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Bounds for Domination Parameters in Cayley Graphs

on Dihedral Group

T. Tamizh Chelvam, G. Kalaimurugan

Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, India Email: tamche59@gmail.com

Received November 3,2011; revised December 18, 2011; accepted December 25, 2011

ABSTRACT

In this paper, sharp upper bounds for the domination number, total domination number and connected domination number for the Cayley graph G = Cay(D2n, Ω) constructed on the finite dihedral group D2n,and a specified generating

set Ω of D2n. Further efficient dominating sets in G = Cay(D2n, Ω) are also obtained. More specifically, it is proved that

some of the proper subgroups of D2n are efficient domination sets. Using this, an E-chain of Cayley graphs on the

dihe-dral group is also constructed.

Keywords: Cayley Graph; Dihedral Group; Domination; Total Domination; Connected Domination; Efficient Domination

1. Introduction and Notation

Design of interconnection networks is an important inte-gral part of any parallel processing of distributed system. There has been a strong interest recently in using Cayley graphs as a model for developing interconnection net-works for large interacting arrays of CPU’s. An excellent survey of interconnection networks based on Cayley graphs can be found in [1]. The concept of domination for Cayley graphs has been studied by various authors [2-7]. I. J. Dejter and O. Serra [3] obtained efficient do- minating sets for Cayley graphs constructed on a class of groups containing permutation groups. The efficient domination number for vertex transitive graphs has been obtained by Jia Huang and Jun-Ming Xu [4]. A neces-sary and sufficient condition for the existence of an in-dependent perfect domination set in Cayley graphs has been obtained by J. Lee [5]. Total domination in graphs was introduced by Cockayne, Dawes, and Hedetniemi [2] and is now well studied in graph theory. T. Tamizh Chelvam and I. Rani [6-8] have obtained bounds for various domination parameters for a class of Circulant graphs.

Let Γ be a finite group. Let Ω be a generating set of Γ satisfying e and a implies a−1. The Cayley graph corresponding to Γ is the graph G = (V, E), where

V(G) = Γ and E(G)={(x, xa)xV(G), aΩ} and it is denoted by G = Cay(Γ, ). Let G= (V, E), be a finite, simple and undirected graph. We follow the terminology of [9]. A set SV of vertices in a graph G is called a

dominating set if every vertex vV is either an element

of S or adjacent to an element of S. A dominating set S is a minimal dominating set if no proper subset of S is a dominating set. The domination number(G)of a graph

G is the minimum cardinality of a dominating set in G

and the corresponding dominating set is called a -set. A set SV is called a total dominating set if every vertex v

V is adjacent to an element u (v) of S. The total domination number t(G) equals the minimum cardinality

among all the total dominating sets in G and the corre-sponding total dominating set is called a t-set. A

domi-nating set S is called a connected dominating set if the induced subgraph S is connected. The connected domi-nation numberc(G) of a graph G equals the minimum

cardinality of a connected dominating set in G and a cor-responding connected dominating set is called a c-set. A

set SV is called an efficient dominating set (E-set) if for every vertex vV, |N[v]∩S|=1.

An E-chain is a countable family of nested graphs, each of which has an E-set. We say that a countable fam-ily of graphs G = {Gi, i 1} with each Gi has an E-set Si

is an inclusive E-chain if for every i  1, there exists a surjective map fi: Gi+1  Gi such that fi 1

(S

i)  Si+1.

And also we define that a finite family of graphs G = {Gi,

i 0} is an inductive E-chain if every Gi+1 is a spanning

subgraph of Gi and each Gi has an E-set Si. Let V(Gi) be

any finite group and if, for each i 0, there exists a bi-jective map i: V(Gi)  V(Gi+1) such that i(Si)  Si+1

and Si is the subgroup of V(Gi) then we say that G is an

inductive subgroups E-chain.

A graph is called a covering of G with projection if there is a surjection

GG

:

(2)

such that p N v  :N v

 

 N

 

v is a bijection for any vertex v V G

 

and vp1

 

v

D

. We use the covering function to show the inclusive E-chain.

In this paper, we obtain upper bounds for domination number, total domination number and connected

domi-nation number in a Cayley graph con-

structed on the dihedral group 2n, for and a

generating set . Further, we obtain some E-sets in . Note that the dihedral group 2n

with identity e is the group generated by two elements r

and s with

2n,Ω

G Cay D

3

n

2n,Ω

ay D

G CD

,o s

 

2 1, , , 2, r s sr sr

is a gen

r n

2 3

, , , ,

D e r r r

2n,Ω

ay D

 and . From these

defining relations, one can take

1 rs sr 

1

2 , ,

n n

n sr

 

 

G C , where Ω erating set of and D2n.

ughout this paper, n3 be integer, Γ D2n

Thro an  ,

1 2

and k, t

t n. We take t

1, 2, ,ra ,rn

1 2

1aa

b 

n

m be i hat

in the form at

where

1

ntegers such t

1

, , , kak n akrn

 

k

a m

  and

n Let

1 k m,

1  he generating set Ω th

1 1 2

, , , , bt

a a a b b

r r r sr sr sr

,



t

b

1 2

0 b 1. d1a1,di a ai i for

2 i k , 1 1,d j bj j1

d dj

. Some of

nc

k-regul d b

1i k,1 j t

d    

r further re

b

 and

the resu isted

e. Theorem 1 [4] Let G be a

 

,

i

fere

for 2 j t

lts are l max

below fo

ar graph. Then

 

 

1

V G G

k

 

 , with th nd only if G has an

efficient dominating set.

G be a covering and let S

e equality if a

:G tion set of

Theorem 2 [5] Let p

be a perfect domina G. Then p1

 

S is a perfect domination set of G. Moreover, if S is inde-pendent, then p1

 

S is in pendent.

Theorem 3 ry subgroup of th

de

[10] Eve e dihedral group

2n is cyclic or dihedral. A complete listing of the

sub-ps is as follows: 1) cyclic subgroup D

grou

s rd , where d divides n, with

in-de

edral subgroups x 2d.

2) dih r r sd, i , where d divides n

an x d. Eve

In t e

domina-Lemma 4 Let be an integer,

d 0  i d 1 with inde ry subgroup of D2n

occurs exactly once in this listing.

2. Domination, Total Domination and

Connected Domination Numbers

his section, we obtain upper bounds for th

tion number, total domination number and connected domination number of graph G Cay D

2n,Ω

. Also

whenever the equality occurs we onding

sets.

give the corresp

and k,

t are integers such that 1 k m, 1 t

3

n 1

2

n m 

n

    . Let

r ra1, a2, ,rak,rn ak,r 1 1 1 srb2,

, , , , ,

k t

n a  rn asrb srb

  

2n,Ω

G Cay D .

i k

and If d1a d1, i ai ai1 for

2  , d1b d1, jbjbj1 for 2 j t and

j

dmax1 i k,1 j t d di,  , then

 

1 b

2

2 2 k t

G d

d a b

  n

 

Proof. Let

.

1

2 k 2 t

xad b b and l n

x

 

   . Con- sid

1 . er the set

 

ix g, n ak d b ix g1 : 0 1 and 0

S

r sr     i l g d

     

2

Sdl and

Clearly

 

1

1 0

] l ix g n ak d b ix g

i N sr

      

    

[

N S N r ,

where 0  i l 1 and 0  g d 1 We have to e that

prov

 

[ ]

V GN S

r one vertex of the

0 c n 1

. If v V G

 

, then we can write

form v rv

as eithe c or v srn c b ( t),

where    . By algorithm,

c xi j

the division

  , where 0  i l 1 a  j

Supp c nd 0 x 1

 .

v r .

ose We have the following cases:

. Suppose

Case 1 0  i l 1 and 0 j ak  d 1.

Subcase 1.1 If 0 j a1, then by the definition of ,v S N S[ ]

d   .

Subcase 1.2 If j amg, for some integers m, g

with 1 m k and 0  g d 1 then v rix amg

whereas rix gS an N r[ ix g]N S[ ].

Ca se 0 1

d so v i l

se 2. Suppo    and

1

k k t

a  d j a d  b b d1. se ex

In this ca , there ists an integer h with 1   h bt b1 d 1

Subcase 2.1 If hΩ2  n

, ,

b b the

such that

1

k

n a d b h

v sr sr   

b, , ( ix).

1 2  t

( ix)

N S[ ]

 

ppose h bm g

1

k

n a d b

v N sr   

Subcase 2.2 Su   , for some integers m,

g with 1 m t and 1  g d 1. ea

]. 0 i

In this case,

1

( )

m k

b n a d b ix g

sr sr     

 , which m ns that

)

(ak d b ix g [

n

vsr      N S

2

l v

Case

1

3. Suppose    and

k t

1

k t 1 k 1

a      d b b d j a     d b b d a  .

xists an in with

In this case, there e teger h 1 h ak

such that v r hr( 1)ix.

Subcase 3.1 If hΩ1

a a1, , ,2ak

, then

( 1i )x

v N r  N S[ ].

ppose h a g

Subcase 3.2 Su  m , for some integers m,

g, with 1 m k and 1  g d 1. In this case,

( 1) m

a i x g

r r

v    , which means that

(3)

Case 4. Supp

.

Then there exists an integer h with such

.

Subcas Su ers m,

g an 1 . In this

ose i l 1 and

j

1

k t k t 1 k 1

a     d b b d a     d b b d a  1 h ak

 

0

that

0

h

v r r

Subcase 4.1 If hΩ1, then v  [ ]S .

e 4.2

N r N

g

  , for some integ

ppose m

with 1 m k 1g case,

h a

d  d

m

a g

v r r which means

 

rg N S[ ].

Suppose v s bt. We lowing cases:

Suppose 0  i l 1 that v

have th

N

e f

n c

r

 ol

Case 1. and

1

ase 1.2 Su rs m,

g and In this case,

. In this case, the that

then

for some integers m,

g with and . In this case,

1. In this case,

1 such

[

g with and

1

ith 0  h bt b ch that

2

Ω , then v N r

 

ix N S[ ].

Subc bmg, for ge

0 d

h w

Subcase 1.1 If h

1  . In

1 d su 

t

j b b

   this case, there exists an integer

x

. h i

v srr

ppose h

1  1. ns that v

l and

some inte

with 1 m t g d

m

b ix g

sr r

 , which mea N r

 

ix gN S[ ].

Case 2. 0 i 1

1

t k

b b d a

     re

v  

Suppose

1 1

ists an integer h w ak such

Subcase 2.1 If hΩ

t

b   b d j ex

1

k

n a d b h

v r sr (   ix)ith . 1 h

1

 

ix

 

ppose

1 [ ]

k

n a d b

v N sr    N S .

Subcase 2.2 Su h amg,

1  g d 1

hich mea .

0 1 m k

1

d b ix

  

3. Suppose

1

( )

m k

a n a d b ix g

rsr     

 , w ns that

( k [ ]

n a

v srN S

1

i l  and

2 2 v gCase ) 

1 1 k

0 h k

Subcase 3.1 If 0 h a1, the on of

,

d v

t k

b   b d aja

there exists an integer that v rhsrn ak

n by th

t

b b d

   h with

1

( d b xi  ). a  d

e definiti

]

pose h

S N S

  .

Subcase 3.2 Sup amg, for some integers m,

1 m k 0 g d 1

hich mea

g

 . In this case,

The following lemma pr es an upper bound for the

to .

,

1

( )

m k

a n a d b ix g

rsr     

 , w ns that

( k [ ]

n a d

v sr    N S .

set of G. ovid v

)

Thus S atin

1

b ix g 

is a domin

tal domination number in G Cay D

2n,Ω

Lemma 5Let n3 be an integer 1

2

n

m   and

 

k, ch tha . Let

and If 1

t be integers su t 1 k m, 1tn

a1 a2 a n an a 1 1 1 b2

, , , k, k, k , , n a , b, , , bt

r r r r rrsr sr sr

   ,

2n,Ω

G Cay D . d1a d1, i  ai ai for

2 i k, d1b d11, jbjbj1 for 2 j t and d d then

1 ,1

max j t i, j

d i k    ,

 

2

2

t

k

G d

a

n .

d

 

Proof. Let x d 2ak and

n l x        

ix g, 0 0 1

t

Srsrn ix g b  1 :  i land   g d .

2

t

Sd .

Clearly l We have to prove that

 

 

t

V GN S If v V G

 

, then we can write v

n c  as

either on of the form v r or

wher 0 n 1

e vertex

e c

c v sr  b1,

   . By th vision algorithm,

c xi j

e di

  , where 0  i l 1 and 0  j x 1. We have the following cases:

Case 1. Suppose 0  i l 1 and 0 j a . For

some integer g with 0  g d 1 and 1nition

of c

by the defi

d, if v r , then 1 or if

 1

n c b

v sr   , then

x g b 

N S

 

t

srn i  

v N

 

ix g

N r

 

t

v N S .

Case 2 pose 0

. Sup   i l 1

m

a g

and

We can wri 1 k

a  j ad1.

te j  , for some integers m, g with

1 m k and 0  g d 1. If ix g

rv r

m

ix g a

v r   whereas St and so

c, then

 

 

St or if

ix g

v N r  N v srn c b  1, then 1

m

g a b

   whe n ix g b 1

 

n ix

v sr 

. Consider the set

reas sr St

   and so

 

1

v N sr N

Case 3. Suppos

 

St . n ix g b  

e 0  i l 2 and

2

k k

d a   j d a .

h w

In this case, there exists an integer

ith 1 h ak su r

( 1) 1

n i x b

h    ch that

x or

( 1) . h i

v r  

v rsr . ase 3.1 S

Subc uppose h 

a a , , ak

and if

c

v r 1 1 2

Ω ,

 , then

( 1)ix

N

 

v s

t

S

  or if

v r rn c b  1,

then v N sr

n (ib1

case 3.2

 

N S  . 1)x t

Sub Suppose h amg, for some integers m,

th

g wi 1 m k and 1  g d 1. In this case, if

c

v r , then v ramr i1x g, which means that

 

N

 

 

1

i x g

t

v N r   S or if v srn c b  1, then

i1x g bam   1

v r ,

 

n

sr

 which implies that

n i 1x g b1

v N sr     

Case 4. Suppose

 

t

N S .

1

i l  and d ak ith 1 h a

2

j d a

    .

k such that k

Then there exists an integer h w

0

h

vrr or v rhsrn b1.

Subcase 4.1 When hΩ1, and if r

c

v , then

 

r0 N S

 

  or if n c

sr

t

v N vb1, then

n b1

 

t

N srN S

  .

pose h v

Subcase 4.2 Sup amg, fo ntegers m,

g with

r some i

1 m k and 1  g d 1. In this case, if

c

r

v and v ramrg, which means that

 

N S

 

t

g

v N r  or if v sb1, then

 

n

sr

 h means that

c n

r

1

m g b

a

v r  ,

  whic

n g b1

v N sr   

Thus St is a t

 

t

N S .

f G. otal dominating set o

2

S d

 

t t l

 .

nnected do

(4)

Lemma 6 Let n3 be an integer, 1

2

mn 

at . Let

,

and If 1

  and k, t be integers such th 1 k m, 1 t n

r1,r 2, , r k,r k,r k1 1 1 2, ,

, , , , t

a n a n a b

a a    rn asr srb b sr

2n,Ω

G Cay D . d1a11,di ai ai

1

for 2 i k, d1b d11, jbjbj for 2 j tand

1 ,1

max i k  j t i, j

dd d , then

 

1

d

2

c

k

G d

a

.

Proof. Let

n

1and

k

x a  d l n

x

 

   . Consider the set

1  

In a 5 1 = 1 an

and is a total dominating set. Since

each with , we have paths

note

)and

Hence the

3. Subgroups as Efficie

obta in

n

 

ix g, n ix g b1 : 0 1 and 0

Strsr      i lg d

.

the notation of Lemm , a d x ak d 1

r  and for

c

S

i 0  i l 1

1 ( 1)

, , ,

ix ix ix d

r r   r   and

 1,  1 1, ,  1 1

n i n ix b n ix d b

sr   sr     sr     . Also that

( 1)

ix d

r   andrix r i1x, srn ix d(  1

x b

d

 

ix d 1ak b1

n n

sr     sr induced subgraph

1ak

1

  are

c

S

  is connected

1

b

connected.

( 1)ix b

.

nt Domination Sets

In this section, we in some E-sets

2n,Ω

G Cay D . Moreover we have identified certain

subgroups of D2n which are also efficient dominatio sets in G.

Theorem 7Let n3 be an integer, 1

2

n

m   

  and k, t be integers such that and d is an

and Then

1 k m,

2k

d k n

1 t n

Let

  

integer such that  t 1

dividesn.

r r d, 2 , ,r d r, ( kd),rn k( 1)d, ,

   

d

( ) 2

rn d ,sr srd, d, , sr dt

2n,Ω

G Cay D .

 

2

2 1

n G

k t

 

  . In this

case, has an E-set.

. Let

G

Proof

22 1

n l

d k t  . In

 and

ta a 4, ’s and ’s are same,

for and d all

2 1

x dk t 

the no tion of Lemm

 all 1 i k di for di

i

a id bjj 1 j t.

Letx d

2k t 1

and l n x

     

 

rix g ,srn k  g : 0 0 g d . By Lemma 4,

1

dominating set nce

1,

d ix

S   i l

is a and he

 

G  2n .

2k t 1

  Since

is regular, by Theorem 1, one can conc e

m 3 identifies

gr th

G 2k t lud

that S is an E-set in G.

Remark 8 Note that Theore all

sub-oups of e dihedral group D2n. Now we us identify

some f the subgroups efficient dominating sets. o as

Theorem 9Let n3 be an integer, 1

2

n m   

  and

k, t be integers such that 1 k m, 1 t n and

2k t 1 divides n . Let H r sra, n b be

he dihedral group where a2k t

0 b k 1

a subg

1

an

roup

d

of t D2n, b,

   Then, there exists a generating set of 2n

D such that H is an efficie ti r the Cayley graph

nt domina ng set fo

2n,Ω

.

G CD

Proof. Let

ay

r, ,k,rn k ( 1), ,rn1, ,sr sr2, ,srt

,   r2 , ,r rk n  

2k t 1

n l

  and x2k t 1. By taking d1 in

Theorem 7,

      

0, , , 1 k, n k x, , n k l 1x

S rxrlx,srnsr    sr   

g r th omin

r

is an

is an efficien set of

0 Unde e assumptions of Theorem 9,

efficient d ating set for the Cayley graph

t dominatin G.

Remark 1

. S x

2n,Ω

y D

G Ca for all x D2n.

4. E-Chains in Cayley Graphs

orem 7 and 9 provide a tool to produce E-sets and

the subg s

The

visuali e some ofz roups a E-sets in

2n,Ω

Cay D . We use this tool to ob

chain and inductive subgroups

E-tain an inclusive chain of Cayley graphs on the dihedral group.

Theorem 11 Let n3 be an integer, 1

2 n m   

 

and k be an integers such that1 k m,

0 2n, 2n { }

GCay D De ,

2

, , , ,2 k n n (k 1), , n1, , , , srn bi

i r r r r r sr sr

   

  kd,r

and GiCay D

2n,Ωi

(i1) Assume that i 1

divides n and i1 1 divides i 1. Then the finite

family of graphs { ,G ii 0} is inductive subgroups

E-. Let

chain.

oof

Pri  Ωi 1. By the assumption i1.

divides i. Define the map i:V G

 

iV G

i1

by

 

i v v

  for all v Gi. By Theor

is of the form

em 9, Gi has an

efficient dominating set and it

1 2

0 , i, i,

n

n

S r e r r  

  

   

  

( )

1 ( 2 )

, , ,

, , , i

i i

i i i

n k i

n k

n k n k

r sr

sr sr sr

 

  

   

    

    

  

(5)

and also ’s are subgroups. It implies that

for every . Hence the family of

is ve subgroups E-chain.

sive E-chain of ayley graphs is based on the following lemma.

Lemma 12 Let n3 be an integer,

i

S

SS

e const

 

1

i i i

graphs

Th r

1

i

inducti an inclu

{ ,G ii 0}

 

uction of C

1 2

n

m   

 

be in ch that 1 k m, 1 t n and d is an in- teger su d

2k t 1

divides n . For i1, let

, k, t

tegers su ch that

i . The is

surjective m

i

d, 2 , , k , 2in d, 2in 2d, ,

ir r d r d rr

  

2 2

r inkd,srd,sr d, , sr dt

and GiCay D

2i ,Ω

n i1 a covering of

i

G. n

Proof. Define the

G

ap

1

:

i i

f V G V G by

 

2

i

j j mod n

i

f rr and

 

j j mod2in i

f srsr for all j, where 0 j 2i

n

.

rphism from D2i1n

r with

1 1

a group homomo

ts Note that fi is

2in

D . Let ,

adj ent in Gi1. Th

onto i 1 . Suppose u and v are

ac u v G k

 

en, there exis

1

n

k   

  or t

1

2

 

  sr with 1 t n1 such that

or group homomor

. k

u v r  u v sr . k. Since

i

f is a -

phism and

 

k k mod2in k

 

t t mod2in t

i i

f rr  srsr , we have

 

 

k

i i

f u  f v r  or

 

 

t

i i

f u  f v sr  and so

,

r f sr

 

i

f u

and f vi

 

 are adjacent in Gi . Consider th

 

e map

 

 

:

i N v

fN v N v for any vertex v V G

i 1

and v V G

 

i . Claim fi N v  is bijection. Any

ele-ment x in N v

 

 as either ne vertex of the form

e o

x r or x sr e

 

. ,

, where 0 j 2i1

n1

. Let

x y N v 

1. Let

Then we have following three cases:

Case x re1 and y re2 with

1 2

ee . Sup-pose f xi

 

i

 

1 d 2i 2m

e n

e e

 

f y

 , i.e.

1 2mod 2i

e e n

r r r  e. i.e.

 

1 2 2

i

o r mod n ich is a contradiction to

 

o rn. There

 

mo

n

od 2i

e n

fore

, wh

 

i i

f xf y .

2. Let e1

Case x r andy sre2. Suppose

 

 

i i

f xf y , Th ns

e e1 2

r  e e n

i.e. e1mod 2ine2mod 2in is mea

or

r sr

i

mod 2in

e s sr 12mod 2 e

ore

 , which is a

f

contradiction. There f xi

 

f yi

 

1

e .

Case 3. Let x sr and y sre2with

1 2

ee . Sup-

 

pose f xi

f yi , i.e.

1 2

1mod 2i 2 e

e n e mod 2in emod 2in

srsrr  e. i.e.

  

1 2 2

i

o ree n n

Therefore i

 

i

 

mod which is a contradiction.

f x f y . H

pe

 t elements of

 

N v are distinctly map

ence distinc

d onto N v

and so

 

i N v is a req jection

The n3

f uired bi

1 2

n

m   

 

.

orem 13 Let be an integer, ,

k, t, be integers such that such that

1 k m, 1 t n and d is an integer d

2k t 1

divides n . For

1

ilet

2in kd 2 t

2 2

, , , ,

d k

ir r d r d r

  , 2 2 ,

, , , ,

in d in d

d

r

r sr sr d sr d

 

,

and

.

for Gi. Then

{ ,G ii 1} 2i ,

i n i

GCay D be an effi

g set

Let cient domi-natin th inite family of graphs

S

e fi   is an inclu

s in.

Proof. Since by above Lemma, is a covering of

ive E-cha 1

i

G

, 1

i

G i . Since by Theorem 2, i

 

i Si

1

fS  1

1} is an . Hence

ily of graphs

inclu-Th d

sistance P /2007) of Un

warded to the undaranar -2012.

91(93)90054-O

the finite fam sive E-chain.

{ ,G ii

5. Acknowledgements

b

e work reported here is supporte y the Special

As-rogramme (F510-DRS-I iversity

Grants Commission, India a Department of

Mathematics, Manonmaniam S University for

the period 2007

REFERENCES

[1] S. Lakshmivarahan, J. S. Jwo and S. K. Dhall, “Symme-try in Interconnection Networks Based on Cayley Graphs of Permutation Groups: A Survey,” Parallel Computing, Vol. 19, No. 4, 1993, pp. 361-407.

doi:10.1016/0167-81

[2] E. J. Cockayne, R. M. Dawes and S. T. Hedetniemi, “To-tal Dominatio s, Vol. 10, No. 3, 1980, pp. 211-2

n in Graphs,” Network

19. doi:10.1002/net.3230100304

[3] I. J. Dejter and O. Serra, “Efficient Dominating Sets in Cayley Graphs,” Discrete Applied Mathematics, Vol. 129, No. 2-3, 2003, pp. 319-328.

doi:10.1016/S0166-218X(02)00573-5

[4] R. J. Huang and J.-M. Xu, “The Bondage and Efficient Domination of Vertex Transitive Graphs,” Discrete Mathe- matics, Vol. 308, No. 4, 2008, pp. 571-582.

doi:10.1016/j.disc.2007.03.027

[5] J. Lee, “Independent Perfect Domination Sets in Cayley Graphs,” Journal of Graph Theory, Vol. 37, No. 4, 2000, pp. 219-231.

[6] T. Tamizh Chelvam and I. Rani, “Dominating Sets in Cayley Graphs on Zn,” Tamkang Journal of Mathematics,

Vol. 37, No. 4, 2007, pp. 341-345.

[7] T. Tamizh Chelvam and I. Rani, “Independent Domina-tion Number of Cayley Graphs on Zn,” The Journal of

Combinatorial Mathematics and Combinatorial Comput-ing, Vol. 69, 2009, pp. 251-255.

(6)

athematics, Vol. 20, ral Groups II,” 2009. Domination Numbers of Cayley Graphs on Zn,”

Ad-vanced Studies in Contemporary M New

2010, pp. 57-61.

[9] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, “Fun-damentals of Domination in Graphs,” Marcel Dekker,

York, 1998. [10] K. Conrad, “Dihed

References

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