TELKOM
NIKA
, Vol. 11, No. 9, September 20
13, pp.
5316
~53
2
1
ISSN: 2302-4
046
5316
Re
cei
v
ed
Jan
uary 23, 201
3
;
Revi
sed
Jun
e
10, 2013; A
c
cepted
Jun
e
21, 2013
Some Results of Bonda
ge Number of (n,k)-Star Graphs
Yunchao
We
i*, Hongxian
Zhu, Junli Han
Coll
eg
e of Information T
e
chno
log
y
, Sha
n
g
hai
Ocean Univ
er
sit
y
, Sha
ngh
ai,
2013
06, P. R. Chin
a
*
Corresp
on
din
g
author, e-ma
i
l
:
y
c
w
e
i@sh
ou.
edu.cn
A
b
st
r
a
ct
In the co
mpute
r
netw
o
rk, bo
n
dag
e n
u
m
b
e
r i
s
one
of th
e
most i
m
porta
nt p
a
ra
meters t
o
me
asur
e
the co
ntrol
the
o
ry of th
e c
o
mputer netw
o
rk, den
oted
by
()
bG
for
a
netw
o
rk gr
a
ph
G
. So com
p
uting
()
bG
of some partic
u
lar kn
ow
n gr-
aphs is extr
e
m
e
l
y val
uab
le.
In this paper,
w
e
determi
ne
,2
()
n
bS
and the
precis
e low
e
rb
oun
d of
()
bG
of
(,
)
nk
-star graph
s, denoted by
,
nk
S
, followed by som
e
relati
ve
conclusions of
n
-star, denote
d
by
n
S
as the iso
m
orph
ism
of
,1
nn
S
.
Ke
y
w
ords
:
(,
)
nk
-star grap
h, bon
d
age n
u
m
b
e
r, combi
natori
e
s.
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
It is wi
dely
known that
bo
ndag
e n
u
mb
er i
s
one
of
the mo
st imp
o
rtant
param
eters to
measure the resili
en
ce of grap
hs
of
co
mputer n
e
two
r
k. Fo
r the p
a
rticul
ar
kno
w
n g
r
aph
s, so far
th
e
re
su
lts
o
f
th
is as
p
e
c
t
a
r
e a
few
s
u
c
h
a
s
b
o
n
d
a
g
e
nu
mb
er
o
f
d
e
Br
u
ijn a
n
d
Ka
u
t
z
d
i
g
r
aphs
[9], bonda
ge
numbe
r i
n
o
r
i
ented
gra
p
h
s
[10]. Esp
e
ci
ally, Hua
ng
a
nd Xu
[11] g
o
t a g
ood
lo
wer-
boun
d an
d a
good
upp
er-b
ound
of bo
nd
age n
u
mb
er
of
vertex-tra
n
s
itive graph
s,
but the
pre
c
i
s
e
lowe
r-boun
d
of bon
dag
e n
u
mbe
r
of
(,
)
nk
-sta
r g
r
ap
hs (ve
r
tex-tran
sitive
grap
hs)
and
,2
()
n
bS
can
not be got by their re
sult
s. Next, we se
e con
c
e
p
tion of
bonda
ge nu
mber:
Defini
tion 1.1.
Let
G
be a graph, an
d
S
be
a nonempty sub
s
et of
()
VG
, then
S
is one
dominatin
g set
of
G
if all n
o
des of
G
is eith
er i
n
S
, or
adja
c
ent to
a
nod
e of
S
. Moreov
er,
w
e
call that
S
is dominating n
u
m
ber of
G
if
S
is minimum in all dominatin
g sets of
G
, denoted
by
()
G
.
Defini
tion 1.
2.
Let
G
be a
u
ndire
ct g
r
ap
h, and
B
be a no
nempty edg
e-sub
s
et of
()
E
G
,
then minimu
m
B
is bonda
g
e
numbe
r of
G
if
()
(
)
GB
G
, denoted by
()
bG
.
In a network grap
h, pred
ece
s
sors h
a
ve sh
own tha
t
comp
uting
()
bG
are
extreme
l
y
diffic
u
lt. So
computing
()
bG
of
some
p
a
rticul
ar
kn
own
g
r
a
phs
is very va
luable.
For ex
ample, th
e
(,
)
nk
-sta
r graph
s wa
s first p
r
opo
sed in
1
995 by W.K
Chian
g
et
al [1]. Becau
s
e of go
od
topologi
cal p
r
opertie
s
of
,
nk
S
, its many p
r
op
erties
have b
een resea
r
ch
ed such a
s
di
ameter
and
con
n
e
c
tivity [1, 8], pancy
c
licity [2],
(1
)
()
s
G
and
(2
)
()
s
G
[3-7], fault hamiltonicity and fault
hamiltoni
city con
n
e
c
tivity [4, 12], indep
ende
nt
numb
e
r an
d do
min
a
ting num
be
r [13] and
so
on.
In this pap
er,
we dete
r
min
e
()
bG
of
(,
)
nk
-sta
r g
r
aph
s, so th
at can
get
()
n
S
an
d the goo
d
lowe
r-boun
d of
()
n
bS
of
n
-sta
r, denoted by
n
S
as the isomo
r
p
h
i
s
m of
,1
nn
S
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Som
e
Result
s of Bondag
e
Num
ber of
(n
,k)-Sta
r Grap
hs (Yu
n
chao
Wei)
5317
2. Preliminaries
For given
i
n
tegers
n
a
nd
k
, where
11
kn
, let
1
,
2
,
....,
n
Jn
and
let
(
Pn
,)
k
be the set
of
k
-
p
er
mu
ta
tio
n
s
on
n
J
for
11
kn
, that is
,
12
,{
.
.
.
k
Pn
k
p
p
p
:,
,
1
}
in
i
j
pJ
p
p
i
j
k
.
Defini
tion 2.1.
The
(,
)
nk
-star
grap
h, denot
ed by
,
nk
S
, is an undire
cted
grap
h with
v
e
rtex
-s
et
,
Pn
k
. The adj
acen
cy is d
e
fined
a
s
follows: a ve
rtex
12
...
...
ik
p
pp
p
is
adja
c
ent to
a vertex
(1)
12
1
1
1
ii
k
p
pp
p
p
p
, where
2
ik
( sw
ap
1
p
with
i
p
).
(2)
2
...
k
x
pp
, where
:1
ni
x
Jp
i
k
( repla
c
e
1
p
by
x
).
Figure 1 sh
o
w
s a
(4,2)-sta
r gra
ph
4,2
S
.
Figure 1. The
Structure of a (4,2)-star G
r
aph
4,2
S
The ed
ge
s of type (1) are referred to a
s
i
-edg
es (
2
ik
), an
d the edge
s
of type (2)
are
refe
rre
d to as
1-edge.
The verti
c
e
s
of type (1
) are refe
rred to
as
swap-adja
c
ent ve
rtice
s
,
and
the vertice
s
of type (2) are referre
d
to
as un
swap
-adja
c
ent verti
c
e
s
. We al
so
call
i
-edge a
s
swap-edg
e, a
nd call 1
-
ed
g
e
s a
s
un
swa
p
-ed
ge.
Clea
rly, every vertex in
,
nk
S
has
(1
)
k
swap-
adja
c
ent verti
c
e
s
and
()
nk
unswap
-
adj
acent
vertices. Usually, if
12
...
k
vp
p
p
is a vertex in
,
nk
S
, we c
a
ll that
i
p
is
the
i
-th bit for ea
ch
i
1
,
2
,
...
k
.
By Definition
2.1, we
kno
w
,1
nn
n
SS
a
nd
,1
nn
SK
wh
er
e
n
S
is
n
-sta
r g
r
a
p
h
and
n
K
is compl
e
te grap
h with
order
n
. So
,
nk
S
is
a
gene
rali
zatio
n
of
n
S
. It has been
sh
own
by
Chia
ng an
d Che
n
[1] that
n
S
is an
1
n
-regul
ar,
1
n
-conn
ecte
d vertex-tra
n
s
itive gra
p
h
with
!!
nn
k
ver
t
ic
es
.
The follo
wing
conte
n
t, we
mainly dete
r
mine the
dom
inating n
u
mb
er of
,
nk
S
for obta
i
ning
main re
sults
of Section 3. Since
,1
nn
SK
, we only consi
d
e
r
the ca
se
2
k
in the following
discu
ssi
on.
Lemma 2.2.
,
1!
!
nk
n
S
nk
for
21
kn
.
Proof.
Let
,
nk
SV
S
b
e
a minim
u
m
dominatin
g set of
,
nk
S
, then
,
()
nk
SS
by
Definition
1.1
.
By Definitio
n
2.1,
we
ha
ve kn
own tha
t
,
nk
S
is a
(1
)
n
-regul
ar gra
ph, so
each
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TELKOM
NIKA
Vol. 11, No
. 9, September 201
3: 531
6 – 5321
5318
vertex of
S
ca
n at most dominate
(1
)
n
vertices in
,
nk
SS
. If
,
()
nk
S
(1
)
!
1
()
!
n
nk
then
S
can at mo
st dominate
(1
)
!
11
()
!
n
n
nk
vertices in
,
nk
SS
. Thus, we can get
:
,
,
(1
)
!
(1
)
!
()
1
1
1
()
!
(
)
!
!!
()
()
!
(
)
!
nk
nk
nn
SV
S
S
n
nk
nk
nn
nV
S
nk
nk
It is contra
ry to the definitio
n of dominati
ng numb
e
r.
□
Theorem 2.3
.
,
(1
)
()
()
nk
n
S
nk
！
！
for
21
kn
.
Proof.
By Le
mma 2.2,
we
have sho
w
n
,
(1
)
!
()
()
!
nk
n
SS
nk
. Thu
s
, by Definition 1.1,
Theo
rem 2.3
can b
e
prove
d
if we can
co
nstru
c
t a dom
inating set
S
, s
o
that
S
(1
)
!
()
!
n
nk
.
We now split
,
()
nk
VS
into
thre
e
v
e
rtex
-s
ub
set
s
:
n(
1
,
1
)
}
,
n
VP
n
k
｛
n
V
{(
1
,
)
}
Pn
k
and
12
1
1
1
{,
2
}
.
na
a
k
i
n
Vp
p
p
n
p
p
p
J
a
It is
eas
y to verify
that
n
V
,
n
V
and
n
V
have no inte
rse
c
tion, an
d
'"
nn
,
()
nn
k
VV
V
V
S
!
()
!
n
nk
since
'
n
n-
1
!
(
1
)
,
n-
k
!
(
1
)
n
n
VV
nk
（）
！
（）
！
and
"
n
(1
)
!
(1
)
()
!
n
Vk
nk
.
Let
12
p
k
p
p
be any one vertex of
n
V
,
then all neigh
borin
g-edge
s
of
12
p
k
p
p
must
have one u
n
swap
-ed
ge co
nne
cted to
2
k
np
p
of
n
V
.
Let
12
1
1
aa
k
p
p
p
np
p
be a
n
y
one ve
rtex of
n
V
, then
all
n
e
ighb
orin
g-e
d
ges o
f
12
1
a
p
pp
n
1
ak
p
p
must have
one
swap-ed
ge conn
ecte
d
to
21
1
1
aa
k
np
p
p
p
p
of
n
V
.
Thus
,
we can let
n
VS
, and
n
(1
)
!
!
n
V
nk
（）
.
□
Corollar
y
2.4
.
In n-s
t
ar graph
n
S
,
()
(
1
)
!
n
Sn
.
Corollar
y
2.
5
If let
12
1
\}
(
)
x
kj
n
n
Vx
p
p
p
p
J
x
x
J
｛
, then each
x
V
is a
minimum do
minating set of
,
nk
S
for
1,
2
,
,
x
n
.
Lemma 2.6.
If
S
is a minimum domin
a
t
ing set of
,
nk
S
,
then any two
vertices of
S
aren
't adja
c
e
n
t in
,
nk
S
, and any two neighb
oring
-
verti
c
e
s
of
S
are
n't co
mmon.
Proof.
Let
1
v
and
2
v
be any two vertice
s
of
S
, if
1
v
and
2
v
are a
d
jacent in
,
nk
S
, then
1
v
and
2
v
can at
most domin
ate
(2
4
)
n
vertices of
,
nk
SS
sin
c
e eit
her
1
v
or
2
v
only
dominate
(2
)
n
ver
t
ic
es
of
,
nk
SS
. Thus,
we
ca
n get that
S
can
at mo
st
domin
ate
[(
2
)
(
1
)
2
4]
Sn
n
ver
t
ic
es of
,
nk
SS
, and
ca
n get
(|
|
2
SS
,,
)(
1
)
2
4
2
2
nk
nk
n
n
n
S
VS
VS
, a contra
dicti
on.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Som
e
Result
s of Bondag
e
Num
ber of
(n
,k)-Sta
r Grap
hs (Yu
n
chao
Wei)
5319
If
there exist
two neig
hbo
ring
-vertices of
S
who are comm
on,
the
n
S
can
at
m
o
st
dominate
11
Sn
ver
t
ic
es
of
,
nk
SS
Therefore, we
have
(1
)
1
|
(
SS
n
V
,,
)|
1
nk
nk
SV
S
, a contra
dicti
on.
□
3. The Importan
t
Res
u
lts
of Bon
d
age
Number o
f
,
nk
S
In this se
ction
,
we mainly consi
der the b
onda
ge num
b
e
r of
,
nk
S
..
Lemma 3.1.
If let
12
1
{\
}
x
kj
n
n
Vx
p
p
p
p
J
x
x
J
, then:
(a) Any
two v
e
rtice
s
of
x
n
Vx
J
aren't adjacent.
(b) Any
two
set
x
V
and
,,
yn
Vx
y
J
x
y
have no intersection, an
d
,
nk
VS
12
n
VV
V
.
(c) Any one vertex of
x
n
Vx
J
has
exactly a neig
hbor
re
spe
c
ti
vely in
(
yn
Vy
J
\)
x
.
Proof.
In fact
, the con
c
lu
si
on (a
) is the
same
as
L
e
m
m
a 2.6. By Definition 2.1, it is easy
to verify that
(b) is corre
c
t. Next, we p
r
ove con
c
lu
si
on
(c). Let
23
1
x
k
vx
p
p
p
b
e
any
one
vertex of
x
V
. If element
y
isn't
in
x
v
, then
x
v
is
only adja
c
e
n
t to
12
yk
vy
p
p
p
of
y
V
by
a
swap-edg
e
x
y
vv
. If element
y
is in
x
v
, i.e the
t
-th bit
t
p
y
for each
\{
1
}
k
tJ
, then
x
v
is
only adjacent
to
12
1
yk
vy
p
p
p
of
y
V
by an
swap-edg
e
x
y
vv
, and
the
t
-th bit
t
p
x
of
y
v
.
□
Corollar
y
3.2.
The in
du
ced subg
rap
h
,
[]
nk
x
y
SV
V
of any two
set
x
V
and
(,
y
Vx
y
,)
n
Jx
y
is a
bip
a
rtite gra
ph, d
enoted
by
[,
,
]
x
xy
y
VE
V
. Moreover,
x
y
E
is a uniqu
e
compl
e
te mat
c
hing of
,
[]
nk
x
y
SV
V
.
Lemma 3.3.
,2
n
S
has exa
c
tly
n
minimum do
minating sets, which a
r
e
{
x
Vx
p
p
\}
(
)
nn
Jx
x
J
.
Proof.
By
Co
rolla
ry 2.5,
we only
prov
e
no oth
e
r dom
inating
set
s
e
x
cept
12
,
,
...,
n
VV
V
.
Let X be a minimum d
o
minating
se
t and differe
nt from
Xn
Vx
J
, and let nonem
pty
,
(,
2
,
)
mm
ii
b
m
n
XX
V
m
J
b
n
i
J
and
mt
ii
XX
for
mt
by th
e con
c
lu
sion
(b)
of Lemma 3.1
.
By the con
c
l
u
sio
n
(a
) of
Lemma
3.1
and L
e
mma
2.6, each ve
rtex of
11
ii
VX
ha
s
exactly on
e
n
e
ighb
or in
1
i
XX
si
nce
1
i
VX
must
be
dominate
d
by
1
i
XX
. By
Lemma
2.6,
we kno
w
that all neighbo
ring
-vert
i
ce
s of
11
ii
VX
aren't com
m
o
n
, so we
have
11
1
,2
1!
2!
ii
i
n
n
XX
V
X
S
n
. Now, let
1
i
U
be a sub
s
et of
,2
(
n
S
N
11
)
ii
VX
, and
denote
that neigh
bors of
each
ve
rtex of
11
ii
VX
only h
a
ve
one
in
1
i
U
, then
11
ii
XX
U
,
clea
rly
,
11
1
ii
i
UV
X
.
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Vol. 11, No
. 9, September 201
3: 531
6 – 5321
5320
Next, by the proof
of Th
eore
m
2.3,
we let
1
1
{,
2
,
}
i
VP
n
i
,
1
1
{
i
Vp
i
1
\}
n
p
Ji
and
11
1
1
ii
i
i
VX
X
X
, where
1
11
ii
iX
X
UU
U
, and let
1
i
X
U
be in
1
i
V
,
1
i
X
U
be in
1
i
V
, c
l
early
,
1
1
i
Xi
UX
and
1
1
i
Xi
UX
.
If
1
i
X
U
, then nei
g
hbori
n
g
-
vertices of
1
i
X
in
1
i
V
can
'
t
be domin
ate
d
sin
c
e it i
s
e
a
sy
to verify
11
,2
,2
ni
n
i
SV
SV
, a contradi
ction.
If
1
i
X
U
, then neighb
orin
g-ve
rtice
s
of
1
i
X
in
1
i
V
can't be dominate
d
,
a
contradi
ction.
If
1
i
X
U
and
1
i
X
U
, then
1
i
X
U
can
exa
c
tly d
o
minate
neig
hbori
n
g
-
vertices
of
1
i
X
and
1
i
X
in
1
i
V
exce
pt
1
i
X
. Therefore
,
we have:
11
1
1
1
1
22
3
2
2
i
X
ii
i
i
i
U
n
Xn
Xn
Xn
Xn
X
(1)
In addition, we have kn
own:
11
1
1
1!
1
2!
ii
i
i
n
XX
X
V
n
n
(2)
By (
3
.1) (3.2), w
e
can
get
11
11
ii
Xn
X
n
, a
contradi
ction fo
r
1
0
i
X
an
d
1
0
i
X
. Thus
,
X
doe
s not exist.
□
Theorem 3.4
.
,
()
2
nk
n
bS
, and
,2
()
2
n
n
bS
for
3
n
.
Proof.
Let
B
be a minimu
m
bonda
ge
set
of
,
nk
S
. If
,
()
2
nk
n
bS
, then there at le
ast
exists a
x
V
, all neigh
bo
ring
-edge
s
of wh
ich
are
n't in
B
such that
x
V
is
still a
mi
nimum
dominatin
g set
of
,
nk
S
by Corolla
ry 3.2
sin
c
e e
a
ch
edge
of
B
ca
n exactly
co
nne
ct two
element
s of
{:
}
x
n
Vx
J
, that is
,
2
Bn
, a contradi
ction. So we have
,
()
2
nk
n
bS
.
Next, we
c
o
ns
truc
t a set
B
su
ch t
h
at
,2
()
2
n
n
Bb
S
. Let
x
y
e
b
e
a
n
y o
ne e
dge
of
,2
[]
nx
y
SV
V
, then
x
yx
y
eE
by
Corolla
ry
3.2. Now, we let
12
3
4
1
,,
,
}
nn
Be
e
e
for even
n
, or
12
34
21
,
1
,,
,
,
}
nn
n
n
Be
e
e
e
for odd
n
. It
is
eas
y
to verify that
,2
,2
()
(
)
1
nn
SB
S
by Corolla
ry 3.2 and Lem
ma 3.3.
□
4. Conclusio
n
In fac
t, we
conjec
ture
,
()
2
nk
n
bS
, b
u
t need to fin
d
a suitable
method fo
r p
r
oving the
conj
ectu
re.
In any ca
se,
in Graph T
h
e
o
ry, it is rath
er
difficult to
comp
ute bo
n
dage
numb
e
r of the
grap
hs.
Up t
o
no
w, the concl
u
si
on
s in
this re
sp
ect
are
confin
ed
only to a few spe
c
ific
gra
p
h
s
su
ch a
s
cub
e
,
de Bruijn an
d Kautz digra
phs an
d
so o
n
. Thus, the pape
r is very valuable si
nce it
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TELKOM
NIKA
ISSN:
2302-4
046
Som
e
Result
s of Bondag
e
Num
ber of
(n
,k)-Sta
r Grap
hs (Yu
n
chao
Wei)
5321
solve
s
bo
nda
ge num
be
r of
(,
2
)
n
-sta
r graph
s
and the p
r
e
c
i
s
e lo
we
r- b
o
u
nd of bo
ndag
e numb
e
r
of
(,
)
nk
-star g
r
a
p
h
s
and
n
-sta
r graph
s.
Ackn
o
w
l
e
dg
ement
The wo
rk wa
s su
ppo
rted b
y
"973" Program of China
(No. 201
2CB3
1620
0).
R
e
fe
re
nc
es
[1]
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K
Chiang, R
J
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he
(,
)
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-star gragh: A gen
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Internati
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a
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S
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N
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i
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Broadc
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-Star Graph
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Procee
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n
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n
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