael Rao, Stephan Thomase, The Domination Number of Grids, Ar xiv: 1102.2506 VI
[CS.DM] 25 feb 2011.
[4]. Arumugan. S. and Kala, R.Domination parameters of star graph, ARS combinatoria, 44 (1996), 93-96.
[5]. Arumugam,S. and Kala, R. Domination parameters of Hypercubes, Journal of the Indian math Soc., (1998),31-38.
[6]. ohdan Zelinka, Domatic number and bichromaticity of a graph, Lecture Notes in Methematics .Dold and Eckman, Ed.Pragan (1981)
1018.
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(2011),PP23-31.
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[12]. Harary, F. Graph Theory, Adison-wesly, Reading Mass, 1972.
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[16]. Hedetniemi, M. and Hedetniemi, S.T. Laskar, C. Lisa Markus, Pater J. Slater, Disjoint Dominating sets in Graphs, Proc, of ICDM,
(2006), 87-100.
[17]. Kala, R. and Nirmala Vasantha , T.R Restrained double domination number of a graph, AKCE J Graph Combin ,5, No.1. (2008)
PP.73-82.
[18]. LaskR, r. AND Walikar, H.B. On domination related concepts in graph theory , Proceedings of the international Sysposium, Indian
Statistical Institute, Calculta, 1980, Lecture notes in Mathematics No 885, Springer- Verlag, Berlin 1981, 308-320.
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Journal of combinatorics, 18 (2011).
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