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AdvancesinAppliedMathematics
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,2022,11(1),288-301
PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.111036
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The Firefighter Problem onCayleyGraph
JinjunHan
1
∗
,HongBian
1
†
,HaizhengYu
2
,LinaWei
1
1
SchoolofMathematicalSciences,XinjiangNormalUniversity,UrumqiXinjiang
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[J].
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,2022,11(1):288-301.
DOI:10.12677/aam.2022.111036
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2
CollegeofMathematicsandSystemSciences,XinjiangUniversity,UrumqiXinjiang
Received:Dec.24
th
,2021;accepted:Jan.14
th
,2022;published:Jan.26
th
,2022
Abstract
Let
G
be aconnectedgraphwith
n
vertices.Assume thata firebreaks outat avertex
v
of
G
.Afirefighterchoosesasetof
k
verticesnotyetonfiretoprotect(oncea
vertexhasbeenchosenbythefirefighter,itisconsideredprotectedorsafefromany
furthermovesofthefire).Thefirefighterandthefirealternatelymoveonthegraph.
Theprocessendswhenthefirecannolongerspread.Afterthefirefighter’smove,
thefiremakesitsmovebyspreadingtoallverticeswhichareadjacenttothevertices
onfire,exceptforthosethatareprotected.Inthispaper,wediscussthefirefighter
problemsintheCayleygraphsofadditivegroupofintegersmodulo
n
withthe
k
-
element
(
k
≥
2)
inverseclosedsubset.Firstlyweconsiderthefirefighterproblemson
the
2
-elementinverseclosedsubsetofCayleygraph,whichdeterminesthestructure
ofCayleygraphandputsforwardthestructurealgorithmandthe
matlab
languageof
thealgorithm.Secondly,westudythefirefighterproblemsonthe
3
-elementinverse
closedsubsetofCayleygraph, whichalsodeterminesthestructureofCayleygraphand
considerssurvivingrate,edgesurvivingrateand
MVS
problem.Finally,wediscuss
the firefighter problemsonthe
k
-elements
(
k
≥
4)
inverse closedsubsetofCayley graph,
inwhichwedrawtheCayleygraphinorderofpoints,anddividetheCayleygraph
intotwocategoriesaccordingtotheorderoftheinverseclosedsubset.Moreover,
whileanyvertexofCayleygraphisonfire,weconsiderthesufficientandnecessary
conditionsthatafirefightercancontrolthefire.
Keywords
FirefighterProblem,CayleyGraph,InverseClosedSubset,SurvivingRate,
EdgeSurvivingRate
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
DOI:10.12677/aam.2022.111036289
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DOI:10.12677/aam.2022.111036291
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Z
n
,S
) =
{
0
,ia,b
}
;
C
CG
(
Z
n
,S
)
=
Maxi
+1;
C
(
CG
(
Z
n
,S
)) = 1
Ä
K
ˆ
£
(1),
†
•
3
m
(
m
∈
Z
+
),
¦
m
∗
(
Maxi
+1) =
n
Ñ
Ñ
CG
(
Z
n
,S
) =
{
0
,ia,b
}
,
{
1
,ia
+1
,b
+1
}···{
m
−
1
,ia
+
m
−
1
,b
+
m
−
1
}
;
C
CG
(
Z
n
,S
)
=
Maxi
+1;
C
(
CG
(
Z
n
,S
)) =
m
Ä
K
d
Maxia
−
b
+
j
1
a<n
,
(
½
Maxj
1
(
j
1
∈
N,j
1
<Maxj
1
,i
≤
Maxi
)
Ú
S
0
,ia,Maxia
−
b
+
j
1
a
(2)
X
J
Maxia
−
b
+
Maxj
1
a
=
b
,
Ñ
Ñ
S
0
,ia,Maxia
−
b
+
j
1
a,b
P
0 = 1(3)
ia
=
i
+1
Maxia
−
b
+
j
1
a
=
Maxi
+
j
1
+1
b
=
Maxi
+
Maxj
1
+1+1
X
J
1
∗
(
Maxi
+
Maxj
1
+1+1) =
n
Ñ
Ñ
C
CG
(
Z
n
,S
)
=
{
0
,ia,Maxia
−
b
+
j
1
a,b
}
;
C
CG
(
Z
n
,S
)
=
Maxi
+
Maxj
1
+1+1;
C
(
CG
(
Z
n
,S
)) = 1
Ä
K
ˆ
£
(3),
†
•
3
m
(
m
∈
Z
+
),
¦
m
∗
(
Maxi
+
Maxj
1
+1+1) =
n
Ñ
Ñ
CG
(
Z
n
,S
) =
{
0
,ia,Maxia
−
b
+
j
1
a,b
}
,
{
1
,ia
+1
,Maxia
−
b
+
j
1
a
+1
,b
+1
}···{
m
−
1
,ia
+
m
−
1
,Maxia
−
b
+
j
1
a
+
m
−
1
,b
+
m
−
1
}
;
C
CG
(
Z
n
,S
)
=
Maxi
+
Maxj
1
+1+1;
C
(
CG
(
Z
n
,S
)) =
m
Ä
K
ˆ
£
(2),
†
•
3
Maxj
p
¦
Maxia
−
b
+
Maxj
1
a
−
b
+
Maxj
2
a
−
b
+
···
+
Maxj
p
a
=
b
(
j
p
∈
N,j
p
<
Maxj
p
)
Ñ
Ñ
S
0
,ia,Maxia
−
b
+
j
1
a,
···
,Maxia
−
b
+
Maxj
1
a
−
b
+
···
+
j
p
a,b
P
0 = 1(4)
ia
=
i
+1
Maxia
−
b
+
j
1
a
=
Maxi
+
j
1
+1
Maxia
−
b
+
Maxj
1
a
−
b
+
···
+
j
p
a
=
Maxi
+
Maxj
1
+
···
+
j
p
b
=
Maxi
+
Maxj
1
+
···
+
Maxj
p
+
p
+1
X
J
1
∗
(
Maxi
+
Maxj
1
+
···
+
Maxj
p
+
p
+1) =
n
Ñ
Ñ
C
CG
(
Z
n
,S
)
=
{
0
,ia,Maxia
−
b
+
j
1
a,
···
,Maxia
−
b
+
Maxj
1
a
−
b
+
···
+
j
p
a,b
}
;
C
CG
(
Z
n
,S
)
=
Maxi
+
Maxj
1
+
···
+
Maxj
p
+
p
+1;
DOI:10.12677/aam.2022.111036292
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C
(
CG
(
Z
n
,S
)) = 1
Ä
K
ˆ
£
(4),
†
•
3
m
(
m
∈
Z
+
),
¦
m
∗
(
Maxi
+
Maxj
1
+
···
+
Maxj
p
+
p
+1) =
n
Ñ
Ñ
CG
(
Z
n
,S
) =
{
0
,ia,Maxia
−
b
+
j
1
a,
···
,Maxia
−
b
+
Maxj
1
a
−
b
+
···
+
j
p
a,b
}
,
{
1
,ia
+1
,Maxia
−
b
+
j
1
a
+1
,
···
,Maxia
−
b
+
Maxj
1
a
−
b
+
···
+
j
p
a
+1
,b
+1
}···{
m
−
1
,ia
+
m
−
1
,Maxia
−
b
+
j
1
a
+
m
−
1
,
···
,
Maxia
−
b
+
Maxj
1
a
−
b
+
···
+
j
p
a
+
m
−
1
,b
+
m
−
1
}
;
C
CG
(
Z
n
,S
)
=
Maxi
+
Maxj
1
+
Maxj
2
+
···
+
Maxj
p
+
p
+1;
C
(
CG
(
Z
n
,S
)) =
m.
3ù
‡
Ž
{
Ä
:
þ
,
·
‚
¢
y
T
Ž
{
matlab
Š
ó
.
CZnXS
´
Z
n
3
_
4
f
8
S
=
{
¯
a,
¯
b
}
(
a<b
…
¯
a
−
1
=
¯
b
)
þ
p
4
ã
,CcznXS
´
p
4
ã
ë
Ï
©
|
:
ê
,CCZnXS
´
p
4
ã
ë
Ï
©
|
ê
,
2
L
«
I
Ñ
\
n
Ú
a
.
Ñ
\
n,a,b
(
e
n
•
Û
ê
K
a
≤b
n
2
c
,
e
n
•
ó
ê
K
a
≤
n
2
−
1)
Ñ
Ñ
CZnXS,CcznXS,CCZnXS
CZnXS=0;CcznXS=0;CCZnXS=0;
n=
2
;a=
2
;b=n-a;c=mod(b,a);
ifc==0;
Max(1)=fix((n-0.001)/a)
end
i=0;
while(c=0)
i=i+1;
Max(i+1)=fix((b+c-0.001)/a);
Max(1)=fix((n-0.001)/a);
c=mod(b+c,a);end
i;%
(
½
I
‡
^
Maxj
A
.
Max;%Maxi,Maxj.
d=0;%1*(d).
forj=1:1:i+1
d=d+Max(j)+1;
end%
O
Ž
1*(d)
u
õ
.
d;j=1;
while(j*d=n)
j=j+1;
DOI:10.12677/aam.2022.111036293
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end
j;dj=[0];%
(
½õ
*(d)=n.
fory=1:1:Max(1)
dj=[dj,y*a];
end
b=b+c;
forx=1:1:i
abc=0;%abc
•
?
¿
·¶
¥
m
C
þ
.
forz=1:1:x
abc=abc+Max(z)*a-b;
end
fory=0:1:Max(x+1)
dj=[dj,y*a+abc];
end
end
dj=[dj];der=dj;%der
•
?
¿
·¶
¥
m
C
þ
.
forp=2:1:j
dj=[dj,der+(p-1)];
end
CZnXS=dj,CcznXS=d,CCZnXS=j.
2.2.
p
4
ã
3
n
_
4
f
8
þ
ž
“
¯
K
½
n
2
CG
(
Z
n
,S
)
´
Z
n
3
n
_
4
f
8
S
=
{
¯
a,
¯
b,
¯
c
}
(¯
a
−
1
=¯
c,
¯
b
−
1
=
¯
b,a<b<c
)
þ
p
4
ã
,
d
´
n
,
a
,
b
,
c
o
‡
ê
•
Œ
ú
Ï
ê
.
(1)
3
n
8
{
a
d
,
b
d
,
c
d
}
¥
,
e
a
d
,
c
d
´
Û
ê
…
|
a
d
−
b
d
|6
=1,
|
b
d
−
c
d
|6
=1,
K
é
A
CG
(
Z
n
,S
)
U
x
•
d
‡
ã
1
Ø
¿
.
(2)
3
n
8
{
a
d
,
b
d
,
c
d
}
¥
,
e
a
d
,
c
d
´
ó
ê
…
|
a
d
−
b
d
|6
=1,
|
b
d
−
c
d
|6
=1,
K
é
A
CG
(
Z
n
,S
)
U
x
•
d
‡
ã
2
Ø
¿
.
(3)
3
n
8
{
a
d
,
b
d
,
c
d
}
¥
,
e
|
a
d
−
b
d
|
= 1,
|
b
d
−
c
d
|
= 1,
K
é
A
CG
(
Z
n
,S
)
U
x
•
d
‡
ã
3
Ø
¿
.
a
'
_
4
f
8
þ
p
4
ã
(
Ž
{
,
·
‚
N
´
n
_
4
f
8
þ
p
4
ã
(
Ž
{
.
ã
1
Ú
ã
2
3
n
_
4
f
8
þ
(
Ž
{
a
q
u
_
4
f
8
,
ã
3
3
n
_
4
f
8
þ
(
Ž
{
I
‡
r
_
4
f
8
Ž
{
¥
a
†
•
p
4
ã
:
ê
n
Ú
a,b,c
ù
o
‡
ê
•
Œ
ú
Ï
ê
,
ù
p
·
‚
Ì
DOI:10.12677/aam.2022.111036294
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‡
•
Ä
n
_
4
f
8
S
=
{
¯
a,
¯
b,
¯
c
}
(¯
a
−
1
=¯
c,
¯
b
−
1
=
¯
b,a<b<c
)
þ
ž
“
¯
K
.
Ï
•
ã
1,
ã
2,
ã
3
´
é
A
n
_
4
f
8
S
þ
p
4
ã
,
l
Ã
I
½
Â
ù
n
«
ã
.
e
¡
·
‚
x
Ñ
14
‡
:
ù
n
«
ã
,
ã
1,
ã
2,
ã
3,
é
A
S
8
©
O
´
{
¯
1
,
¯
7
,
¯
13
}
,
{
¯
2
,
¯
7
,
¯
12
}
,
{
¯
6
,
¯
7
,
¯
8
}
,
•
•
B
x
ã
,
·
‚
P
ã
º:
¯
i
•
i
(
i
∈
N,i<n
).
·
K
1[6]
G
´
˜
‡
r
K
ã
,
e
˜
‡
»
3
ã
G
¥
-
,
K
r
−
1
‡
ž
“
Œ
±
3
ü
‡
ž
m
ü
S
›
›
ù
‡
»
,
-
º:
•
ê
8
•
2.
Figure1.
CayleyGraph
CG
(
Z
14
,
{
¯
1
,
¯
7
,
¯
13
}
)
ã
1.
p
4
ã
CG
(
Z
14
,
{
¯
1
,
¯
7
,
¯
13
}
)
Figure2.
CayleyGraph
CG
(
Z
14
,
{
¯
2
,
¯
7
,
¯
12
}
)
ã
2.
p
4
ã
CG
(
Z
14
,
{
¯
2
,
¯
7
,
¯
12
}
)
·
‚
Œ
±
w
n
_
4
f
8
S
=
{
¯
a,
¯
b,
¯
c
}
(¯
a
−
1
=¯
c,
¯
b
−
1
=
¯
b,a<b<c
)
þ
p
4
ã
´
n
K
ã
,
d
·
K
1,
·
‚
Œ
±
Ñ
˜
‡
»
3
n
_
4
f
8
S
þ
p
4
ã
¥
-
,2
‡
ž
“
Œ
±
3
ü
‡
ž
m
ü
S
›
›
ù
‡
»
,
-
º:
•
ê
8
•
2,
e
¡
·
‚
•
Ä
˜
‡
»
3
n
_
4
f
8
S
þ
p
4
ã
¥
-
,1
‡
ž
“
›
›
œ
¹
.
½
n
3
CG
(
Z
n
,S
)
´
Z
n
3
n
_
4
f
8
S
=
{
¯
a,
¯
b,
¯
c
}
(¯
a
−
1
=¯
c,
¯
b
−
1
=
¯
b,a<b<c
)
þ
p
4
ã
,
é
n>
6,
k
MVS
(
CG
(
Z
n
,S
)
,v
;1)
≥
n
−
6.
DOI:10.12677/aam.2022.111036295
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Figure3.
CayleyGraph
CG
(
Z
14
,
{
¯
6
,
¯
7
,
¯
8
}
ã
3.
p
4
ã
CG
(
Z
14
,
{
¯
6
,
¯
7
,
¯
8
}
y
²
:
d
´
n
,
a
,
b
,
c
o
‡
ê
•
Œ
ú
Ï
ê
,
e
d
= 1,
n
= 8
ž
,
k
MVS
(
CG
(
Z
8
,S
)
,v
;1) =
3
>
8
−
6 = 2,
n>
8
ž
,
n
= 14
†
n
Ã
•
Œ
J
˜
,
é
ã
1,
ã
2,
ã
3,
Œ
±
ï
áa
q
þ
ã
¥
o
¡
“
»
p
¦
»
Ã
{
ø
ò
™
X
»
Ù
{
º:
,
Ï
•
ã
1,
ã
2,
ã
3,
ä
k
D
4
5
,
Ï
d
•
•
Ä
˜
‡
:
X
»
œ
¹
.
b
½
ã
1,
þ
0
:
X
»
,
¤
k
›
›
»
•{
„
ä
G
ãã
4.
e
d
6
=1,
é
n>
6,
w
,
˜
‡
»
•
U
u
)
3
CG
(
Z
n
,S
)
,
‡
ë
Ï
©
|
þ
,
X
J
ë
Ï
©
|
:
ê
u
6,
(
Ø
w
,
¤
á
.
X
J
Œ
u
6,
?
Ø
•{
Ó
d
= 1,
n
þ
é
n>
6,
MVS
(
CG
(
Z
n
,S
)
,v
;1)
≥
n
−
6.
Figure4.
Atreechartthatcontrolsafireat0
ã
4.
›
›
0
:
X
»
ä
G
ã
½
n
4
CG
(
Z
n
,S
)
´
Z
n
3
n
_
4
f
8
S
=
{
¯
a,
¯
b,
¯
c
}
(¯
a
−
1
=¯
c,
¯
b
−
1
=
¯
b,a<b<c
)
þ
p
4
ã
,
K
CG
(
Z
n
,S
)
´
1
−
`
.
y
²
:
d
½
n
3
Œ
•
,
n
(
n
−
6)
n
2
≤
ρ
(
CG
(
Z
n
,S
))
≤
1,
lim
n
→∞
n
(
n
−
6)
n
2
=
lim
n
→∞
1
−
6
n
=1,
¤
±
lim
n
→∞
ρ
(
CG
(
Z
n
,S
)) = 1.
½
n
5
CG
(
Z
n
,S
)
´
Z
n
3
n
_
4
f
8
S
=
{
¯
a,
¯
b,
¯
c
}
(¯
a
−
1
=¯
c,
¯
b
−
1
=
¯
b,a<b<c
)
þ
p
4
ã
,
K
CG
(
Z
n
,S
)
´
1
−
Ð
.
DOI:10.12677/aam.2022.111036296
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y
²
:
d
½
n
4,
Ï
•
ρ
(
CG
(
Z
n
,S
))
≥
n
(
n
−
6)
n
2
= 1
−
6
n
,
n>
6,
‡
¦
•
3
~
ê
c
¦
ρ
(
CG
(
Z
n
,S
))
≥
c>
0,
•
I
n
= 8,
l
c
=
1
4
.
½
n
6
CG
(
Z
n
,S
)
´
Z
n
3
n
_
4
f
8
S
=
{
¯
a,
¯
b,
¯
c
}
(¯
a
−
1
=¯
c,
¯
b
−
1
=
¯
b,a<b<c
)
þ
p
4
ã
,
é
n>
8
k
ρ
(
CG
(
Z
n
,S
)
,e
;(1
,
1))
≥
1
−
8
n
.
y
²
:
a
q
u
½
n
3
y
²
Œ
±
ï
á
›
›
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[1]Hartnell,B. (1995) Firefighter!An Application of Domination.
Presentationatthe25thMani-
tobaConferenceonCombinatorialMathematicsandComputing
, University ofManitoba, Win-
nipeg,Canada.
[2]Cai,L.andWang,W.(2009)TheSurvivingRateofaGraphfortheFirefighterProblem.
SIAMJournalonDiscreteMathematics
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23
,1814-1826.
https://doi.org/10.1137/070700395
[3]Stephen,F.andGary,M.(2009)TheFirefighterProblem:ASurveyofResults,Directions
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AustralasianJournalofCombinatorics
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[4]Wang,W.,Finbow,S.andWang,P.(2010)TheSurvivingRateofAllInfectedNetwork.
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https://doi.org/10.1016/j.tcs.2010.06.009
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[5]Kong,J.,Wang,W.andZhu,X.(2012)TheSurvivingRateofPlanarGraphs.
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