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AdvancesinAppliedMathematics
A^
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Ð
,2022,11(1),278-287
PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.111035
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ResearchontheDoubleRoman
DominationNumberofSome
SpecialGraphs
ShashaLiu
1
∗
,HongBian
1
†
,HaizhengYu
2
,LinaWei
1
1
SchoolofMathematicalSciences,XinjiangNormalUniversity,UrumqiXinjiang
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[J].
A^
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Ð
,2022,11(1):278-287.
DOI:10.12677/aam.2022.111035
4
ââ
2
CollegeofMathematicsandSystemSciences,XinjiangUniversity,UrumqiXinjiang
Received:Dec.24
th
,2021;accepted:Jan.14
th
,2022;published:Jan.26
th
,2022
Abstract
Let
G
=(
V
(
G
)
,E
(
G
))
beasimpleconnectedgraph,afunction
f
:
V
(
G
)
−→{
0
,
1
,
2
,
3
}
satisfieswiththepropertythat1)if
f
(
v
)=0
,thenvertex
v
mustexistatleasttwo
neighbors
v
1
,v
2
suchthat
f
(
v
1
)=
f
(
v
2
)=2
oroneneighbor
u
suchthat
f
(
u
)=3
;2)
if
f
(
v
)=1
,thentheremustexistatleastoneneighbor
u
of
v
suchthat
f
(
u
)=2
or
3,and
f
iscalledadoubleRomandominationfunction(DRDF).Theweightofa
DRDFis
f
(
V
(
G
)) =
P
u
∈
V
(
G
)
f
(
u
)
.TheminimumweightofaDRDFon
G
isthedouble
Roman dominationnumber, denoted by
γ
dR
(
G
)
.A double Roman dominationfunction
withtheweightof
γ
dR
(
G
)
iscalleda
γ
dR
-functionofG.Inthispaper,wepresentthe
exactvaluesofthedoubleRomandominationnumbersofsomespecialgraphs,such
as
P
m
P
n
(
m
= 2
,
3)
,
P
n,t
,
K
∗
n
,
M
(
C
n
)
,
M
(
P
n
)
.
Keywords
DoubleRomanDominationFunction,DoubleRomanDominationNumber,
StrongProduct,ThornGraph,MiddleGraph
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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V
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›
›
¼
ê
-
•
f
(
V
(
G
))=
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u
∈
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(
G
)
f
(
u
).
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G
¤
k
V
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DOI:10.12677/aam.2022.111035280
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3.1
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P
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P
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2
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n
r
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γ
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y
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P
2
P
n
1
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1
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2
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n
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1
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n
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ã
1
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e
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5
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f
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i
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≤
f
(
v
i
).
Figure1.
P
2
P
n
ã
1.
P
2
P
n
•
Ä
¼
ê
f
:
V
(
P
2
P
n
)
→{
0
,
2
,
3
}
,
n
≡
0(
mod
3)
ž
,
f
(
v
3
i
−
1
) = 3,1
≤
i
≤
n
3
,
Ù
{
º:
D
Š
•
0;
n
≡
1(
mod
3)
ž
,
f
(
v
3
i
−
1
) = 3,1
≤
i
≤
n
−
1
3
,
f
(
v
n
) = 3,
Ù
{
º:
D
Š
•
0;
n
≡
2(
mod
3)
ž
,
f
(
v
3
i
−
1
) = 3,1
≤
i
≤
n
−
2
3
,
f
(
v
n
) = 3,
Ù
{
º:
D
Š
•
0.
w
,
f
•
P
2
P
n
˜
‡
V
Ûê
›
›
¼
ê
,
¦
γ
dR
(
P
2
P
n
)
≤
3
d
n
3
e
.
‡
L
5
,
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8
?
1
8
B
,
w
,
é
u
n
≤
5
¤
á
,
b
é
u
n
≥
6
¿
…
é
u
?
¿
u
n
(
Ø
Ñ
¤
á
.
-
f
= (
V
0
,V
2
,V
3
)
•
P
2
P
n
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‡
γ
dR
-
¼
ê
.
©
±
e
3
«
œ
¹
?
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:
1)
f
(
v
n
)+
f
(
u
n
) = 3
ž
.
d
ž
f
(
v
n
)=3,
f
(
u
n
)=0.
d
V
Ûê
›
›
¼
ê
½
Â
Œ
•
:
f
(
v
n
−
1
)=
f
(
u
n
−
1
)=0,
K
k
f
(
v
n
−
2
) +
f
(
v
n
−
3
)
≤
3.
Ä
K
e
f
(
v
n
−
2
) +
f
(
v
n
−
3
)
>
3.
½
Â
¢
Š
¼
ê
g
:
V
(
P
2
P
n
)
−→
{
0
,
2
,
3
}
,
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g
(
v
n
)=
g
(
v
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−
3
)=3,
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f
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P
2
P
n
˜
‡
V
Ûê
›
›
¼
ê
,
¦
ω
(
g
)
<ω
(
f
),
g
ñ
.
½
Â
¢
Š
¼
ê
h
:
V
(
P
2
P
n
−
3
)
−→{
0
,
2
,
3
}
,
¦
h
(
v
n
−
3
)=
f
(
v
n
−
2
)+
f
(
v
n
−
3
),
h
(
u
n
−
3
)=
f
(
u
n
−
2
)+
f
(
u
n
−
3
),
Ù
{
:
D
Š
†
f
ƒ
Ó
,
K
k
γ
dR
(
P
2
P
n
)=
ω
(
f
) =
ω
(
h
)+3
≥
γ
dR
(
P
2
P
n
−
3
)+3 = 3
d
n
3
e
.
2)
f
(
v
n
)+
f
(
u
n
)
<
3
ž
.
Ø
”
f
(
v
n
)=2,
f
(
u
n
)=0,
K
k
f
(
u
n
−
1
)
≥
2
½
f
(
v
n
−
1
)
≥
2.
Ø
”
f
(
u
n
−
1
)
≥
2.
½
Â
¢
Š
¼
ê
g
:
V
(
P
2
P
n
)
−→{
0
,
2
,
3
}
,
¦
g
(
v
n
−
1
)=3,
g
(
v
n
)=0,
Ù
{
:
D
Š
†
f
ƒ
Ó
,
w
,
g
•
P
2
P
n
˜
‡
V
Ûê
›
›
¼
ê
,
¦
ω
(
g
)
<ω
(
f
),
g
ñ
.
3)
f
(
v
n
)+
f
(
u
n
)
>
3
ž
.
DOI:10.12677/aam.2022.111035281
A^
ê
Æ
?
Ð
4
ââ
œ
/
1:
f
(
v
n
)=3,
f
(
u
n
)=2.
½
Â
¢
Š
¼
ê
g
:
V
(
P
2
P
n
)
−→{
0
,
2
,
3
}
,
¦
g
(
v
n
)=3,
g
(
u
n
) = 0,
Ù
{
:
D
Š
†
f
ƒ
Ó
,
w
,
g
•
P
2
P
n
˜
‡
V
Ûê
›
›
¼
ê
,
¦
ω
(
g
)
<ω
(
f
),
g
ñ
.
œ
/
2:
f
(
v
n
)=2,
f
(
u
n
)=2.
½
Â
¢
Š
¼
ê
g
:
V
(
P
2
P
n
)
−→{
0
,
2
,
3
}
,
¦
g
(
v
n
)=3,
g
(
u
n
) = 0,
Ù
{
:
D
Š
†
f
ƒ
Ó
,
w
,
g
•
P
2
P
n
˜
‡
V
Ûê
›
›
¼
ê
,
¦
ω
(
g
)
<ω
(
f
),
g
ñ
.
½
n
3.2
-
P
3
P
n
•
P
3
†
P
n
r
†
È
,
Ù
¥
ê
n
≥
2
.
K
γ
dR
(
P
3
P
n
) = 3
d
n
3
e
.
y
²
•
•
B
?
Ø
,
P
P
3
P
n
1
˜
º:
•
u
1
,u
2
,
···
,u
n
,
1
º:
P
•
v
1
,v
2
,
···
,v
n
,
1
n
º:
•
w
1
,w
2
,
···
,w
n
(
g
e
þ
).
•
Ä
¼
ê
f
:
V
(
P
3
P
n
)
−→{
0
,
2
,
3
}
,
n
≡
0
,
2(
mod
3)
ž
,
f
(
v
i
) = 3,
i
≡
2(
mod
3),
Ù
{
º:
D
Š
•
0;
n
≡
1(
mod
3)
ž
,
f
(
v
i
) = 3,
i
≡
2(
mod
3),
f
(
v
n
) = 3,
Ù
{
º:
D
Š
•
0;
w
,
f
•
P
3
P
n
˜
‡
V
Ûê
›
›
¼
ê
,
¦
γ
dR
(
P
3
P
n
)
≤
3
d
n
3
e
.
‡
L
5
,
é:
8
?
1
8
B
,
w
,
é
u
n
≤
4
¤
á
,
b
é
u
n
≥
5
¿
…
é
u
?
¿
u
n
(
Ø
Ñ
¤
á
.
b
f
= (
V
0
,V
2
,V
3
)
•
P
3
P
n
˜
‡
γ
dR
-
¼
ê
.
d
P
3
P
n
(
A
:
9
V
Ûê
›
›
ê
½
Â
,
Œ
±
P
3
P
n
•
k
D
Š
•
3
±
9
D
Š
•
0
.
Ä
K
b
•
3
D
Š
•
2
.
œ
/
1:
f
(
w
n
)=2,
f
(
v
n
)=
f
(
u
n
)=0
ž
.
d
V
Ûê
›
›
ê
½
Â
•
f
(
v
n
−
1
)=3,
K
k
f
(
u
n
−
1
)=0,
f
(
w
n
−
1
)=0.
½
Â
¢
Š
¼
ê
g
:
V
(
P
3
P
n
)
−→{
0
,
2
,
3
}
,
¦
g
(
w
n
)=0,
Ù
{
:
D
Š
†
f
ƒ
Ó
,
w
,
g
•
P
3
P
n
˜
‡
V
Ûê
›
›
¼
ê
,
¦
ω
(
g
)
<ω
(
f
),
g
ñ
.
œ
/
2:
f
(
v
n
)=2,
f
(
w
n
)=
f
(
u
n
)=0
ž
.
d
V
Ûê
›
›
ê
½
Â
•
f
(
v
n
−
1
)
≥
2,
K
k
f
(
w
n
−
1
)=0,
f
(
u
n
−
1
)=0.
½
Â
¢
Š
¼
ê
g
:
V
(
P
3
P
n
)
−→{
0
,
2
,
3
}
,
¦
g
(
v
n
−
1
)=3,
g
(
v
n
) = 0,
Ù
{
:
D
Š
†
f
ƒ
Ó
,
w
,
g
•
P
3
P
n
˜
‡
V
Ûê
›
›
¼
ê
,
¦
ω
(
g
)
<ω
(
f
),
g
ñ
.
œ
/
3:
f
(
u
n
) = 2,
f
(
w
n
) =
f
(
v
n
) = 0
ž
.
du
P
3
P
n
é
¡
5
Œ
•
Ú
œ
/
1
a
q
.
e
5
?
Ø
f
(
w
n
)+
f
(
v
n
)+
f
(
u
n
) = 3
†
f
(
w
n
)+
f
(
v
n
)+
f
(
u
n
) = 0
ž
œ
/
.
(i)
f
(
w
n
)+
f
(
v
n
)+
f
(
u
n
) = 3
œ
/
1:
f
(
w
n
) = 3,
f
(
v
n
) =
f
(
u
n
) = 0
ž
.
d
V
Ûê
›
›
ê
½
Â
•
f
(
w
n
−
1
) =
f
(
v
n
−
1
) = 0,
K
k
f
(
u
n
−
1
) = 3.
½
Â
¢
Š
¼
ê
g
:
V
(
P
3
P
n
)
−→{
0
,
2
,
3
}
,
¦
g
(
v
n
−
1
) = 3,
g
(
w
n
) = 0,
Ù
{
:
D
Š
†
f
ƒ
Ó
,
w
,
g
•
P
3
P
n
˜
‡
V
Ûê
›
›
¼
ê
,
¦
ω
(
g
)
<ω
(
f
),
g
ñ
.
œ
/
2:
f
(
v
n
)=3,
f
(
w
n
)=
f
(
u
n
)=0
ž
.
d
V
Ûê
›
›
ê
½
Â
•
f
(
w
n
−
1
)=
f
(
v
n
−
1
)=
f
(
u
n
−
1
)=0.
½
Â
¢
Š
¼
ê
g
:
V
(
P
3
P
n
−
2
)
−→{
0
,
2
,
3
}
,
¦
g
(
w
n
−
2
)=
f
(
w
n
−
2
) +
f
(
w
n
−
1
),
g
(
v
n
−
2
) =
f
(
v
n
−
2
)+
f
(
v
n
−
1
),
g
(
u
n
−
2
) =
f
(
u
n
−
2
)+
f
(
u
n
−
1
),
Ù
{
:
D
Š
†
f
ƒ
Ó
,
K
k
γ
dR
(
P
3
P
n
) =
ω
(
f
) =
ω
(
g
)+3
≥
γ
dR
(
P
3
P
n
−
2
)+3 = 3
d
n
3
e
.
DOI:10.12677/aam.2022.111035282
A^
ê
Æ
?
Ð
4
ââ
œ
/
3:
f
(
u
n
) = 3,
f
(
w
n
) =
f
(
v
n
) = 0
ž
.
du
P
3
P
n
é
¡
5
Œ
•
Ú
œ
/
1
a
q
.
(ii)
f
(
w
n
)+
f
(
v
n
)+
f
(
u
n
) = 0
d
ž
=
f
(
w
n
)=0,
f
(
v
n
)=0,
f
(
u
n
)=0.
d
V
Ûê
›
›
ê
½
Â
•
f
(
v
n
−
1
)=3,
K
k
f
(
w
n
−
2
)=
f
(
w
n
−
1
)=
f
(
v
n
−
2
)=
f
(
u
n
−
2
)=
f
(
u
n
−
1
)=0.
½
Â
¢
Š
¼
ê
g
:
V
(
P
3
P
n
−
3
)
−→
{
0
,
2
,
3
}
,
¦
g
(
w
n
−
3
)=
f
(
w
n
−
3
)+
f
(
w
n
−
2
),
g
(
v
n
−
3
)=
f
(
v
n
−
3
)+
f
(
v
n
−
2
),
g
(
u
n
−
3
)=
f
(
u
n
−
3
)+
f
(
u
n
−
2
),
Ù
{
:
D
Š
†
f
ƒ
Ó
,
K
k
γ
dR
(
P
3
P
n
) =
ω
(
f
) =
ω
(
g
)+3
≥
γ
dR
(
P
3
P
n
−
3
)+3 = 3
d
n
3
e
.
½
n
3.3
-
P
n,t
´
˜
‡
e
ã
,
Ù
¥
n
≥
3
,
t
≥
3
.
K
γ
dR
(
P
n,t
) =
(
n
+3
,n
≡
0
,
2(
mod
3);
n
+2
,n
≡
1(
mod
3)
.
y
²
•
•
B
?
Ø
,
é
P
n,t
?
1
·
I
Ò
,
X
ã
2
¤
«
.
Figure2.
P
n,t
ã
2.
P
n,t
•
Ä
¼
ê
f
:
V
(
P
n,t
)
−→{
0
,
2
,
3
}
,
n
≡
0(
mod
3)
ž
,
f
(
u
i
) = 3,
i
≡
2(
mod
3),
f
(
u
n
) = 3,
Ù
{
º:
D
Š
•
0;
n
≡
1(
mod
3)
ž
,
f
(
u
i
) = 3,
i
≡
2(
mod
3),
Ù
{
º:
D
Š
•
0;
n
≡
2(
mod
3)
ž
,
f
(
u
i
)=3,
i
≡
2(
mod
3)
…
i
6
=
n
−
1,
f
(
u
n
−
2
)=2,
f
(
u
n
)=3,
Ù
{
º:
D
Š
•
0.
w
,
f
•
P
n,t
˜
‡
V
Ûê
›
›
¼
ê
,
¦
γ
dR
(
P
n,t
)
≤
(
n
+3
,n
≡
0
,
2(
mod
3);
n
+2
,n
≡
1(
mod
3)
.
‡
L
5
é
n
?
1
8
B
,
w
,
é
u
n
≤
6
Ñ
¤
á
,
y
b
é
u
n
≥
7
±
9
¤
k
:
ê
u
n
(
Ø
Ñ
¤
á
.
b
g
= (
V
0
,V
2
,V
3
)
•
P
n,t
˜
‡
γ
dR
-
¼
ê
.
du
P
n,t
(
A
:
,
5
½
g
(
u
1
) =
g
(
u
n
) = 3,
g
(
l
1
) =
g
(
l
2
) =
...
=
g
(
l
t
−
1
) = 0,
g
(
k
1
) =
g
(
k
2
) =
...
=
g
(
k
t
−
1
) = 0.
?
Ø
g
(
u
n
−
2
)
ù
‡
:
:
1)
g
(
u
n
−
2
)=0
ž
,
Œ
g
(
u
n
−
3
)=3.
½
Â
¢
Š
¼
ê
h
:
V
(
P
n
−
3
,t
)
−→{
0
,
2
,
3
}
,
Ù
¥
P
n
−
3
,t
DOI:10.12677/aam.2022.111035283
A^
ê
Æ
?
Ð
4
ââ
´
í
u
n
−
1
,
u
n
−
2
,
u
n
−
3
2
ë
u
n
−
4
†
u
n
.
w
,
g
•
P
n
−
3
,t
˜
‡
V
Ûê
›
›
¼
ê
.
γ
dR
(
P
n,t
) =
ω
(
g
) =
ω
(
h
)+3
≥
γ
dR
(
P
n
−
3
,t
)+3
≥
(
n
+3
,n
≡
0
,
2(
mod
3);
n
+2
,n
≡
1(
mod
3)
.
2)
g
(
u
n
−
2
)=2
ž
,
½
Â
¢
Š
¼
ê
h
:
V
(
P
n
−
2
,t
)
−→{
0
,
2
,
3
}
,
Ù
¥
P
n
−
2
,t
´
í
u
n
−
1
,
u
n
−
2
,
2
ë
u
n
−
3
†
u
n
.
w
,
g
•
P
n
−
2
,t
˜
‡
V
Ûê
›
›
¼
ê
.
γ
dR
(
P
n,t
) =
ω
(
g
) =
ω
(
h
)+3
≥
γ
dR
(
P
n
−
2
,t
)+2
≥
(
n
+3
,n
≡
0
,
2(
mod
3);
n
+2
,n
≡
1(
mod
3)
.
3)
g
(
u
n
−
2
)=3
ž
,
g
(
u
n
−
3
)=0.
½
Â
¢
Š
¼
ê
h
:
V
(
P
n
−
3
,t
)
−→{
0
,
2
,
3
}
,
Ù
¥
P
n
−
3
,t
´
í
u
n
−
1
,
u
n
−
2
,
u
n
−
3
2
ë
u
n
−
4
†
u
n
.
w
,
g
•
P
n
−
3
,t
˜
‡
V
Ûê
›
›
¼
ê
.
γ
dR
(
P
n,t
) =
ω
(
g
) =
ω
(
h
)+3
≥
γ
dR
(
P
n
−
3
,t
)+3
≥
(
n
+3
,n
≡
0
,
2(
mod
3);
n
+2
,n
≡
1(
mod
3)
.
•
•
B
?
Ø
,
‰
Ñ
ã
K
4
e
ã
º:
I
P
•
ª
,
X
ã
3
¤
«
.
Figure3.
K
∗
4
ã
3.
K
∗
4
½
n
3.4
e
K
∗
n
´
˜
‡
ã
K
n
e
ã
,
Ù
¥
n
≥
3
,
l
i
≥
1
,i
= 1
,
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[1]Berge,C.(1962)TheoryofGraphsandItsApplications.Methuen,London.
[2]Ore,O.(1962)TheoryofGraphs.AmericanMathematicalSociety,Providence.
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[3]Stewart,I.(1999)DefendtheRomanEmpire.
ScientificAmerican
,
281
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https://doi.org/10.1038/scientificamerican1299-136
[4]Beeler,R.A.,Haynes, T.W. and Hedetniemi, S.T.(2016) Double Roman Domination.
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[5]Anu,V.andLakshmanan,S.A.(2018)DoubleRomanDominationNumber.
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[7]
•
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ã
V
Ûê
›
›
[D]:[
a
¬
Æ
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©
].
x²
:
x²
Œ
Æ
,2018.
[8]
Ú
ûw
.
‚
f
ã
V
Ûê
›
›
8
[J].
×
²
Æ
Æ
,2021,23(2):54-57.
[9]Nazari-Moghaddam,S.andVolkmann,L.(2020)CriticalConceptforDoubleRomanDomi-
nationinGraphs.
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https://doi.org/10.1142/S1793830920500202
[10]Bonchev,D.I.andKlein,D.J.(2002)OntheWienerNumberofThornTrees,Stars,Rings,
andRods.
CroaticaChemicaActa
,
75
,613-620.
[11]
4
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pebbling
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[J].
Œ
ë
°
¯
Œ
ÆÆ
,2006,32(4):
125-128.
DOI:10.12677/aam.2022.111035287
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