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 eΩ and aΩ implies a−1Ω. The Cayley graph corresponding to Γ is the graph G = (V, E), where
V(G) = Γ and E(G)={(x, xa)xV(G), aΩ} and it is denoted by G = Cay(Γ, Ω). 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 number c(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
G G
:
such that p N v :N v
N
v is a bijection for any vertex v V G
and vp1
vD
. 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,ΩG Cay D
3
n
Ω
2n,Ωay D
G C D
,o s
2 1, , , 2, r s sr sr is a gen
r n
2 3
, , , ,
D e r r r
2n,Ωay 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, n3 be integer, Γ D2n
Thro an ,
1 2
and k, t
t n. We take t
1, 2, ,ra ,rn
1 2
1a a
b
n
m be i hat
in the form at
where
1
ntegers such t
1
, , , k ak n ak rn
k
a m
and
n Let
1 k m,
1 he generating set Ω th
1 1 2
Ω
, , , , 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 ofnc
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 p1
S is a perfect domination set of G. Moreover, if S is inde-pendent, then p1
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
. Alsowhenever 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,Ω
, , , , ,
k t
n a rn a srb srb
2n,Ω
G Cay D .
i k
and If d1a d1, i ai ai1 for
2 , d1b d1, jbjbj1 for 2 j t and
j
dmax1 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
x a d b b and l n
x
. Con- sid
1 . er the set
ix g , n a k d b ix g1 : 0 1 and 0S
r sr i l g d
2
S dl 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 G N S
r one vertex of the
0 c n 1
. If v V G
, then we can writeform v r v
as eithe c or v sr n 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 a mg, for some integers m, g
with 1 m k and 0 g d 1 then v r ix amg
whereas rix g S 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 d1. se ex
In this ca , there ists an integer h with 1 h bt b1 d 1
Subcase 2.1 If hΩ2 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
v sr 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)i x.
Subcase 3.1 If hΩ1
a a1, , ,2 ak
, 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
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
0that
0
h
v r r
Subcase 4.1 If hΩ1, then v [ ]S .
e 4.2
N r N
g
, for some integ
ppose m
with 1 m k 1g 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 sr r
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 g N 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 hΩ
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 a mg,
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
r sr
, w ns that
( k [ ]
n a
v sr N S
1
i l and
2 2 v g Case )
1 1 k
0 h k
Subcase 3.1 If 0 h a1, the on of
,
d v
t k
b b d a j a
there exists an integer that v r hsrn a k
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
r sr
, 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 Ca y D
2n,Ω
Lemma 5Let n3 be an integer 1
2
n
m and
k, ch tha . Let
and If 1
t be integers su t 1 k m, 1tn
a1 a2 a n a n a 1 1 1 b2
Ω
, , , k, k, k , , n a , b, , , bt
r r r r r r sr sr sr
,
2n,Ω
G Cay D . d1a d1, i ai ai for
2 i k, d1b d11, jbjbj1 for 2 j t and d d then
1 ,1max j t i, j
d i k ,
22
t
k
G d
a
n .
d
Proof. Let x d 2ak and
n l x
ix g, 0 0 1
tS r srn ix g b 1 : i l and g d .
2
t
S d .
Clearly l We have to prove that
tV G N S If v V G
, then we can write vn 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 gN r
tv N S .
Case 2 pose 0
. Sup i l 1
m
a g
and
We can wri 1 k
a j a d1.
te j , for some integers m, g with
1 m k and 0 g d 1. If ix g
r v r
m
ix g a
v r whereas St and so
c, then
St or ifix g
v N r N v sr n 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 r sr . ase 3.1 S
Subc uppose h
a a , , ak
and ifc
v r 1 1 2
Ω ,
, then
( 1)i x
N
v st
S
or if
v r rn c b 1,
then v N sr
n (i b1
case 3.2
N S . 1)x tSub Suppose h a mg, for some integers m,
th
g wi 1 m k and 1 g d 1. In this case, if
c
v r , then v r am r i1x g , which means that
N
1
i x g
t
v N r S or if v sr n c b 1, then
i1x g b am 1
v r ,
n
sr
which implies that
n i 1x g b1
v N sr
Case 4. Suppose
tN 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
vr r or v r h srn b1.
Subcase 4.1 When hΩ1, and if r
c
v , then
r0 N S
or if n c
sr
t
v N v b1, then
n b1
t
N sr N 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 r am rg, which means that
N S
tg
v N r or if v s b1, 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
tN S .
f G. otal dominating set o
2
S d
t t l
.
nnected do
Lemma 6 Let n3 be an integer, 1
2
m n
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, ,
Ω, , , , t
a n a n a b
a a rn a sr srb b sr
2n,Ω
G Cay D . d1a11,di ai ai
1
for 2 i k, d1b d11, jbjbj for 2 j tand
1 ,1max i k j t i, j
d d 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 0St r sr i l g d
.the notation of Lemm , a d x a k 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 i1x, srn ix d( 1
x b
d
ix d 1ak b1
n n
sr sr induced subgraph
1ak
1
are
c
S
is connected
1
b
connected.
( 1)i x 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 n3 be an integer, 1
2
n
m
and k, t be integers such that and d is an
and Then
1 k m,
2kd 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 .
22 1
n G
k t
. In this
case, has an E-set.
. Let
G
Proof
22 1
n ld k t . In
and
ta a 4, ’s and ’s are same,
for and d all
2 1
x d k t the no tion of Lemm
all 1 i k di for di
i
a id bj j 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 n3 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 a2k 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 C D
Proof. Let
ay
r, , k,rn k ( 1), ,rn1, ,sr sr2, ,srt
, r2 , ,r rk n 2k t 1
n l
and x2k t 1. By taking d1 in
Theorem 7,
0, , , 1 k, n k x, , n k l 1x
S rx rl x,srn sr 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 D 2n.
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 n3 be an integer, 1
2 n m
and k be an integers such that1 k m,
0 2n, 2n { }
G Cay D D e ,
2
Ω , , , ,2 k n n (k 1), , n1, , , , srn bi
i r r r r r sr sr
kd,r
and Gi Cay D
2n,Ωi
(i1) Assume that Ωi 1divides n and Ωi1 1 divides Ωi 1. Then the finite
family of graphs { ,G ii 0} is inductive subgroups
E-. Let
chain.
oof
Pr i Ωi 1. By the assumption i1.
divides i. Define the map i:V G
i V G
i1
by
i v v
for all v G i. 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
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 n3 be an integer,
i
S
S S
e const
1i 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 i1, let, k, t
tegers su ch that
i . The is
surjective m
i
d, 2 , , k , 2in d, 2in 2d, ,i r r d r d r r
2 2
r inkd,srd,sr d, , sr dt
and GiCay D
2i ,Ω
n i1 a covering ofi
G. n
Proof. Define the
G
ap
1
:i i
f V G V G by
2i
j j mod n
i
f r r and
j j mod2in if sr sr for all j, where 0 j 2i
n
.rphism from D2i1n
r with
1 1
a group homomo
ts Note that fi is
2in
D . Let ,
adj ent in Gi1. 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 n1 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 ti i
f r r sr sr , we have
ki i
f u f v r or
ti 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
f N v N v for any vertex v V G
i 1
and v V G
i . Claim fi N v is bijection. Anyele-ment x in N v
as either ne vertex of the forme o
x r or x sr e
. ,, where 0 j 2i1
n1
. Letx y N v
1. Let
Then we have following three cases:
Case x r e1 and y r e2 with
1 2
e e . 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 2i
o r mod n ich is a contradiction to
o r n. There
mo
n
od 2i
e n
fore
, wh
i i
f x f y .
2. Let e1
Case x r andy sr e2. Suppose
i i
f x f y , Th ns
e e1 2
r e e n
i.e. e1mod 2in e2mod 2in is mea
or
r sr
i
mod 2in
e s sr 12mod 2 e
ore
, which is a
f
contradiction. There f xi
f yi
1
e .
Case 3. Let x sr and y sr e2with
1 2
e e . Sup-
pose f xi
f yi , i.e.1 2
1mod 2i 2 e
e n e mod 2in emod 2in
sr sr r e. i.e.
1 2 2i
o r e e 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 n3
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 . For1
i let
2in kd 2 t
2 2
, , , ,
d k
i r r d r d r
, 2 2 ,
, , , ,
in d in d
d
r
r sr sr d sr d
,
and
Ω
.for Gi. Then
{ ,G ii 1} 2i ,
i n i
G Cay 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
, 1i
G i . Since by Theorem 2, i
i Si1
f S 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
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