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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(1),278-287
PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.111035
AÏãVÛê››êïÄ
444ââââââ
1∗
§§§>>>ùùù
1†
§§§uuu°°°
2
§§§ŸŸŸwwwAAA
1
1
#õ“‰ŒÆêƉÆÆ§#õ¿°7à
2
#õŒÆêƆXÚ‰ÆÆ§#õ¿°7à
ÂvFϵ2021c1224F¶¹^Fϵ2022c114F¶uÙFϵ2022c126F
Á‡
-G=(V(G),E(G))´˜‡{üëÏ㧼êf:V(G)−→{0,1,2,3}÷vµ1)XJf(v)=0§
@o–•3vü‡:v
1
,v
2
§¦f(v
1
) = f(v
2
) = 2§½–•3˜‡:u¦f(u) = 3¶
2)XJf(v)=1§@o–•3v˜‡:u¦f(u)=2½3"K¡f•ãG˜‡VÛ
ê››¼ê(DRDF)"˜‡VÛê››¼êŠ•f(V(G))=
P
u∈V(G)
f(u)"ãGVÛê
››¼ê•Š¡•ãGVÛê››ê§PŠγ
dR
(G)"Š•γ
dR
(G)VÛê››¼ê
¡•Gγ
dR
-¼ê"©Ì‡‰Ñ˜AÏãXµP
m
P
n
(m=2,3)§P
n,t
§K
∗
n
§M(C
n
)§
M(P
n
)VÛê››ê(ƒŠ"
'…c
VÛê››¼ê§VÛê››ê§rȧe㧥mã
ResearchontheDoubleRoman
DominationNumberofSome
SpecialGraphs
ShashaLiu
1∗
,HongBian
1†
,HaizhengYu
2
,LinaWei
1
1
SchoolofMathematicalSciences,XinjiangNormalUniversity,UrumqiXinjiang
∗1˜Šö"
†ÏÕŠö"
©ÙÚ^:4ââ,>ù,u°,ŸwA.AÏãVÛê››êïÄ[J].A^êÆ?Ð,2022,11(1):278-287.
DOI:10.12677/aam.2022.111035
4ââ
2
CollegeofMathematicsandSystemSciences,XinjiangUniversity,UrumqiXinjiang
Received:Dec.24
th
,2021;accepted:Jan.14
th
,2022;published:Jan.26
th
,2022
Abstract
LetG=(V(G),E(G))beasimpleconnectedgraph,afunctionf:V(G)−→{0,1,2,3}
satisfieswiththepropertythat1)iff(v)=0,thenvertexvmustexistatleasttwo
neighborsv
1
,v
2
suchthatf(v
1
)=f(v
2
)=2oroneneighborusuchthatf(u)=3;2)
iff(v)=1,thentheremustexistatleastoneneighboruofvsuchthatf(u)=2or
3,andfiscalledadoubleRomandominationfunction(DRDF).Theweightofa
DRDFisf(V(G)) =
P
u∈V(G)
f(u).TheminimumweightofaDRDFonGisthedouble
Roman dominationnumber, denoted byγ
dR
(G).A double Roman dominationfunction
withtheweightofγ
dR
(G)iscalledaγ
dR
-functionofG.Inthispaper,wepresentthe
exactvaluesofthedoubleRomandominationnumbersofsomespecialgraphs,such
asP
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/
1.Úó
ã››¯K´ãØ¥˜‡-‡ïÄ+•,T¯KåuIS–Ú¯K.››êVg•@
dBerge[1]3Ùч5ãØ6¥JÑ,ÉBergeé«,OysteinOre[2]ªJÑ››ê9Ùƒ
'âŠ.1999c,StewartÄg3©z[3]¥JÑÛê››Vg,ùã¥Ûê2I¯,Û
ê2I˜‡¢½XJvk7Sè,KT¢½vko.˜‡vko¢½3Éôž,
ƒ¢½Œ±è|,•k˜|è¢½Ã{•NÄ,eNÄÑ,T¢½KØ
o.XJz˜‡ ¢½Ñ7S˜|è,¯s¤p.‡^¦ŒUè5o‡Ûê2I,é
DOI:10.12677/aam.2022.111035279A^êÆ?Ð
4ââ
AÛê››ê.‘XéÛê››¯KïÄ,ÅìÑyéõÛê››í2/ª.VÛê››
Vg´dBeeler[4]<•@JÑ,=Ûê2I˜‡¢½XJ;‘'<ôž,ekü|è
o,ÒV-o,Ûê2IÒPk•r“”üÑ.
Beeler3©z[4]¥•@JÑVÛê››Vg,¿ïÄVÛꛛꆛ›êÚÛê›
›ƒm'X,Óž‰Ñ'uãGêVÛê››êþ.,¿…•xˆþ.ãa.
Lakshmanan3©z[5]¥‰Ñ˜‡ãMycielskianãVÛê››ê.±9éu˜‡ã\˜
^>éãVÛê››êK•.3©z[6]¥AhangarJÑéuÜãÚuãV Ûê›
›êä¯K´NP-,‰ÑP
n
ÚC
n
VÛê››ê(ƒŠ,¿…‰ÑäkV
Ûê››êã•x.•`3[7]¥‰ÑP
2
C
n
,C
3
C
n
VÛê››ê.Úûw3[8]¥‰Ñ
P
2
P
n
VÛê››ê.Nazari-Moghaddam3[9]‰ÑVÛê››ê.5Vg¿•xÑ
ƒ'ã.
©Ì‡‰Ñ˜AÏãX:P
m
P
n
(m= 2,3),P
n,t
,K
∗
n
,M(C
n
),M(P
n
)VÛê››ê
(ƒŠ.
2.ÄVg
-G= (V(G),E(G))´˜‡{üã,º:8•V(G),>8•E(G),|V(G)|= n•ãGº:
‡ê.éuv∈V(G),½Âvm••N(v)={u∈V:uv∈E},v4••N[v]=N(v)∪v.
w,,º:vÝdeg(v) = N(v).
XJ¼êf: V(G) −→{0,1,2,3}÷v:
1)XJf(v) = 0,@ o–•3vü‡:v
1
,v
2
¦f(v
1
) = f(v
2
) = 2,½–•3v˜
‡:u¦f(u) = 3;
2)XJf(v) = 1,K–•3v˜‡:u¦f(u) = 2½3.K¡f•G˜‡VÛê››
¼ê(DRDF).
˜‡VÛê››¼ê-•f(V(G))=
P
u∈V(G)
f(u).ãG¤kVÛê››¼ê•
Š¡•ãGVÛê››ê,PŠγ
dR
(G).Š•γ
dR
(G)VÛê››¼ê¡•Gγ
dR
-¼ê.
-V
i
={v∈V: f(v)=i},i={0,1,2,3},KVÛê››¼êŒ±P•f=(V
0
,V
1
,V
2
,V
3
),¼ê
•ω(f) = f(V) = |V
1
|+2|V
2
|+3|V
3
|.
éu?¿ü‡{üãGÚH,GH¡•ãGÚHrÈã,Ùº:8•V(G)×V(H),
:(u
1
,v
1
)Ú(u
2
,v
2
)ƒ…=eu
1
u
2
∈E(G),v
1
=v
2
½u
1
=u
2
,v
1
v
2
∈E(H)½u
1
u
2
∈
E(G),v
1
v
2
∈E(H).n{üãGeã[10]´•3ãGz‡º:v
i
þ©OV\l
i
(i=
1,2,...,n)‡“fº:¤ã,PŠG
∗
.˜‡e•´dn‡º:|¤´,3´ü‡à:ˆV
\t−1‡]!:¤|¤ã,P•P
n,t
.ãG¥mã[11] ´•3ãGz^>þÑ\˜‡#
º:,2rãG¥ƒ>þ\#:ƒëã¡•ãG¥mã,P•M(G).^C
n
ÚP
n
©OL«n‡º:Ú´.
·K2.1[4]éu?¿ãG,f= (V
0
,V
1
,V
2
,V
3
)´ãGγ
dR
-¼ê,KV
1
= ∅.
DOI:10.12677/aam.2022.111035280A^êÆ?Ð
4ââ
3.̇(J
½n3.1 -P
2
P
n
•P
2
†P
n
r†È,Ù¥ên≥2.K
γ
dR
(P
2
P
n
) = 3d
n
3
e.
y²••B?Ø,PP
2
P
n
1˜º:•u
1
,u
2
,···,u
n
, 1º:P•v
1
,v
2
,···,v
n
,
Xã1¤«(le–þ),Ø”˜„5,5½f(u
i
) ≤f(v
i
).
Figure1.P
2
P
n
ã1.P
2
P
n
•Ä¼êf: V(P
2
P
n
) →{0,2,3},
n≡0(mod3)ž,f(v
3i−1
) = 3,1 ≤i≤
n
3
,Ù{º:DŠ•0;
n≡1(mod3)ž,f(v
3i−1
) = 3,1 ≤i≤
n−1
3
,f(v
n
) = 3,Ù{º:DŠ•0;
n≡2(mod3)ž,f(v
3i−1
) = 3,1 ≤i≤
n−2
3
,f(v
n
) = 3,Ù{º:DŠ•0.
w,f•P
2
P
n
˜‡VÛê››¼ê,¦γ
dR
(P
2
P
n
) ≤3d
n
3
e.
‡L5,éº:8?18B,w,éun≤5¤á,béun≥6¿…éu?¿un(Ø
Ѥá.-f= (V
0
,V
2
,V
3
)•P
2
P
n
˜‡γ
dR
-¼ê.©±e3«œ¹?Ø:
1)f(v
n
)+f(u
n
) = 3ž.
džf(v
n
)=3,f(u
n
)=0.dVÛê››¼ê½ÂŒ•:f(v
n−1
)=f(u
n−1
)=0,K
kf(v
n−2
) +f(v
n−3
)≤3.ÄKef(v
n−2
) +f(v
n−3
)>3.½Â¢Š¼êg:V(P
2
P
n
)−→
{0,2,3},¦g(v
n
)=g(v
n−3
)=3,Ù{:DŠ†fƒÓ,w,g•P
2
P
n
˜‡VÛê››
¼ê,¦ω(g)<ω(f),gñ.½Â¢Š¼êh: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{:DŠ†fƒÓ,Kkγ
dR
(P
2
P
n
)=
ω(f) = ω(h)+3 ≥γ
dR
(P
2
P
n−3
)+3 = 3d
n
3
e.
2)f(v
n
)+f(u
n
) <3ž.
Ø”f(v
n
)=2,f(u
n
)=0,Kkf(u
n−1
)≥2½f(v
n−1
)≥2.Ø”f(u
n−1
)≥2.½Â
¢Š¼êg:V(P
2
P
n
)−→{0,2,3},¦g(v
n−1
)=3,g(v
n
)=0,Ù{:DŠ†fƒÓ,w,g
•P
2
P
n
˜‡VÛê››¼ê,¦ω(g) <ω(f),gñ.
3)f(v
n
)+f(u
n
) >3ž.
DOI:10.12677/aam.2022.111035281A^êÆ?Ð
4ââ
œ/1:f(v
n
)=3,f(u
n
)=2.½Â¢Š¼êg:V(P
2
P
n
)−→{0,2,3},¦g(v
n
)=3,
g(u
n
) = 0,Ù{:DŠ†fƒÓ,w,g•P
2
P
n
˜‡VÛê››¼ê,¦ω(g) <ω(f),gñ.
œ/2:f(v
n
)=2,f(u
n
)=2.½Â¢Š¼êg:V(P
2
P
n
)−→{0,2,3},¦g(v
n
)=3,
g(u
n
) = 0,Ù{:DŠ†fƒÓ,w,g•P
2
P
n
˜‡VÛê››¼ê,¦ω(g) <ω(f),gñ.
½n3.2 -P
3
P
n
•P
3
†P
n
r†È,Ù¥ên≥2.K
γ
dR
(P
3
P
n
) = 3d
n
3
e.
y²••B?Ø,PP
3
P
n
1˜º:•u
1
,u
2
,···,u
n
, 1º:P•v
1
,v
2
,···,v
n
,
1nº:•w
1
,w
2
,···,w
n
(geþ).
•Ä¼êf: V(P
3
P
n
) −→{0,2,3},
n≡0,2(mod3)ž,f(v
i
) = 3,i≡2(mod3),Ù{º:DŠ•0;
n≡1(mod3)ž,f(v
i
) = 3,i≡2(mod3),f(v
n
) = 3,Ù{º:DŠ•0;
w,f•P
3
P
n
˜‡VÛê››¼ê,¦γ
dR
(P
3
P
n
) ≤3d
n
3
e.
‡L5,é:8?18B,w,éun≤4¤á,béun≥5¿…éu?¿un(ØÑ
¤á.bf= (V
0
,V
2
,V
3
)•P
3
P
n
˜‡γ
dR
-¼ê.
dP
3
P
n
(A:9VÛê››ê½Â,Œ±P
3
P
n
•kDŠ•3±9DŠ
•0.ÄKb•3DŠ•2.
œ/1:f(w
n
)=2,f(v
n
)=f(u
n
)=0ž.dVÛê››ê½Â•f(v
n−1
)=3,K
kf(u
n−1
)=0,f(w
n−1
)=0.½Â¢Š¼êg:V(P
3
P
n
)−→{0,2,3},¦g(w
n
)=0,Ù{:
DŠ†fƒÓ,w,g•P
3
P
n
˜‡VÛê››¼ê,¦ω(g) <ω(f),gñ.
œ/2:f(v
n
)=2,f(w
n
)=f(u
n
)=0ž.dVÛê››ê½Â•f(v
n−1
)≥2,K
kf(w
n−1
)=0,f(u
n−1
)=0.½Â¢Š¼êg:V(P
3
P
n
)−→{0,2,3},¦g(v
n−1
)=3,
g(v
n
) = 0,Ù{:DŠ†fƒÓ,w,g•P
3
P
n
˜‡VÛê››¼ê,¦ω(g) <ω(f),gñ.
œ/3:f(u
n
) = 2,f(w
n
) = f(v
n
) = 0ž.duP
3
P
n
é¡5Œ•Úœ/1aq.
e5?Øf(w
n
)+f(v
n
)+f(u
n
) = 3†f(w
n
)+f(v
n
)+f(u
n
) = 0žœ/.
(i)f(w
n
)+f(v
n
)+f(u
n
) = 3
œ/1:f(w
n
) = 3,f(v
n
) = f(u
n
) = 0ž.dVÛê››ê½Â•f(w
n−1
) = f(v
n−1
) = 0,
Kkf(u
n−1
) = 3.½Â¢Š¼êg: V(P
3
P
n
) −→{0,2,3},¦g(v
n−1
) = 3,g(w
n
) = 0,Ù{:
DŠ†fƒÓ,w,g•P
3
P
n
˜‡VÛê››¼ê,¦ω(g) <ω(f),gñ.
œ/2:f(v
n
)=3,f(w
n
)=f(u
n
)=0ž.dVÛê››ê½Â•f(w
n−1
)=f(v
n−1
)=
f(u
n−1
)=0.½Â¢Š¼êg: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{:DŠ†fƒÓ,Kkγ
dR
(P
3
P
n
) =
ω(f) = ω(g)+3 ≥γ
dR
(P
3
P
n−2
)+3 = 3d
n
3
e.
DOI:10.12677/aam.2022.111035282A^êÆ?Ð
4ââ
œ/3:f(u
n
) = 3,f(w
n
) = f(v
n
) = 0ž.duP
3
P
n
é¡5Œ•Úœ/1aq.
(ii)f(w
n
)+f(v
n
)+f(u
n
) = 0
dž=f(w
n
)=0,f(v
n
)=0,f(u
n
)=0.dVÛê››ê½Â•f(v
n−1
)=3,K
kf(w
n−2
)=f(w
n−1
)=f(v
n−2
)=f(u
n−2
)=f(u
n−1
)=0.½Â¢Š¼êg: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{:DŠ†fƒÓ,Kkγ
dR
(P
3
P
n
) = ω(f) = ω(g)+3 ≥γ
dR
(P
3
P
n−3
)+3 = 3d
n
3
e.
½n3.3 -P
n,t
´˜‡eã,Ù¥n≥3,t≥3.K
γ
dR
(P
n,t
) =
(
n+3,n≡0,2(mod3);
n+2,n≡1(mod3).
y²••B?Ø,éP
n,t
?1·IÒ,Xã2¤«.
Figure2.P
n,t
ã2.P
n,t
•Ä¼êf: V(P
n,t
) −→{0,2,3},
n≡0(mod3)ž,f(u
i
) = 3,i≡2(mod3),f(u
n
) = 3,Ù{º:DŠ•0;
n≡1(mod3)ž,f(u
i
) = 3,i≡2(mod3),Ù{º:DŠ•0;
n≡2(mod3)ž,f(u
i
)=3,i≡2(mod3)…i6=n−1,f(u
n−2
)=2,f(u
n
)=3,Ù{º:D
Š•0.
w,f•P
n,t
˜‡VÛê››¼ê,¦
γ
dR
(P
n,t
) ≤
(
n+3,n≡0,2(mod3);
n+2,n≡1(mod3).
‡L5én?18B,w,éun≤6Ѥá,ybéun≥7±9¤k:êun(ØÑ
¤á.bg= (V
0
,V
2
,V
3
)•P
n,t
˜‡γ
dR
-¼ê.duP
n,t
(A:,5½g(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.?Øg(u
n−2
)ù‡::
1)g(u
n−2
)=0ž,Œg(u
n−3
)=3.½Â¢Š¼êh:V(P
n−3,t
)−→{0,2,3},Ù¥P
n−3,t
DOI:10.12677/aam.2022.111035283A^êÆ?Ð
4ââ
´íu
n−1
,u
n−2
,u
n−3
2ëu
n−4
†u
n
.w,g•P
n−3,t
˜‡VÛê››¼ê.
γ
dR
(P
n,t
) = ω(g) = ω(h)+3 ≥γ
dR
(P
n−3,t
)+3 ≥
(
n+3,n≡0,2(mod3);
n+2,n≡1(mod3).
2)g(u
n−2
)=2ž,½Â¢Š¼êh:V(P
n−2,t
)−→{0,2,3},Ù¥P
n−2,t
´íu
n−1
,u
n−2
,
2ëu
n−3
†u
n
.w,g•P
n−2,t
˜‡VÛê››¼ê.
γ
dR
(P
n,t
) = ω(g) = ω(h)+3 ≥γ
dR
(P
n−2,t
)+2 ≥
(
n+3,n≡0,2(mod3);
n+2,n≡1(mod3).
3)g(u
n−2
)=3ž,g(u
n−3
)=0.½Â¢Š¼êh:V(P
n−3,t
)−→{0,2,3},Ù¥P
n−3,t
´í
u
n−1
,u
n−2
,u
n−3
2ëu
n−4
†u
n
.w,g•P
n−3,t
˜‡VÛê››¼ê.
γ
dR
(P
n,t
) = ω(g) = ω(h)+3 ≥γ
dR
(P
n−3,t
)+3 ≥
(
n+3,n≡0,2(mod3);
n+2,n≡1(mod3).
••B?Ø,‰ÑãK
4
eãº:IP•ª,Xã3¤«.
Figure3.K
∗
4
ã3.K
∗
4
½n3.4 eK
∗
n
´˜‡ãK
n
eã,Ù¥n≥3,l
i
≥1,i= 1,2,...,n.K
1)l
i
= 1,i= 1,2,...,nž,γ
dR
(K
∗
n
) = 2n+1;
2)l
i
≥2,i= 1,2,...,nž,γ
dR
(K
∗
n
) = 3n.
y²1)½Â¢Š¼êf:V(K
∗
n
) −→{0,2,3},-f(v
i
) =3,f(v
i1
) =0,1≤i≤n•{ :DŠ
•2.w,f•K
∗
n
˜‡VÛê››¼ê,Kkγ
dR
(K
∗
n
) ≤3+2(n−1) = 2n+1.
‡L5,én?18B.bf=(V
0
,V
2
,V
3
) •K
∗
n
˜‡γ
dR
-¼ê,Œ•‡¦K
∗
n
Š
Ú•,w,¤k]!:DŠ•U•0½2,ÄK,eÙ¥kDŠ•3:,Ø”ù‡:•v
i1
,
DOI:10.12677/aam.2022.111035284A^êÆ?Ð
4ââ
=f(v
i1
)=3.dVÛê››¼ê½Â•:f(v
i
)=0,K| :¥Øv
i
7,„k˜‡:D
Š•3,Ø”•v
j
,=f(v
j
)=3.Kf(v
j1
)=0.½Â˜‡¢Š¼êg:V(K
∗
n
)−→{0,2,3},¦
g(v
i
)=3,g(v
i1
)=0,g(v
j
)=0,g(v
j1
)=2.Ù{:DŠ†fƒÓ.w,g•K
∗
n
˜‡VÛ
ê››¼ê,¦ω(g)<ω(f)gñ.Óž¤k| :DŠ•U•0½3,ÄK,eÙ¥kDŠ
•2:,Ø”ù‡:•v
i
,=f(v
i
)=2.dVÛê››¼ê½Â•:f(v
i1
)=2.½Â˜‡¢
Š¼êh:V(K
∗
n
)−→{0,2,3},¦h(v
i
)=3,h(v
i1
)=0,Ù{:DŠ†fƒÓ.w,h•K
∗
n
˜‡VÛê››¼ê,¦ω(h)<ω(f)gñ.w,f(v
i
)=3ž,kf(v
i1
)=0.f(v
i
)=0ž,
kf(v
i1
) = 2.Œ±K
∗
n
¥–k˜‡:DŠ•3.bK
∗
n
¥TÐk1+t‡:DŠ•3(t•
ê), KK
∗
n
TÐkn−(1+t)‡:DŠ•2.Kkω(f) = 3(1+t)+2[n−(1+t)] = 2n+1+t≥2n+1.
2)dã/A:Œ•,z‡| :–'éü‡]!:,‡¦K
∗
n
ŠÚ•,•Iò¤k|
:D3,¤k]!:D0=Œ.¤±γ
dR
(K
∗
n
) = 3n.
½n3.5 -M(C
n
)ÚM(P
n
)©O´C
n
ÚP
n
¥mã,Ù¥n≥2.
Kγ
dR
(M(C
n
)) = γ
dR
(M(P
n
)) = n+d
n
2
e.
y²ŠâM(C
n
)ã/(Œ•:ã¥:Ý©•deg(v)=2†deg(v)=4ü«.l?¿˜
‡deg(v) = 2:Uì^ž••éã?1IÒv
1
,v
2
,···,v
2n
,Xã4¤«.
Figure4.M(C
n
)
ã4.M(C
n
)
¢Š¼êf:V(M(C
n
)) −→{0,2,3},n•óêž,f(v
i
) =3,i≡2(mod4),Ù{º:DŠ
•0.n•Ûêž,f(v
i
)=3,i≡2(mod4),f(v
2n−1
)=2,Ù{º:DŠ•0.w,f•M(C
n
)
˜‡VÛê››¼ê.¦γ
dR
(M(C
n
)) ≤n+d
n
2
e.
‡L5,én?18B.bg=(V
0
,V
2
,V
3
)•M(C
n
)˜‡γ
dR
-¼ê.dã/IÒ9A:Œ
:deg(v
i
) = 2,i≡1(mod2),deg(v
i
) = 4,i≡0(mod2).
Äk?Øv
2n
,v
2n−1
ùü‡::
1)g(v
2n
)∈{0,3}.ÄKg(v
2n
)=2.dVÛê››¼ê½ÂŒ•:g(v
1
)=g(v
2n−1
)=0,
Kkg(v
2
)≥2,g(v
2n−2
)≥2.½Â¢Š¼êh:V(M(C
n
))−→{0,2,3},-h(v
2n
)=3,h(v
2
)=
DOI:10.12677/aam.2022.111035285A^êÆ?Ð
4ââ
h(v
2n−2
)=0,Ù{:DŠ†gƒÓ,w,h•M(C
n
)˜‡VÛê››¼ê,¦ω(h)<ω(g)g
ñ.
2)g(v
2n−1
)∈{0,2}.ÄKg(v
2n−1
)=3.n•óêž,dVÛê››¼ê½ÂŒ•:
g(v
2n
)=g(v
2n−2
)=0.½Â¢Š¼êh
0
:V(M(C
n
))−→{0,2,3},-h
0
(v
2n−1
)=2,Ù{:DŠ
†gƒÓ,w,h
0
•M(C
n
)˜‡VÛê››¼ê,¦ω(h
0
) <ω(g)gñ.n•Ûêž,dVÛ
ê››¼ê½ÂŒ•:g(v
2n
)=g(v
2n−2
)=g(v
1
)=g(v
2n−2
)=0,Kkg(v
2
)=3,g(v
2n−4
)=3,
g(v
2n−5
) = 0.½Â¢Š¼êh
0
: V(M(C
n
)) −→{0,2,3}, -h
0
(v
2n−2
) = 3,h
0
(v
2n−1
) = h
0
(v
2n−4
) =
0, h
0
(v
2n−5
) = 2, Ù{:DŠ†gƒÓ, w,h
0
•M(C
n
)˜‡VÛê››¼ê,¦ω(h
0
) <ω(g)
gñ.
Ùg,•Ä±en«œ¹:
1)g(v
2n
)+ g(v
2n−1
)=3ž,=g(v
2n
)=3,g(v
2n−1
)=0.w,Œ±g•M(C
n−2
)˜
‡VÛê››¼ê¿…-u½uω(g)−3.Šâ8Bbγ
dR
(M(C
n−2
)) ≥n−2+d
n−2
2
e=
n+d
n
2
e−3.Ïdkγ
dR
(M(C
n
)) = ω(g) ≥γ
dR
(M(C
n−2
))+3 = n+d
n
2
e.
2)g(v
2n
) + g(v
2n−1
)<3ž.©•g(v
2n
)=0,g(v
2n−1
)=2†g(v
2n
)=0,g(v
2n−1
)=0.
g(v
2n
)=0,g(v
2n−1
)=2 ž,w,Œ±g•M(C
n−1
) ˜‡VÛê››¼ê¿…
-u½uω(g)−2.Šâ8Bbγ
dR
(M(C
n−1
))≥n−1+d
n−1
2
e=n−1+d
n
2
e.Ïd
kγ
dR
(M(C
n
)) = ω(g) ≥γ
dR
(M(C
n−1
))+2 = n+d
n
2
e+1 ≥n+d
n
2
e.g(v
2n
) = 0, g(v
2n−1
) = 0
ž,w,Œ±g•M(C
n−2
)˜‡VÛê››¼ê¿…-u½uω(g)−3.Š
â8Bbγ
dR
(M(C
n−2
))≥n−2+d
n−2
2
e=n+d
n
2
e−3.Ïdkγ
dR
(M(C
n
))=ω(g)≥
γ
dR
(M(C
n−2
))+3 = n+d
n
2
e.
3)g(v
2n
)+g(v
2n−1
) >3ž,=g(v
2n
) =3,g(v
2n−1
) =2.½Â¢Š¼êh
00
: V(M(C
n
)) −→
{0,2,3},-h
00
(v
2n
) = 3,h
00
(v
2n−1
) = 0,Ù{:DŠ†gƒÓ,w,h
00
•M(C
n
)˜‡VÛê››
¼ê,¦ω(h
00
) <ω(g)gñ.
éN´z˜‡M(C
n
)˜‡γ
dR
-¼ê•´M(P
n
)˜‡γ
dR
-¼ê,KéuM(P
n
)(Ø
Ó¤á.
Ä7‘8
I[g,‰ÆÄ7‘8(11761070,61662079,12071194);2021c#õ‘Æg£«g,Ä7é
Ü‘8(2021D01C078);2020c#õ“‰ŒÆ˜6;’!˜6‘§‘8]Ï.
ë•©z
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[2]Ore,O.(1962)TheoryofGraphs.AmericanMathematicalSociety,Providence.
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4ââ
[3]Stewart,I.(1999)DefendtheRomanEmpire.ScientificAmerican,281,136-139.
https://doi.org/10.1038/scientificamerican1299-136
[4]Beeler,R.A.,Haynes, T.W. and Hedetniemi, S.T.(2016) Double Roman Domination.Discrete
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https://doi.org/10.1016/j.dam.2018.03.026
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https://doi.org/10.1016/j.dam.2017.06.014
[7]•`.ãVÛê››[D]:[a¬Æ Ø©].x²:x²ŒÆ,2018.
[8]Úûw.‚fãVÛê››8[J].ײÆÆ,2021,23(2):54-57.
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nationinGraphs.DiscreteMathematics,AlgorithmsandApplications,12,1-12.
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