Graphes parfaits : structure et algorithmes Nicolas Trotignon To cite this version: Nicolas Trotignon. Graphes parfaits : structure et algorithmes. Autre [cs.OH]. Université JosephFourier - Grenoble I, 2004. Français. �tel-00008558� HAL Id: tel-00008558 https://tel.archives-ouvertes.fr/tel-00008558 Submitted on 23 Feb 2005 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. 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F F R e1 , e 2 , . . . , e k R G G G ei ei ej |i − j| > 1 R ej F G F G G v1 , . . . v k F F vi i = 1, . . . , k i ∈ {1, . . . , k − 1} ui ∈ V (R) v1 u1 uk 0≤i<j≤k u i = uj R u0 = uk ui 0 ≤ i ≤ k F R uk−1 P R vi vk P R vi+1 u0 2 G {a, b} a a G b b G k/k P + 7 [ H < #AKJ E [email protected]/[email protected];/W8a'< JS? e SH E ACH>68AC<>7Q68=´D7FE 38W< ? jB=Y3k<>68;[email protected] <>; H[4a-a'<QG8<[N+E 7 >; <[NBG8< ? jB; <>=YAbc=64;< < <35W<>VI; 4;/687 7 O>a-; [email protected]; <[O[; NmN¢G8G8<><{6SAKV´J E 7 <W>Vf; ;/< Sc7 O>4a-?=^P [email protected] ; 7 EO[ACNQH>H[68<G8AC<'#B<£AKAK68JJ [email protected]<£DM>4;HF<W7/[email protected];/FCW8=Sa'<È4<=G8<OS=?KR4S; <3k4687 ; 4<>;68; AC<[N2=G84<>; 6SO[<>V <; ; <>=G^SVG8a'<94I<[ACN/Gf4;È68=8AC68D4 =6J<°<>68 3? 7{E Z [email protected]?57 E <E [email protected]< H>J 68E<>=a-;< 7 ?C<>NÈ[email protected]?\<BG8E < =cH[<[; NH ? WZ GE [email protected]# ==4<[=NS=H[J 4W< AKVf=J E?CN G8N/?C;G8Gf<O>[email protected];/7 ?CN<>4; 79=4AK38JG8<>35<[=<>NÄN 7{<>[email protected]<>=am;Gf:8?C7 ACH[O>< <[3kN{4N=J G8Gf<>=S7<[<NM E 7 ?>;Z <[; N¢438G83k<£<>AK7ÄJ 68<>=;¢<97 G8<>;<£4H[68<[7/N¢=HF<>W7GE ACFC<9=N <[4N 6SMNbcMK68D?7FE<[38N/;WE<B6G8N < N/?868='[email protected]=<>[email protected]<>;am3:8E AC7 <ÈG8<£NF4a-a'<>; NG8< ? ? ! Z ?¢AKJ E [email protected]/[email protected];/W8a'<#GJ ¢Gfa'4=G8Nm;/7 468LM<'68=<'H WGE FC=< G8< AC4=8D6<>687 3N/68E 7 [email protected] <S E AC4? 7 jBN=y4===44; ; < < N <[NE 7 ><>; <[; N·S GE =NAKJ 47 Gf7 <4'2<>[email protected]<[NE 383? 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K A J E @ A M D 4 / 7 @ ? / ; 8 W ' a È < G J ¢Gfa'4=G8N=<£;/7 468LM<{3EN¢G8<£HFWGE FC=< S EN/68AC4387 3kN44N <1=<>L=E´LIa'64<1=VG;/7 J 68<>7=<1bc6H[4< =c;/7FEGf?CH>;/<[?C4N/;1=68AKJ=W< <YVf?CN/3; E <>[email protected]=7 <`H[<GGJ E E a-=N ?CNG8< GJ 68=?oL H W4<>68a-7[email protected][= <>A\EfS4G8= < ACAC4<[NB=8DN 64a-<>68a'7 <>[email protected]; a-NÈ3G8E < [email protected] <¡S<>=cG;/E 7 =< NÈAKJ 4<>7 ; Gf7 <m? 4 jB' = [email protected]=4E ; 38< 3E 7FE ?CN N <>=c;BS N/687S ? ? ? ?4S A <[NÈSN 4a-a'<>; N S · S N 4 c = ; E 6 N / N M ? 8 G [ < N E 7 >; <[NG8< bc68?KS GJ E 38j7 T[NAC< AC<>a-a'< OS?KE S 3k<>68LM<>=V; >;/7 <mLc6<[NBH[4a-a'<AC<[NBE 7 >; <[NBGJ 68=<mHFWEGFC=<2G8<°AC4=8D6<>687 3E [email protected] <mG8< ? _`E ?CN E AC47 N [ < / N m ; K A J > < = N > < 2 a 8 : C A < 8 G [ < Q N E 7 >; <[N GJ 68=<1H WGE FC=< 8 G ' < C A 4 8 = D 6 > < 8 6 7 3 E @ ? 7 < ? bc6<+u AKJ E [email protected]/[email protected];/W8a'<QE 687FE [email protected];BG G8O>; <[H>; <>72]f68=<H[4=c;/?7FEZ Gf?CH>;/?C4= =O[<°34E =7BH H[<>[email protected] EAC<[email protected]<+7/[email protected];/JW8E a'[email protected]<°47/[email protected];/4W8=8a'=<< A\OSE?KRS:k? 4=8=<B7 O>354=N < E2H[4a-38ACW< [email protected]; O+<[N/;:8?C<>=G84a-N? M L(R) ua,2 a b R R b a ub,1 ub,2 i ∈ {1, 2} R \ {ua,i+1 , ua,j+1 } j ∈ {1, 2} Ci,j ua,i {a, b} G= ua,1 ub,j G P G Q = P ∪ {a, b} G n3 Ci,j e1 , e2 , . . . , e2α e0 = a e2α+1 = b Ci,j Q = {e0 , e1 , . . . , e2α+1 } R \ {ua,i+1 , ua,j+1 } R 2α + 2 G Ci,j Q G a Q b Ci,j {a, b} G G a b v0 = a v1 P0 k = 0, . . . , 2α+1 v2α P0 P0 v2α+1 = b vk R R {v1 , . . . , v2α } Ci,j 2 F ; E V = Ä ; 8 G 4 8 = = [ O N 8 G > < SV-N 4a-a'<>; N GJ 68='D7FE 38W< SM4=^G8OW\[email protected];ÄAKJ 435O>7FE ;/?C4=1G8< 6 =<>*,; 4; O ? jB*=' <>= ;B^4;7 G8<<>[email protected]< ?CO #'<>; HFWACE<£] bV4D6=27F<QE <>38N=8W4AUa-<ÈT>LM4a'< :8<>; ;+<><>=cG8; 6 < ? jBG8< =^[email protected]Æ;<>;bc64=< bcEG68r ?4LM68,4; <Ä[email protected]; 68 AK=2J 68 ==`4E686YLM< a'EG6 E [email protected] =N 4N+a-G8a'< <>; =4; 4=N bc6< 7 > < 8 3 7 [ O N > < c = ; < E 6 F N / N ? ? R A<[N/;mH>A\E [email protected] bV6<^3546872; 468;mD7FE 38W< <>; ;O>4;FE68=c; <;m3G8E [email protected]=87 =<O[G8<-<Ä68N 4=a-< a'<>; N%=4=-M EH[4G ACr 4E7FH[E <>;/=V?C;4N =G8< S4=-SE 4=¡35<>68;+eKEH>[email protected]<>a'<>=V? ;Q 4=-:8<; k<<>=8>;[email protected] S 68=<>< ; S3868?CN4M=^H[4G8AC44=87F=E ;/< ?C4# =G8<>< ; A\]E4H[='4DV68E AC<>7 68G87Ä<9GfA\E°6'H[N 4468a-AC<>a'687Ä<>;G8H[<[4N=VN ;/47Fa-EH>a'; O <>? ; NÄjBGf=#?!kO4>:87 ;/<>?C<>=c=V; ;{NÄG8G8<< ? l <[N[email protected]=O>DVE [email protected][email protected]; O[N3k<>68LM<>=V; >;/7 <BN ;/7/?CH>; <[NB]IN/? a >a'<BAKJ [email protected]=O>DVE [email protected][email protected]; O / x y x, y x y xy y G/xy xy x V V \ {x, y} xy x y χ(G/xy) χ(G/xy) y G G x y ω(G) ≤ ω(G/xy) x G/xy G {x, y} χ(G) ≤ χ(G/xy) G/xy G 8 kQ [<E N/AC4;7 68N =2HFW<>[email protected]='E[LM<><[; Hbc6E ;/7 <N 4a-;FE a'=<>Gf; ?CN NBS·bcN ? 6< <>; N 4=V;AC<[N<WVf<>;/;7 O>[email protected]; O[N%G8<H[<HFW? <>UMa-< [email protected] == S X7FE [email protected]<>38AC4; =<>N°;+A\E1UM< 38E 7 =S<>M 68L LM?C<-<>7/H>7 A\< ENFBN/?CW8bV7i6H <SkH R E 7°[email protected]=cA ;B=3847 64N°68LMNF<>O2amAC<:8AC;/<'WbcO[467j<QT>a'AKJ E < 7/DN/[email protected]'LE <>=V=V;+;2G8G4J O[=VHF; W=E =84DM6<N :8W< VI?CH [email protected]; a^O°E G;/E ?Cbc=6N£<h68=68;/H[[email protected]@4?C=cNFO ; W< 3kVI;4<687^[email protected] N£4DM68O>LM=<>O>7^7FEA\E´A H[=44=6H>N£[email protected]=N/4?C46= N9W< gVfE38'[email protected]?CEbc6G8<><[7 N14=H NÈWE N/=68H[7£<[H[N1<°G3kJ 4>;/[email protected]=V7 <; #QA\E N <[H>;/?C4= [email protected]> H[4=NFEH>7 O[<QE S6 V13E 7/;/[email protected];/?C4=NÈ3E [email protected] <[N ? x χ(G) = 2 ω(G) = 2 " ` *p +! ' 252 $* " -6_.#7# `. y χ(G/xy) = 3 ω(G/xy) = 3 G - + {x, y} - $ G .0 . - * .3 . G - .0 1 *,-/.0- x y G G/xy . 1 1 *1 #- * G G/xy *. 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GE =N S5H[<2bc68?a'4=c;/7 < bV6< A x G y x y X X ∪Y Y G x y (X, Y ) G[X ∪ Y ] x G x G[X ∪ Y ] G[X ∪ Y ] {x, y} y x x χ(G) x G G ω(G) ≥ ω(G/xy) G G/xy K ω(G/xy) K \ {xy} a K \ {xy} y x b y a K x y y K \ {xy} G ω(G/xy) xy K \ {xy} x−b−a−y x y G y ω(G/xy) ω(G/xy) x χ(G) ≥ χ(G/xy) G/xy A χ(G) x y χ(G) G/xy χ(G) G y y ω(G/xy) G/xy G b G/xy x k9 xy x y L 4687/;FE =V; AC<+D7FE 38W< <[N/; ? @ a 3 E 7 K e E @ ? B ; [ E M L [ < H 68=<°3E [email protected] <GJ E a-?CN 3E 7 eKE [email protected]; ? 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H>Z [email protected][email protected]?C=bc46= < a^GE Vfl [email protected] G8< ? 68G=J E <^a-3?C<[N#E N{[email protected]>E <';[email protected][email protected];'J7/E [email protected];/W8=?Ca'N238<[G7 N{4E :83k=UA Nm4T>N a'68<>==V<Y;DGfG87F3? <>2^E 6S38H>V'[email protected]< 38< 7 4 jB38:87AU=T>4a':8EUA <[T>G8N a'O ]Vr< ;#4OS68LI?KO6´; GS bcJ 6a^E :k<-E 4?CA\7N#EGbV7SI<[6H[H <4Wa-<>AC<>7 a'68HFW7#<><=VG8;{G8O>;/;<-7 <[43H>68;/E [email protected]<>= <7 B=D7FE 38W< G G/xy {x, y} H0 H G/xy x y V (H 0 ) 0 0 H H 0 χ(H) = χ(H ) = ω(H 0 ) = ω(H) H = H 0 /xy H 0 /xy = G G/xy G {x, y} G/xy x y 2 G G xy G v G G {a, b} K G G ab G K (K \ ab) ∪ {a} K G (K \ ab) ∪ {b} G/ab 8 k!e 38G 7E 4=:8NÈAUT>AC<[a'N <2D7F3kE 438N WO2<[3N+E G87+< AKJ 0{E [email protected]<>DM7/4DM7/<[email protected];/<[W8N/a';È<-7 O <[E N/[email protected];?CNTE<>=:8H[AC<Q47 <>< =`4; 68<>LMa-<>37/;'N ]535H[[email protected]=a'4a-<>=V?\;E A H[?$4A AC4<27/G8?C<><>7°6SVf<12^?UT>a'H EG< M 35H[<><>a'7/a'/ <>=V<>;;/;/687FE=^[email protected];ÈD7FG8E <m38H[W4< AC4G87/<?C<>0{79<>; 7/4DM68<È;9NTDE 7F=E N383WE <°[email protected] 3<BE G7 eKJ E E [email protected];È?C<>N = ; iÈ<>a-O[NF34N£6Gf3k7 4<È[email protected][=<9438a-7 4?\E :8A AU? 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E¡AC<[N#a [email protected]>=a'Gf68<[?CNN <>N =V4;ma-G8a'<[N2<>; H>[email protected]?CbVbV66< <[NmH<>=8? =Rd<>ABa-<[?CN<[;#NmH>G8A\< E [email protected]#bc<>6;m< bVV6< (K ) ∩ V (H) V (K ) ∩ V (H) ?/u 4=HS3k4687¢3E 7/LM<>[email protected] #°=4N\=N [email protected]=46NÄN/62'H;G8<£LMO>7/?N\<>H7ÄbV6=< H ?4L 4687 H[<>A\EfS4= ) NFE =N£=46N9NF46H>?C<>7ÈG8<[N9N 46N KM D7FE 38W<[N ) = ω(G χ(G 2 1 K1 ≡K2 0 K1 ≡K2 0 1 2 0 a'H[44AC4=V7F;/E 7 ;/<°?C4bc=h6J G8#m< 3E 7/;/[email protected]£GJ 68=E[LM<+<[H[H 4AC47FE ;/?C4=YG8< H[4E[68LMAC<[<>68H 7 N2] H[468AC<>687 N Sf4=;/7 468LM<+68=< B j = [ H 4 a ' a > < = [ H < 3 E 7 [ H 4 C A 4 / 7 C ? > < 7 C A [ < N N 4 a ' a > < ; N c b 8 6 ^ ? = < N 4 V = ; 8 = 1 ? G E = N 8 = ? Z ? S O K ? / J O GE =N ? SE AC47 N'4=NTE [email protected];'GJ E 387jT[N-AC<AC<>a-a'< bc6< (K1 ∩H)≡(K2 ∩H) K1 ≡K2 K1 ≡K2 G GK1 ≡K2 ω(G) ω(GK1 ≡K2 ) K1 K2 ω(GK1 ≡K2 ) > ω(G) /J ? jB=)EªG84=H ? @ = f G 8 6 @ ? ; K A J 8 6 8 = C ? 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E =N 3k4687 \=E AC<>a'<>=c;94:8; <>[email protected]{68=< M H[4AC47FE ;/?C4= G8< Z ? <>; NF4=V;BG8<>S6 VYNF46N M H>[email protected]?CbV6<[NBGf?CN r/[email protected]=V; <[N GJ 68=<2H>[email protected]?Cbc6< SE AC47 N <>[email protected]<[N NAC<>4a-=c;'a'<><=8N=68<>[email protected] ?C<[=cN ;9? Lu E E G=E N'=NÈH[<AC<°H NE<>N =S N94=G8<°[email protected]<°;'H[bc46s= Jr/<><[[email protected]>AC<;/68N 7 4< =VOS; ?KJ/ J ] .01 Q<>=8=<>a-?C<[N ? A < V (K1 ) ∪ V (K2 ) # ω(G K1 ≡K2 ) − ω(G) ω(GK1 ≡K2 ) − ω(G) ω(G) V (K1 ) ∪ V (K2 ) G v2 K2 ω(GK1 ≡K2 ) GK1 ≡K2 K1 K2 GK1 ≡K2 V (K1 ) ∪ V (K2 ) v1 C G G v2 ∈ C 1 G C C1 v1 v1 C1 K1 K1 v2 K2 V (K1 ) ∪ V (K2 ) v1 v2 C1 C1 v2 v1 K1 ∪ K 2 ω(GK1 ≡K2 ) GK1 ≡K2 2 K1 K2 K ! ' U] 1 $ G - 1 $ $ + . K K 1- *. #- 1 1 + . 1 1 ω(G) G .0 1 2 |K1 |+|K2 | 6= Z 68383k4N 4=N |K |+|K | = ω(G) ? iB<>a^E 7 bV64=N%GJ E :k47 G-bV6<GJ E 387j[T N S O K ? / J ? 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[email protected] [email protected]; (r7 r4IH−PACW<<mNFG8−s) 4<a-a'? ∪<>X ; {t}4T; 4\={t} M H[4a-38AC<>;+G8< r−P −sr −P C A ° < 8 3 [email protected]; 6<[NN È N c b 6 < [ H 4 c = / ; C ? > < V = + ; ; 4 8 6 s r −P −s ?<u 4=HS AC3<[E N 7¢TWIH \35{t}4;/W4dM rE[T 7NF><È; G8<[sNÈ<£G87 O[< H>r−P > < £ ; V b 6 < [ H [ < 9 N E 7 > ; [ < £ N N 4 V = £ ; > < = = 4 m a 8 : 7 + < @ ? a 3 E @ ? 7 −s ? <[N/;<>G8; <9AC4=8D6<>68<[7N/;Ä[email protected]<9E [email protected] =8< D6jB<>=^687<>=^[email protected][E [email protected] <[email protected]; ? bVjB6=J <WN/V68E38H>35; <>4a'NF<-<>=VG;J E AK:5J 684=#7 GG868<[bc7/N67 H<>< W=<>H[a-<S [email protected]=N −P−s r−P −r s −P −s ? =¡3E 7/;/?CH>[email protected]?C<>7 S [ < N m ; 8 G < C A 4 8 = D 6 > < 8 6 m 7 @ ? a 3 E @ ? 7 < r −P −r ? r 6= r <>; x 6= r Z ? r a^E =bV6< y SVE AC47 N P = r −P −r−t−y <[N ;¢68=^H W<>[email protected]=G8< G G8<9AC4=8D6<>687 [email protected] a-7 >3; E< [email protected]?.7 <Bu G84=4=cH;ÄS3AC<[E N72<WWIV IH;/7 35O>4a-;/[email protected]; [T O[NFN{<#N G84<^=c; 7 TO[H>\687/{t}7 <>=M H[H[4<a-SAK38J 68AU>T =; <1<[N G8? <[jBN27 H[P4=H>=[email protected] EN ?C34=ENÄN G8 H< T4\6 {t} G M <LM[N/4;m?CN/[email protected]=#E ;/G?CN E eK=E [email protected] ; <-3k4687 ?Pu 4=<>; H T \G84{t}[email protected];LM? 4_`[email protected] E ?CN2SMH[H[<B<>A\bVE 68<[?N/[email protected];[email protected]@?C3kbV46N < N/[email protected]:8AC<^H E 7 t =<>J ; E3EN2G8< ? y s =s=y r =y Z ? r a^E =Tbc6\< {t}x SVE AC47 N Pr = r −P −r−t−x [ < / N ; 8 6 1 = H W > < a [email protected]=G8< G8<9AC4=8D6<>687 M H[4a-38AU>T ; <[N ? jB7 P =J E3GENÄG8< T \ {t}M [email protected] a-7 >3; E< [email protected]?/7 u <B4G8=4H=cS;Ä3ACE<[7NW!<WIVcH ;/3k7 4O>;/[email protected][T ; N O[<£N{G8N <{4=c7 ;O[H>T687/\7 {t} NLMFE 4;/?CN/?CN [email protected]=-eKE [email protected]; E <9=3kN 4687 P ?s<>u; T4=\H {t}G8? 4_`[email protected];%[email protected] H[<><>A\E2S=H[H[<[<{<N/SV;bVAK68<>J [email protected][a-<£47 38G8<[email protected]<[[email protected] a-6H[3k<4=4NH>N/[email protected][email protected]:8N/AC?C<4? =H jBN E 7=' HtEÈ=4G8J 64Em=3H{G E:8N{?C<[G8<>N/= <; T \{t} x r=x ?u r N (t) ∩ P = {x, x , y} <9a >a'<ScN/? s −P −s <[N/;G8<9AC4=8D6<>[email protected] [email protected] <SIE AC47 N4= a'46 4=V;/7 < N (t) ∩ P = {x, y ? , y} ? jB=#E+:8?C<>=#a'4=V;/7 OBbV6< N (t) ∩ V (P ) = {x, x , y} N (t) ∩ P = {x, y , y} [email protected]=c; <>=E =V; S3868?CN bc6< yS<>;%=38J <[68N/?C;9N bc36E<N T M =H[4J <[a-N ;383AC<>E; N [email protected] M HH[4E a-6838=YAC<>;[email protected] a'EB<>68; =-uN ∈4a-Ta'; <><>; A bV6< _`NE (u)∩P = {x, x , y} x T 4 ? u ; > < A c b 6 < 4 = H > < ; v∈T N (v) ∩ P = {x, y , y} u v 35<>7/a'<>;/; <>=c; G8<mNFE ;/?CN eKE [email protected] <+A\E ? 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[email protected] a-E E [email protected]E [email protected]<>7-<AC4G8=< NbV6< 3kMdMdE4E 7N7 >jN>;T[; <[G8<[N N<£S ? 68bcu =#684?==4N H 4am=c:8;-7 <[email protected] >jN 37 T[<>E [email protected]< 7; <[G8E N <6rS%<>a';[email protected]=?ÄN;Md68E47;F=>E < ;[email protected]?C<[NFN <>SI=V<>;-;6868=ª=#==44MdamamE 7 :8:8>;77 <<< c b 8 6 ¢ ? = J [ < / N ; 8 = ? 8 6 = < MdE 7 >; <#=8? 68=< MdE 7 >; < ? jB=´N 6838354N <-NTE =N 3k<>7/; <G8<?<DMu O>4==O>7FH E [email protected][email protected]; O2a^bVE 6=< bV6< 46 S83E 38E 7B3E W< 7FVfE <>?Ca-N N 38<>=cAC< ;BG? 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Q G[T \ {u}] G[T \ {v}] T \ {u} T \ {u} P T \ {v} T \{u} Q 0 P y v yv T \ {v} x, xu , x0u , yv0 , yv , y {x0u , yv0 } ∪ V (Q) T \ {v} T P∗ T \{v} xu x0u T \ {u} P P x0u yv0 x 6= xu {xu , yv0 } ∪ V (Q) 0 yv 6= y {xu , yv } ∪ V (Q) P = xu−x0u−yv0 −yv V (P ) ∪ V (Q) Q 3 G[T \ {u, v}] G[T ] w G[T \ {u, v}] Q∗ w G u v V (Q) ∪ {w} G[T ] Q G[T \ {u, v}] T \ {u, v} P P T \ {u} T \ {v} T \ {u, v} P T \ {u, v} x00 y 00 T \ {u} T \ {v} x, x00 , y 00 , y P 00 ∗ 00 00 y ∈P y u v v y u V (Q) ∪ {y 00 } x00 u x00 y 00 T \{v} x00 v V (Q)∪{x00 } 00 0 00 00 x =x V (Q) ∪ {x , y , x} y 00 = y 0 P = x − x00 − y 00 − y V (Q) ∪ {x00 , y 00 } G T P∗ z x−P −z T x−P −z x − x0 − P − z 0 − z u ∈ T 0 v ∈ T N (u) ∩ V (P ) = {x, x , z} N (v) ∩ V (P ) = {x, z 0 , z} 0 0 V (x −P −z ) ∪ {u, v, y} x−P−z 3 Q G T V (Q) ∪ {y} T x−P −z x−P −z z −P −y T P P 2 e> 6 ? B X 4 6 N 8 G 4 8 = = 4 = ¢ N ^ a E @ ? V = ; > < = E c = ; 8 G > < S 6 V L E / 7 \ ? 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(8)'/78'.(3!) -_ ,="*.( T +1.9#)+)'.%+:Le#73+ (8)K9%4"%+1.%.<,=)&952#+ G > ,=73!"%.L9%!#90+)'.%+:Je#73+b(8)K9%4"%+1.%.<,=)K9 T 0 ,1AB+1/ T ⊂ T 0 AB4"(W-+"<, > C(T 0 ) ⊆ C(T ) !)'/X#)'+X-,=(8"%+02C?[,=:0(3.h.41/(I,=73+J2#+ G 2',=)'. C(T 0 ) .<+"<,V,='.<.( 2',=)'. C(T ) r !(89t2#!)'/ T #)J+)'.<+:Le#73+(8)&9%4"%+1.%.<,=)K9t:^, (8:^,=7 uH+:^,="%$K'!)'.t$&'+p!#"`9%!#9 .%!:0:5+9 z ∈ V (G)\(T ∪C(T )) > [email protected]+)'.<+:Le#73+ N (z)∩C(T ) (8)'2;#(89Z#)'+/78(3$K'+2#+ G > /Q,=".(8)'!) > T ∪{z} .<+"<,=(89q#)5+)'.%+:Le#73+(8)K9%4"%+1.%.<,=)&9#78'.qN"<,=)'2 i )5+ # +9 > T ∪{z} .%+"<,=(89 ,=)&9(3/1!)#)'+ + 8 ##(3.%$K'+ T [email protected]+1.%9+9 z 6∈ C(T ) = +9 C(T ∪{z}) = C(T )∩N (z) #+73!)'. Y5ZPS In`&EHUVJI U 9%!#9g/<O'+:0(8) 7 P "%+78(I,=)&92#+ .%!:0:5+9%.)'!),!2 , /1+)K9%. 2#+ C(T ) +9H2#!)&9 73+1.H.%!:0:5+9%.H(8)&9%4"(3+#"%..%!)&92',=)'. V (G) \ (T ∪ C(T )) > ?[,=#" : .73+*73+:0:5+ 1 (87-)Z? % ,M-,!.2#+ /%O'+:0(8).%!"9<,=)&9g2#+73!)#N'+#"(8:0-,=(8"%+ F> + > #78'. > .( P +1.9H#)d/%O'+:0(8)o.%!"9<,=)&92#+G73!)#N'+#" -,=(8"%+ > ,=73!"%.H.P,\73!)#N'+#"+1.%9 ,= :5!(8)'. 4 i )R+ # +9 7f? #)#(3$&'+J.%!:0:5+9(8)&9%4"(3+#" z 2C? #)o4AB+)&9'+7t/%O'+:0(8)].%!"9<,=)K9 2#+g73!)#N'+#" 6 .%+"<,=(89t9%+7;$K'+ N (z)∩C(T ) )Z? (8)'2;#(89q-,!. #)'+/78(3$K'+ > /1!)K9"%+12;(3.<,=)&9 7I,\:^, (8:^,=78(89%4G2#+ T - )2;(3.%9(8)#N'+G:^,=(8)&9%+)-,=)K92#+ /Q,!. Y a K I¦ \RJ3JG# G/Ia YZPSTI `&EHUVJI U § r !(89 #) ' o + = , 8 ( % " ] + C 2 [ ? = , 0 : 3 ( . . 1 4 / I ( = , 3 7 ] + 2;N"<,=#O'+ G[C(T )] u ,=#+73!)'.$K'+ {a, b} G[C(T )] )[email protected]+1.9\-,!.\#)'+/78(3$&'+/1+V$&#((8:0#78(3$K'[email protected]+ (3.9%+)'/1+2C? #)'+9%+7873+V-,=(8"%+ > -,="JO % p!9O : .%+b2#+5"%41/#""%+)'/1+V.( G[C(T )] )[email protected]+1.9L-,!.L#)'+5"%4#)#(3!)c2;(3. !(8)K9%+b2#+ /78(3$K'+1. > 9"(8AF(I,=73+:5+)&9.( G[C(T )] +1.9*#)'+h"%4#)#(3!)2;(3. !(8)K9%+G2#+G/78(3$K'+1. r !(89 qr P #) /%O'+:0(8)D2#+ G [email protected]+ 9"%4:0(89%41. a +9 b ( P /1!)K9(3+)&90#) .%!:0:5+9 qr t ∈ T ,=73!"%. P = a − t − b > +9 P +1.9J2#+^73!)#N'+#" 2 ( V (P ) ∩ T = ∅ > ,=73!"%. > !)'/ V (P ) ⊂ C(T ) > /Q,=" .%(8)'!) > P /1!)&9(3+)'2;"<,=(89h#)o/%O'+:0(8)].%!"9<,=)&9 P +1.9M2#+ 73!)#N'+#"0-,=(8"%++9 {a, b} +1.%9X#)'+b-,=(8"%+2C?[,=:0(3.02#+ G L> +b#78'. > .Q? (87+ (3.%9%+b#) $K-,!.( 7 #"(3.:5+T.9"(3/9 S [email protected]+ 9"%4:0(89%41. a +9 b 2',=)'. G > ,=73!"%. V (S) ⊂ C(T ) /Q,=" .(8)'!) > 9%!'.H73+1. .%!:0:5+9%.2#+ S .%+G9"%!#A!,=)K9.#"H#)/%O'+:0(8)R"%+78(I,=)&9 a +9 b > (87 +) "%41.#789%+"<,=(89$&'+ S /1!)&9(3+)K9 #)d/<O'+:0(8).%!"9<,=)K9 no,=(3.H/1+7I,0/1!)K9"%+12;(89$K'+ {a, b} +1.9\#)'+-,=(8"%+T2C?[,=:0(3.5.%p41/(I,=73+2#+ G[C(T )] o (8)-,=73+:5+)&9 > {a, b} +1.%9Xe#(3+) #)'+ -,=(8"%+G2C?[,=:0(3. .41/(I,=73+h2#+ G aK Z S `U,ZpGFIY5ZPS I `EHUVJI U § r !(89 #) T # 8 7 '. 1 / ! #" 9 /<O'+:0(8).%!"9<,=)K9 r ,X73!)#N'+#"H+1.9 n + 1 α−z −· · ·−z −β - !9%!)'.5$K1'+T2C?[,=#" : n.^/1+T$K#(*#"%41/ : 2#+ )D)'!9%+ Z = > n +1.%90(8:0-,=(8"^+9 n ≥ 3 - )d2#4sW')#(89 Y z −· · ·−z 1 n A = {v ∈ C(T ) | vz1 ∈ E(G), vzi 6∈ E(G) (i = 2, . . . , n)}, B = {v ∈ C(T ) | vzn ∈ E(G), vzi 6∈ E(G) (i = 1, . . . , n − 1)}. - !9%!)'.G$K'+ A +1.9h)'!)]AF(32#+ > /Q,=" α ∈ A > +9G$K'+ A (8)'2;#(89L#)'+0/78(3$&'+ > /Q,=" H tD g d 1> 2 14 ,="2#4sW')#(89(3!) A ⊆ N (z1 ) ∩ C(T ) +G: :5+ > B (8)'2;#(89#)'+G/78(3$&'+L)'!)dAF(32#+ > > +b#78'. (87g)Z? % ,d-,!.X2C?[,=" 2 9%+ Q , B A 1 + / Q / = , \ " = , 3 7 ! % " . > A∩B = ∅ uv u ∈ A, v ∈ B > {u, z1 , . . . , zn , v} (8)'2;#(8"<,=(89H#)T9"%!d(8:0-,=(8" - !'. ,=7873!)'.:^,=(8)K9%+)-,=)&9t:5!)&9"%+" -,="73+1.73+:0:5+1. j jAd $K'+2#+1.`.%!:0:5+9%. B -,="9(8" a ∈ A > b ∈ B e#(3+)]/<O'!(3.(3. S !":5+)K9#)'+J-,=(8"%+\2C?[,=:0(3.M.p41/(I,=73+X2#+ G 2C? (3/( > 73+1.t"<,=(3.<!)#)'+:5+)&9%..%!)K9,!.%.%+ g#"%K/<O'+1.2#+g7I,#"%+#AB+*2;0/Q,!.q.<,=)'.t/Q,=""%4 ')(*$+, Kco8 C(T ) ∩ N (Z) ⊆ A ∪ B ∪ C(T ∪ A ∪ B) § r & ')( *,+( !(89 w ∈ C(T ) ∩ N (Z) & 7p+ (3.9%+H2#!)'/#)'+M,=" 2 9%+ z w ,QAB+1/ z ∈ Z j j 8 =r ##.%!)'. Y 1 ≤ j ≤ n w ∈ / C(T ∪ A ∪ B) (87 + (3.9%+2#!)'/d#).%!:0:5+9 gr u ∈ A ∪ B ,1AB+1/ uw 6∈ E(G) ,=)'.^p+"9%+T2#+NB4)'4"<,=78(89%4 > !)D+#95.##.%+" Fr F u ∈ A !(89 i 73+*#78'.p+9(89g+)&9(3+" i 9%+7-$K'+ zi w ∈ E(G) 73!"%. u−z1−· · ·−zi−w +1.9t#)5/%O'+:0(8)^.%!"9<,=)K92#+73!)#N'+#" i+1 > 2#!)'/*!)52#!(89,QAB!(8" i = n > +9 w ∈ B 2 GFIR YZPSTI¥ Z0KEIK¤G1ZG/H S P JSTH [email protected] P = u−u0 −· · ·−v 0 −v ¨ K EH ` 0 ZU EG 0 u ∈ A v ∈ B§ u ∈A v ∈ B§ - !9%!)'.J$&'+ & ')( *,+( 1 + . L 9 # 2 ^ + 73!)#N'+#"X,=a:5!(8)'. 3 /Q,="J(87)Z? % ,,='/#)'+ P 2 L > ,=" 9%+T+)&9"%+ A +9 B +#78'. P )'+T/1!)&9(3+)K9V,='/#) .<!:0:5+952#+ T )#(H,='/#) .%!:0:5+9H2#+ C(T ∪ A ∪ B) 8 .%(8)'!) > P .%+"<,=(892#+h73!)#N'+#" 6q=pr ##p.%!)'.H$&'+M73+ _73+:0:5+ jAj .%!(89MSf,= 6Y u0 ∈ / A +9 v 0 ∈ / B !'.L,=7873!)'.hAB!(8"L$K'+0/1+7I,T+)K9"<, 3)'+ #)'+h/1!)&9"<,!2;(3/9(3!) 8P« Za`@`.ZGFK?Za` JHU,Za`ZI U H Z Z&U `EI U ZU Z P ∩ C(T ) z 1 u zn v § i )c+ # +9 .%( 2 zw +1.9L#)'+V,=" 9%+V,QAB+1/ z ∈ Z +9 w ∈ P ∩ C(T ) > ,=73!"%. > ##(3.%$K'+ > "%+:^,="%$K'4 +92C?[,=#" : .*73+M73+:0:5+ j !) > > w∈ / C(T ∪ A ∪ B) /1!:0:5+G!)T7f?[,\2#4 Y B 4 , w ∈ A ∪ B #(3.<$&'+ A +1.9M#)'+X/78(3$K'+ > 73+\/Q,!. w ∈ A +)K9"<, 3)'+ w = u 8 +9 2#!)'/ zw = z u /1!:0:5+0.%!#O-,=(89%4 = ! w = u0 8 +9G2#!)'/ u0 ∈ A > /Q,!.h+ /78o-,=" 1 O % p!9O : .%+ = ! +G/Q,!. w ∈ B +1.9.%(8:0(87I,=(8"%+ R_ +1/(Z#"%!#AB+ 8N<= )G:^,="%$K'+73+1.Z.%!:0:5+9%. 2#+ +)G-,="9(3/#78(3+"`73+1. P $K#(&!)K9Z#)LAB!(3.(8)J2',=)'. Z S B I , U Z H 9 J K ? K Z .%!:0:5+9%.2#+ V (P ) ∩ V (Z) .%!)K9:^,="%$K'41. #p+73!)'. 2#+ P 9%!#9g.%!'. 7 /%O'+:0(8)2#+ P > 2#+h73!)#N'+#",=:5!(8)'. 1 > 2#!)&9 73+1.+ 9"%4:0(89%41. .%!)K9:^,="%$K'41+1.H+9 2#!)&973+1..%!:0:5+9%.(8)K9%4"(3+#"%.)'+73+.%!)&9-,!. q> ?[,=#" : . 8N<= > 73+1..%!:0:5+9%.:^,="%$K'41. 2#+ V (P )\{u, v} .%!)K9t9%!'. 2',=)'. V (P )\C(T ) 94 #(3.%$K'+ u +9 v .%!)&9t:^,="%$K'41. > 73+1. (8)&9%+"A,=7873+1.2#+ P SU!":5+)&9*#)'+M-,="9(89(3!)2#+ P ,=T.%+)'.2#+1.,=" 2 9%+1. uH+:^,="%$K'!)'. $&Z?[,=R:5!(8)'.H#)o.%!:0:5+9M(8)&9%4"(3+#"h2#+ P 2#!(89 2 9"%+L:^,="%$K'4 > /Q,=".%(8)'!) V (Z) ∩ > !)'/ E9` ` $¥ Z;JG P ESTI5` 5 V (P ) = ∅ +9 V (Z)∪V (P ) (8)'2;#(8"<,=(89t#)X9"%!J(8:0-,=(8" P ¥ ZG STI U,ZH $JKTKZ`> +H#78'. -,="73+ 73+:0:5+h2#+ u !'.%.%+7+9uH#e#(3 8 73+:0:5+ 1 F= !) > > E9`a`%$¥ ZGFI IBEPRQH ZpS P JSTHL¥ Z N JH&U,Za` > !)'/ Y .<,=(89$&'+ P T § 81« a K"ZS `U,ZTq¥ JI1` GFI STI U,ZH $JKTKZ G/S EIBU SWZIBU GFI IBEP QH[Z STP JSTHV¥ Z P Q T JH&U,Za` § - !9%!)'.M$K'+\/1!:0:5+ : P .%. 2#+\,=o:5!(8)'.h2#+ (8)&9%+"A,=7873+1. > Q )'+\/1!)&9(3+)K9 # r #r -,!. 7I,MSU!(3. u +9 v ,=)'.+"9%+2#+HNB4)'4"<,=78(89%4 > !).##p.<+ v ∈ / V (Q) !(3+)&9 w KcoI ^ ESTU d9 +9 x 73+1.q+ 9"%4:0(89%41.2#+ Q #+73!)'. w0 8 "%+1. x0 = > 73+*.%!:0:5+9q2#+ Q ,=#-,="9%+)-,=)&9 8 =' )T+#9.%##p.%+" C(T ) +9*/%O'!(3.(,='.<.(#"%K/<O'+h$&'+.%.(8e#73+M2#+ w "%+1. x $K'+ u > w > w0 > x0 > x > +9 v ,=#-,="<,=(3.%.%+)K9X2',=)'.X/1+9J!"%2;"%+b.#" P )6, w0 6= x0 /Q,=" Q /1!)&9(3+)K9J,=o:5!(8)'.M#)'+ T 7 ,=" 2 9%+ _4 !#"G$K'+X9%!#9G.%!(89e#(3+)/7I,=(8" > )'!9%!)'. $K'+ V (Q) ∩ V (Z) = ∅ /Q,="J.(q#)a.%!:0:5+9 Q 49<,=(89L2',=)'. Z (87.%+"<,=(89 > 78#(g,=(8)'.%( $K'+o.%+1.VAB!(3.(8)'.T2',=)'. Q > :^,="%$&'4 > !)'/o!),=#"<,=(89 Q = wx = w0 x0 no,=(3. ,=73!"%. > 7f? #) 2#+ w0 , x0 .%+"<,=(89 7I,dSU!(3.02',=)'. Z +902',=)'. C(T ) > 2#+ +)'.<+:Le#73+1. -,="^2#4sW')#(89(3!) 2;(3. !(8)&9%. !9%!)'.b,='.%.%($&'+T.( w0 = w > ,=73!"%.^/1+T.%!:0:5+9b+1.9 2',=)'. P ∩ C(T ) ∩ N (Z) +9^2#!)'/ > 2C?[,=#" : . 8N<= > w 0 = w = u > ?[,=#9"%+-,="9 > : 8N<= x0 = x +1.9(8:0p.%.%(8e#73+H/Q,=" x .%+"<,=(89,=73!"%.2',=)'. N (Z) ∩ C(T ) > /1+ $&#(p2C?[,=#" . Y Z +)K9"<, 3)'+"<,=(89 v = x #)'+J/1!)K9"<,!2;(3/9(3!) 7 [email protected]+)'.#(89$&'+ Q +1.9 2#+J73!)#N'+#"G,= :5!(8)'. 2 > ?[,=#" : .h7I,V2#4sW')#(89(3!)a2#+1.h.<!:0:5+9%.M:^,="%$&'41. (87q+ (3.9%+X#).%!'. 7 /<O'+:0(8) Z0 > 2#+ Z [email protected]+ 9"%4:0(89%41. z +9 z > 9%+7$K'+ z w +9 z x .%!)&9G2#+1.G,=" 2 9%+1. 8 i < j > i = j i j i j ! i > j .%!)&9Xp.<.(8e#73+ =t )6/%O'!(3.%(89 Z 0 :0(8)#(8:^,=7,QAB+1/b/1+1.X#"%!#"(349%41. > /1+b$K#( Sf,=(89$&'+h.<+1.*4AB+)&9'+73.H.<!:0:5+9%.*(8)&9%4"(3+#"%.H:^,=)'$&'+)K9 w +9 x /4 ,="H/1!)'.%41$K'+)&9 > - !9%!)'.$KZ? (87C)Z? % ,X-,!. H = G[V (Q) ∪ V (Z 0 )] +1.9*#)T9"%! > 4AF(32#+:0:5+)&9H-,=(8" 2#+.%!:0:5+9%.g2#+ C(T ) 2',=)'. V (H) \ V (w 0 −Q−x0) > -,="2#4sW')#(89(3!)T2#+ w0 +9 x0 +9 -,="%/1+*$K'+ Z ⊆ V (G)\(T ∪C(T )) i )\Sf,=(89 > 73+1. T 7 ,=" 2 9%+1.2#+ H .%!)K9+ ,!/9%+:5+)K9 73+1. T 7 ,=" 2 9%+1. 2#+ Q r ( 0 +9 0 )'+5.<!)&9G-,!.G2#+1.J.%!:0:5+9%.L/1!)'.<41/#9(S .J2#+ : w x P > ,=73!"%. H p.<. 2#+ : 2#+ .%!:0:5+9%.)'!)V,!2 ,!/1+)K9%.2#+ C(T ) +9q.%. 2#+!#"9<,=)&9#)0)'!:Le#"%+(8:0-,=(8" 2#+ T 7 ,=" 2 9%+1. Y #)'+/1!)&9"<,!2;(3/9(3!)d,=V73+:0:5+ 1AdF> !)'/ w0 +9 x0 .%!)&9*/1!)'.%41/#9(SU. .#" P - !9%!)'. 8 =C )]2#4sW')#(89h#)]/%O'+:0(8) k = max{i, j} k ≥ 1 Y +)R.<,=)K9 Y = q zk+1−Z−zn−v .%( k < n +9 Y = v .( k = n !9%!)'.$K'+ H ∪ Y +1.9/1!)#)'+ +/Q,=" 1 !'.#"%49%+)'2#!)'.$&'+h/<O-,!$&'+G.<!:0:5+9 zk +1.9*#).%!:0:5+92#+ H ,!2 ,!/1+)&9 Y # i # 2 :5+h:^,=)'$K'+G/1+1.2#+ .%!:0:5+9%.H/Q,=" ^ : = , ' ) K $ '+ + 9 d ) + + 9 8 7 #( Z : > v z∈Y w0 x0 w0 , x0 > x > v .%!)K9J$K-,=9"%+^.%!:0:5+9%.X2;(3.9(8)'/9%.\,=#-,="<,=(3.%.<,=)K9X2',=)'.X/1+9J!"%2;"%+b.#" 5> + P +9*.( z ∈ V (zk+1 −Z−zn ) > ,=73!"%. z :^,=)'$K'+ x0 /Q,=" x0 )[email protected]+1.9-,!.:^,="%$&'4 #78'. > .( z AB!(89 w0 > ,=73!"%. w0 +1.9h:^,="%$K'45+9 w0 = w > +9G2#!)'/ w0 ∈ P ∩ C(T ) no,=(3.L,=73!"%. > /1!:0:5+ z 6= z +9 w0 6= v > 7f?[,=" 2 9%+ zw0 /1!)&9"%+12;(89 8N<=> !)'/5(87+1.9 1 e#(3+)TAF"<,=(C$&'+ z :^,=)'$&'+ w0 +9 x0 o (8)-,=73+:5+)&9 > 73+M9"(8#73+9SU!":54M2;9"%! H > 2; /<O'+:0(8) Y +9 2#[email protected]+)'.%+:Le#73+ T /1!)K9"%+12;(89 73+L73+:0:5+ 1 ##(3.%$KZ? (87)Z? % ,0-,!.2#+ .%!:0:5+9 2#+ C(T ) 2',=)'. H \ {w 0 , x0 } 2 4 !#"%.#(8AB!)'.g7I,#"%+#AB+H2;^9O'41!" : :5+ jA6K )^2#4sW')#(89#)'+*"%+7I,=9(3!) <A .#" A +).<,=)K9 u < u0 .(Z+9.%+#73+:5+)K9 .Q? (87Z+ (3.9%+h#)/<O'+:0(8)2#+h73!)#N'+#" (8:0-,=(8"%+ A 2#+ u AB+"%.#)d.%!:0:5+9H2#+ B 9%+7$K'+ u0 +1.9*73+h2#+ ( : :5+G.%!:0:5+9H2#+h/1+h/<O'+:0(8) - !'.H,=7873!)'.:5!)K9"%+" $K'+ 8 = < +1.%9*#)'+h"%+7I,=9(3!)[email protected]!"%2;"%+ -,="9(3+7 A Za`UiJIBU S `\9P^ U HS G5Z <A § r ##p.%!)'. +)dAF'+L2C? #)'+J/1!)&9"<,!2;(3/9(3!)$KZ? (87+ (3.%9%+L2#+1.H.<!:0:5+9%. & ')(*,+( _> u, v ∈ A 9%+73.h$K'+ u <A v +9 v <A u !)'/\(87t+ (3.9%+X#)/%O'+:0(8)c2#+J73!)#N'+#" KaaRC8 J3H ZK JqU S EI H g tD dj (8:0-,=(8"%+ P = u −· · ·−u ,1AB+1/ u = u > v = u > u ∈ B > p ≥ 3 > p (8:0-,=(8" > +9(87 u 0 p 0 1 p + (3.9%+M#)/%O'+:0(8)2#+G73!)#N'+#"H(8:0-,=(8"%+ P = v −· · ·−v ,QAB+1/ v = v > u = v > v 0 q 0 1 p )d+#9 ,QAB!(8" vq ∈ B > q ≥ 3 > q (8:0-,=(8" up = vq > +9 .(8)'!) up AB!(89 vq ##(3.<$&'+ B +1.9*#)'+G/78(3$K'+ 8P« ¨ G; G/I `EPPRZUg¥ Z I¦ Za`Ugq¥ JI5` V (Pu ) ∪ V (Pv ) \ {u, v} A§ i )+ # +9 #)d.<!:0:5+92#+ ^ : = , ' ) K $ '+ X = , :5!(8)'.7f? #)o2#+ u > V (Pu ) ∪ V (Pv ) \ {u, v} _ 2 +9 v > +9*)'+h+#92#!)'/ 9"%+G2',=)'. A $K#(Z+1.9*#)'+G/78(3$K'+ +1/(Z#"%!#AB+ 8N<= 81« ¨ G;G/I `EPPRZUg¥ Z IC¦ Z`U¥ JI5` V (Pu ) ∪ V (Pv ) \ {up , vq } B§ i )+ # +9 ##(3.%$&'+ 1 + . h 9 #) ' 5 + / 8 7 3 ( K $ '+ #) % . ! 0 : 5 : + L 9 # 2 + B (Pu ∪ Pv \ {up , vq }) ∩ B > > no,=(3. .%( )'+G+#9 $KZ? 2 9"%+ u ! v = , 3 7 ! "%. u −P −u +1.9H#) > up−1 ∈ B p−1 q−1 1 u p−1 /%O'+:0(8)]2#+L73!)#N'+#"M(8:0-,=(8"%+\2#+ A AB+"%. B > +9 73+L73+:0:5+ jAj (8:0#78(3$K'+X!Re#(3+) 8 8N<=[= !e#(3+) 8 u2 ∈ A /1+X$&#( /1!)&9"%+12;(89 up−2 ∈ B /1+J$K#(`/1!)K9"%+12;(89h$&'+ B +1.9 _ = 1 ! +1.9 .(8:0(87I,=(8"%+ +1/(Z#"%!#AB+ 8W6q= #)'+h/78(3$&'+ +G/Q,!. v q−1 ∈ B r !(89 r 73+h#78'.+9(89H+)&9(3+" 9%+7$K'+ ur ∈ V (Pu ) \ {u0 , u1 } ,\#)AB!(3.%(8)d2',=)'. 2 Pv > +9h.%!(89 s 73+L#78'.+9(89M+)K9(3+"h9%+7 $K'+ ur vs +1.9M#)'+X,=" 9%+ > ,1AB+1/ 2 ≤ s ≤ q > +9%+73.+)&9(3+"%.+ (3.%9%+)&9##(3.<$&'+ 7 2 vq 78#( : :5+ ,h#)^AB!(3.(8)^2',=)'. Pu !9%!)'.$&'+ 73+1..<!:0:5+9%. u , . . . , u , v , . . . , v (8)'2;#(3.%+)K9h#)9"%! H > +9M$&'+ r +9 s !)&9h2#+ 1 r 1 s /1+SU,=(89*: 2 :5+L-,="(89%4 8«iI^5 ZGU `aGB E9`.ZH G1ZEG QSWZIa© ª J« ªQ« r=p ] ZU s=q r<p s<q Z&U ] ur+1 −Pu −up = vs+1 −Pv −vq § r !(89 73+h#78'.*N"<,=)'2d+)&9(3+"H9%+7`$&'+ t ur vt ∈ E(G) > ,1AB+1/ 2 ≤ s ≤ t ≤ q r ( t − s +1.9-,=(8"H,=73!"%. u1 −Pu −ur −vt −Pv −vq +1.9#)/%O'+:0(8)2#+M73!)#N'+#" (8:0-,=(8"%+2#+ A AB+"%. B !r !)X2#+")#(3+"t.%!:0:5+9t+1.9 u > +9 .%!)0,1A,=)&9 7 2#+")#(3+".%!:0:5+9 8 = 8 = > ?[,=#" : 2.h73+\73+:0:5+ jAj ,=##78(3$K'4 /1+ w +1.9 vq−1 .%( t < q ! ur .( t = q > /%O'+:0(8) > )'!'.,1AB!)'.*!e#(3+) u ∈ A 8 :^,=(3.*/1+7I,X/1!)&9"%+12;(89 8N<=[= !e#(3+) w ∈ B > ?[,=#" : . 8W6q= /1+h2#+")#(3+"/Q,!.*)[email protected]+1.29*.%.(8e#73+M$&'+h.%( 8 = w = ur = up +92#!)'/ t = q > ,=)'.h/1+0/Q,!. > v −Pv −vs −up +1.%9h#)/%O'+:0(8)a2#+\73!)#N'+#"G(8:0-,=(8"%+^2#+ A AB+"%. : .L731+^73+:0 + J 9 C 2 [ ? = , # " :5+ jAj > !) 2#!(89X,QAB!(8" v ∈ A 8 :^,=(3.L/1+7I,d/1!)&9"%+12;(89 8N<=[= B> 2 _ ! v ∈ B +\2#+")#(3+"M/Q,!.M)[email protected]+1.%9 .%.(8e#73+X$&'+\.( v = v +97I,b/1!)'/78'.(3!) 8 , = s s q 2#+ 8Wq= +1.9*e#(3+)d.<,=9(3.SU,=(89%+ r ( t − s +1.%90(8:0-,=(8"b,1AB+1/ t ≥ s + 3 ,=73!"%. v1 − Pv − vs − ur − vt − Pv − vq +1.9H#)o/%O'+:0(8)2#+J73!)#N'+#"M(8:0-,=(8"%+X2#+ A AB+"%. B pr !)o2#+ ( : :5+\.%!:0:5+9M+1.9 7 8 = 8 = v2 > +9V.%!) ,1A,=)&9 2#+")#(3+".%!:0:5+9 w +1.9 vq−1 .%( t < q ! ur .( t = q > ?[,=#" : . 73+h73+:0:5+ jAj ,=##78(3$K'4 8 /1+L/%O'+:0(8) > !)2#!(89M,1AB!(8" v ∈ A :^,=(3./1+7I, 2 /1!)&9"%+12;(89 8N<=[= ! w ∈ B > ?[,=#" : . 8W6q= /1+2#+")#(3+"b/Q,!.0)[email protected]+1.%95p.<.(8e#73+$K'+.( 8 = > !)'/ w = ur = up +9M2#!)'/ r = p +9 t = q r +9 t .%!)K9(8:0-,=(8"%. > /1+X$K#( +1.9 (8:0p.%.%(8e#73+h/Q,=" t − s +1.9(8:0-,=(8"H+9 r − s +1.9*-,=(8" ! +.<+#7K/Q,!.`"%+1.9<,=)&9t+1.9 t = s+1 > /1+$K#(&+)K9"<, 3)'+g$K'+ u0−Pu−ur−vs+1−Pv−vq +1.9#)/<O'+:0(8)R2#+L73!)#N'+#" (8:0-,=(8"%+ > $K#(+#9 !'+" 73+G" !73+J2#+ P _ +99%+LS !(3. > u d* 7I,0/1!)'/78'.%(3!) 8 e = 2#+ 8Wq= +1.9.P,=9(3.mSU,=(89%+ R_ +1/(C#"%!#AB+ 8Wq= ) =, #p+7873+ STI U,ZH $ JKTK?Z 9%!#9J/%O'+:0(8)c2#+ H > 2#+073!)#N'+#"J,=a:5!(8)'. 1 > 2#!)K9 73+1.+ 9"%4:0(89%41.AB!(3+)K9 z +92#!)&9H73+1.H.%!:0:5+9%.*(8)&9%4"(3+#"%. :^,=)'$K'+)&9 z 1 1 8!« Za`S I U,ZH $JK K?Za`^¥ Z `&EI U JGIBEPRQH Zb¥FZ JGP ESTI5` Z&U STK)`"MEHaPRZI U GFI Z H JHU S U S EIo¥ Z JG `.ZI5`L¥ Za` JHU,Z` § H r !(89 YFr ( 7I,G/1!)'/78'.(3!) 8 , = 2#+ 8Wq= +1.9g.<,=9(3.mSf,=(89%+ #) % / ' O + 0 : 8 ( ) # 2 s 4 ' W # ) p ( # 2 H + I 7 G , % . ! " % 9 + P > K r !)bp.%+ P = z −Z−z (-7I,L/1!)'/78'.(3!) 8 e = 2#+ 8Wq= +1.9.P,=9(3.mSU,=(89%+ > !)b2#4sW')#(89 P 2 n /1!:0:5+ 49<,=)&9#)V/%O'+:0(8)[email protected]+ 9"%4:0(89%41. z +9 u > /1!)&9%+)K2',=)'. {z2 , . . . , zn } ∪ 8 + (3.%9%+J/Q,="[email protected]+)'.%2+:Je#73r+1 5 + ]. %+1.M.%!:0:5+9%.h.%!)&9M9%+)K'.G2#+ {ur+1 , ur+2 , . . . , up } P = ' .%+H9"%!#AB+"*+1.9/1!)#)'+ + )b"%+:^,="%$K'+ $K'+ 73+73+:0:5+ 1 +#9.?[,=##78(3$&'+",= 9"(8#73+9 H, P, {z } Y H +1.9M#)]9"%! > P +1.%9h#)o/<O'+:0(8) > {z } (8)'2;#(89h#)]N"<,=#O'+ 1 1 ,=)K9(3/1!)#)'+ + 2;(3. !(8)K92#+ H ∪ P > G[H ∪ P ] +1.9/1!)#)'+ + > 9%!#9.%!:0:5+9 z ∈ V (P ) :^,=)'$K'+ u +9 v 8 .( z ∈ Z[z , z ] -,="%/1+L$K'+ u, v ∈ A > .( z ∈ V (u 2 n r+1 −Pu −up ) -,="%/1+X$&'+ P +9 P .%!)K92#+1.M/%O'+:0(8)'.M+9 r, s ≥ 2 = > +9+);W') u, v, z .%!)&9M2',=)'. 2 > !u)'/73+v73+:0:5+ 1 :5!)&9"%+d$&Z? #)D.%!:0:5+952#+ C({z1 }) V (H) \ {u, v} +1.%9 2',=)'. C({z }) = N (z ) > !)'/ z p.<. : 2#+0,=]:5!(8)'. 3 AB!(3.(8)'.G2',=)'. H D_ +1/( 1 1 1 #"%!#AB+ 8TF= 8s« E` `%$%¥ Z GFI IBEPRQH ZJSTHL¥ Z NJH&U,Za` § H T u,=#p+73!)'. $&'+ u +9 v .%!)&9 2',=)'. C(T ) !r (&7I,H/1!)'/78'.(3!) 8 , = 2#+ 8Wq= +1.9`.P,=9(3.mSU,=(89%+ > ,=73!"%. u +9 v .<!)&9*2',=)'. C(T ) 'r ( 7I,X/1!)'/78'.(3!) 8 e = 2#+ 8Wq= +1.9.<,=9(3.mSf,=(89%+ > ,=73!"%. r s !)^,=##78(3$K'+73+73+:0:5+ 1Q ,=09"%! H > ,=5/<O'+:0(8) u −P −u +9 [email protected]+)'.%+:Je#73+ u p _ /Q,='.%+\2#+1.h,=" 2 9%+1. 1Q (8r+1 :0#78(3$K'+X$KZ?[,=o:5!(8)'.M7f? #) T uv +9 ur vs > 73+X73+:0:5+ 2#+ u +9 v +1.%9L2',=)'. C(T ) > !)'/ > $K'+07I,T/1!)'/78'.(3!)62#+ 8Wq= .<!(89 8 , = ! 8 e = > r s !)6+#9\, 5":5+"5$K'+ H /1!)&9(3+)&9^,=6:5!(8)'.\9"%!(3.\.%!:0:5+9%.02#+ C(T ) 4 ,="073+ 73+:0:5+ 1Ad > !)d!e#9(3+)&9 7I,\/1!)'/78'.(3!).%!#O-,=(89%41+ R_ +1/(Z#"%!#AB+ 8Wjq= ,="9(8"2C? (3/( > !)p+#9/1!)'.#789%+" 7I,\W'N#"%+ jA uH+:^,="%$K'!)'. $K'+ u−v +1.%9 #)(8)&9%+"A,=7873+X$K#(/1!)K9(3+)&9M#)'+ T 7 ,=" 2 9%+ B> !)'/ > 2C?[,=#" : . 8TF= +9 8Wjq= > (87 + (3.9%+L#) (8)K9%+"A!,=7873+ Q 2#+ H > 2;( # 4"%+)&9X2#+ u−v > $K#(/1!)&9(3+)K9L#)])'!:Je#"%+5(8:0-,=(8"L2#+ T 7 ,=" 2 9%+1. - )d,=#+7873+ x +9 y 73+1.+ 9"%4:0(89%41.H2#+ Q > +9*!)d,=#p+7873+ x0 73+h.%!:0:5+9H2#+ Q ∩ C(T ) 73+#78'.g#"%K/<O'+h2#+ x +9 y 0 73+ .%!:0:5+9*2#+ Q ∩ C(T ) 73+#78'.#"%F/%O'+h2#+ : 4 y > 2#+:^,=)#( "%+$K'+ u, x, x0 , y 0 , y, v ,=#-,="<,=(3.%.%+)K9g2',=)'.g/1+9g!"%2;"%+.#" H #(3.<$&'+ H /1!)&9(3+)&9M,=:5!(8)'.*9"%!(3.*(8)&9%+"A,=7873+1. > !)T+#9.##.%+"*$&Z?[,=T:5!(8)'.7f? #)d2#+ # x +9 y > -,=" + +:0#73+L+9H.P,=)'.+"9%+L2#+GNB4)'4"<,=78(89%4 y > +1.9H2;( 4"%+)&9M2#+ u +92#+ v 4 ,=" /1!)'.%41$K'+)&9 > y :^,=)'$K'+h7f? #)d2#+ u +9 v r ( 1 + . * 9 # 2 + 3 7 !)#N'+#" 1 > !), Q = x−y = x0 −y 0 > +9 y ∈ N (z ) ∩ C(T ) /! + Q 1 j 73+:0:5+ (8:0#78(3$K'+ y ∈ A ∪ B ∪ C(T ∪ A ∪ B) no,=(3. y ∈ A ∪ C(T ∪ A ∪ B) +1.9^(8:0p.%.%(8e#73+T/Q,=" y :^,=)'$K'+7f? #) 2#+ u +9 v > +9 y ∈ B +1.95(8:0.%.(8e#73+T/Q,=" =1> !)'/ y ∈ N (z1 Q +1.92#+G73!)#N'+#" ,=:5!(8)'. 2 > V (Q) ∪ {z1 } (8)'2;#(89H#)T9"%! 7 2 1Ad ,=##78(3$&'4 7I, H1 /1!)&9%+)-,=)K9G#)o)'!:Le#"%+\(8:0-,=(8"h2#+ T ,=" 9%+1. > +9 73+X73+:0:5+ -,=(8"%+ H , T (8:0#78(3$K'+G$K'+ x0 −Q−y 0 +1.92#+M73!)#N'+#" 1 5> !)'/ x0 +9 y 0 .%!)K9*73+1. 1 .%+#73.H.%!:0:5+9%.H2#+ C(T ) 2',=)'. H 1 H g tD dq( Z z2 z1 zn vs ur+1 = vs+1 y y0 x ur 0 x v = v0 = u1 up = vq u = u0 = v1 eifTg"hqjA k ! + :0:5+ jA* .%!:0:5+9%. ! x > x0 > y 0 + 9 y +1..%!:0:5+9%. jk#73+(8)'.l.%!)K92',=)'. > C(T ) Z z1 z2 zn vs = y 0 ur y v = v0 = u1 u = u0 = v1 = x = x0 up = vq eifTg"hj ;[email protected]! +:0:5+ jA* W')02#+g7I,H#"%+#AB+ ! +1.t.%!:0:5+9%.`jk#73+(8)'.mlg.<!)&9q2',=)'. > C(T ) dd r ##.%!)'. 5! 1 ,=##78(3$K'4L,=9"%! x 6= u +M73+:0:5+ H1 > ,=/<O'+:0(8)"%412;#(89 + 9 f 7 @ ? + ' ) % . + J : # e 3 7 + 5 : ! & ) 9 % " M + K $ '+ # 2 ! 8 ( 9 B A ! 8 ( " 1 / + K $ #( 8 ( :0#78(3$K'+ x = x0 = v u T u x0 > 2 )o,^,=73!"%. j 1 > !)'/ x ∈ N (Z) ∩ C(T ) > /1+J$&#(/1!)K9"%+12;(89M73+G73+:0:5+ x = u +9 /1!:0:5+ x +1.92',=)'. C(T ) > x0 = x = u ; )T,=##78(3$&'+:^,=(8)K9%+)-,=)&9*73+ 73+:0:5+ 1 ,=9"%! H > ,=/<O'+:0(8) u −z −Z −z > +9 [email protected]+)'.<+:Le#73+ T _ +1/(:5!)K9"%+J$KZ? (87 p n 2 + (3.9%+b#) 1.%!:0:5+9\2#+ q_ up − zn − Z − z2 $K#(AB!(89X7f? #) 2#+ x0 +9 y 0 +V.%!:0:5+9 )'+\+#9h-,!. 2 9"%+\#) z , (i = 2, . . . , n = > /Q,="G/1+1. z :^,=)'$K'+)&9 x0 = x = u -,=" i i 2#4sW')#(89(3!)a2#+ Z +9h:^,=)'$K'+)&9 y 0 /Q,='.%+02;]73+:0:5+ j 1_ +0.%!:0:5+9G)'+\+#9 2#!)'/ 2 9"%+M$K'+ u > /1+$K#(C+)&9"<, 3)'+ s = 2 +9 u AB!(89 v #_ +7I,X)[email protected]+1.9p.%.%(8e#73+$K'+ p p s .( s = q − 1 ! s = q nR,=(3. s = q +1.9 (8:0p.%.%(8e#73+L/Q,=" !)o,=#"<,=(89M,=73!"%. x0 ∈ A > 2 Y y 0 ∈ B +9 x0 y 0 .%+"<,=(89 #)'+,=" 9%++)K9"%+#)\.%!:0:5+9t2#+ A +9 #)\.%!:0:5+9 2#+ B #)'+ /1!)K9"<,!2;(3/9(3!) F> !)'/ s = q −1 -,="9(8"2#+:^,=(8)&9%+)-,=)&9 > 7I,M.(89-,=9(3!)5"%+1.<.%+:Le#73+ #78#9!9 /1+7873+"%+#"%41.%+)K9%41+W'N#"%+ j 1<! +/<O'+:0(8) u−z −y−P −u +1.9`2#+73!)#N'+#" 1 u r -,=(8"%+J/Q,=" H +1.9#)9"%!d-,=(8" 1> !)'/h73+G/%O'+:0(8) u−z −y−P −u −u +1.9#) 1 1 u r p /<O'+:0(8)52#+73!)#N'+#"t(8:0-,=(8"%+ > "%+78(I,=)&9q2#+ .<!:0:5+9%.q2#+ C(T ) +9 )'+/1!:0!"9<,=)&9 ,='/#)'+ T 7 ,=" 2 9%+ Y #)'+h/1!)&9"<,!2;(3/9(3!)],=73+:0:5+ 1 1 2 KaaRC J3H ZK JqU S EI Za`U U H[JI5`STU S 9Z § r & ')(*,+( ! (3+)K9 u, v, w 9"%!(3.q.%!:0:5+9%.q2#+ A 9%+73.q$&'+ u < v < w 4 #(3.%$K'+ A A v <A w > (87-+ (3.9%+#)V/%O'+:0(8) P = v0−v1−· · ·−vq ,QAB+1/ v0 = v > v1 = w > vq ∈ B > q +1.9*(8:0-,=(8" > q ≥ 3 r ( )Z?[,G-,!.2#+AB!(3.%(8)V.#" u v2 −P −vq ,=73!"%. u−v1 −P −v1 −vq +1.9#)V/<O'+:0(8) 2#+X73!)#N'+#"h(8:0-,=(8"%+02#+ u AB+"%. v $K#( :5!)K9"%+0$&'+ u < w /1!:0:5+0.<!#O-,=(89%4 q A )0+#92#!)'/.##.%+"q$&'+ : u p.%. 2#+#)0AB!(3.%(8) vi .#" v2−P−vq > +9!)5/<O'!(3.(89 8 = )a, i 73+X#78'.hN"<,=)'2o.%.(8e#73+0,1AB+1/^/1+99%+\#"%!#"(349%4 2 ≤ i ≤ q i < q /Q,="h(87 % 2 )Z? ,\-,!.2C?[,=" 9%+1. +)&9"%+ A +9 B = r ( +1.9*(8:0-,=(8" 8 i 3 ≤ i ≤ q − 2 > ,=73!"%. u−vi −P −vq +1.9*#)d/%O'+:0(8)(8:0-,=(8" > : j 2#+ A AB+"%. B ?[,=#" .G73+\73+:0:5+ Aj > )'!'.J,QAB!)'.J,=73!"%. v ∈ A ! v i q−1 ∈ B ! +G#"%+:0(3+" 2#+G/1+1. /Q,!.H+1.9*(8:0.%.(8e#73+G/Q,=" A +1.9H#)'+G/78(3$&'+L+9H2#!)'/ vq−1 ∈ B no,=(3.,=73!"%. > v −u−v −P −v +1.9#)/%O'+:0(8)2#+ 73!)#N'+#"H(8:0-,=(8"%+h2#+ v AB+"%. 0 i q−1 % 8 jA*q= B > $&#(C:5!)K9"%+ v <A u > /1!)&9"%+12;(3.<,=)K9 7f?[,=)K9(3. :549"(3+L2#+ <A 73+:0:5+ r ( +1.9L-,=(8" 8 = i 2 ≤ i ≤ q − 1 ,=73!"%. v − u − vi − P − vq +1.%9J#)6/%O'+:0(8) 2#+ 73!)#N'+#"(8:0-,=(8"%+2#+ v AB+"%.#)V.%!:0:5+92#+ B $K#(-:5!)K9"%+ v < u /1!)K9"%+12;(3.<,=)&9 A )'!#AB+Q,=7f?[,=)&9(3. % :549"(3+L2#+ < 8 73+:0:5+ jA*q= 2 A <A ! +1.R2#+ 73+:0:5+1.#"%41/1412#+)K9%.o:5!)&9"%+)K9]e#(3+) $K'+ <A +1.%9#)'+c"%+7I,=9(3!) [email protected]!"%2;"%+ 8 -,="9(3+7 =H ) 2#4sW')#(892#+d: 2 :5+o#)'+"%+7I,=9(3!) .#" B Y .( u, u0 ∈ B > !)Vp.<+ u < u0 .(p+9.%+#73+:5+)K9*.Q? (87 + (3.9%+#)V/%O'+:0(8)T2#+ 73!)#N'+#"(8:0-,=(8"%+M2#+ B : 5! +1. u AB+"%.g#)V.<!:0:5+9g2#+ A 9%+7$&'+ u0 +1.973+2#+ ( :5+M.%!:0:5+92#+/1+ /<O'+:0(8) +)'.%+:Je#73+1. A +9 B !-,=)K9G2#+1.M"!73+1.h. % :549"(3$&'+1. > !)]AB!(89G$K'+X7I,^"%+7I,=9(3!) < B +1.9,='.%.(C#)'+h"%+7I,=9(3!)[email protected]!"%2;"%+ - !9%!)'.H$K'+ A +9 B 49<,=)K9 2#+1.H+)'.<+:Le#73+1.HW')#(3. > !)d.P,=(89$&Z? (87Z+ (3.9%+G2',=)'. +9H2',=)'. 2#+1.4734:5+)K9%.H:^, (8:^,= _ A B H tD g dr ZU GFI`&EPPRZU_P J STPJK Kco ^ ESTU G/I`&EPP Z&UBP J S P JK#¥ Z a (A, <A ) b ¨ K EHa` ¥ Z Z`U GFI Z*JS H ZX¥¦ JP S ` `f#%S JKZ\¥ Z (B, <B ) § {a, b} G§ r & ')( *,+( ##.%!)'.$KZ? (87G+ (3.9%+R#)/%O'+:0(8)2#+o73!)#N'+#"d(8:0-,=(8"%+ Q = _ /Q,="(87Z)Z? % ,5,='/#)'+ a−a0 −· · ·−b0 −b +G/<O'+:0(8)+1.92#+h73!)#N'+#"M,=:5!(8)'. 2 L ! j A j ,=" 9%++)&9"%+ A +9 B +T73+:0:5+ :5!)K9"%+d$K'+ a0 ∈ A 8 +952#!)'/ a < a0 = A ! b0 ∈ B 8 +9G2#!)'/ b < b0 = > 2',=)'.h9%!'.L73+1.h/Q,!.G!)c,V#)'+0/1!)K9"<,!2;(3/9(3!) 7I, B 1 > :^, (8:^,=78(89%4G2#+ a +9 b !)'/ {a, b} +1.9*e#(3+)#)'+G-,=(8"%+G2C?[,=:0(3.H2#+ G r ##p.<!)'.$&'+ 7 $7 2;(8"%+a$KZ? (87L+ (3.%9%+o#) {a, b} )'+].%!(89-,!..%p41/(I,=73+ > /[email protected]+1.%9 $&-,!.( 7 #"(3.%:5+V.%9"(3/9 S [email protected]+ 9"%4:0(89%41. a +9 b 8 !)6)'!9%+V73+1.X.<!:0:5+9%.0+9X73+1.\/<O'+ 7 :0(8)'.*2;$&-,!.( 7 #"(3.%:5+h/1!:0:5+M(8)'2;(3$&'4W'N#"%+ 1A6q=- )T!#""<,\.%+"%+p!"9%+" 7I, W'N#"%+ jAj -,=NB+ r $&#(q"%+#"%41.<+)&9%+5/1+"9<,=(8)'.J2#+1.h.%!:0:5+9%.L$K'+X)'!'.G2#4sW')#(8"%!)'. ,=d73!)#N2#+L7I,^#"%+#AB+J$K#(`.%#(89 B4 ,="h. % :549"(3+J!)+#9M.##p.<+" $K'+ S +1.9 2#+ 1 73!)#N'+#"J,=]:5!(8)'. 1 > +9G!)a,=#+7873+ a 73+\AB!(3.%(8)2#+ a0 .#" S 4 #(3.%$KZ?[,='/#) 1 1 .%!:0:5+9q2#+ S )'+gAB!(89 7I,HSU!(3. a +9 b > ,='/#)0.%!:0:5+9q2#+ S )[email protected]+1.%9 2',=)'. T 9- !9%!)'. $&'+ a−S −a0 −c−S −c0 −b0 −S −b +9 a−S −a0 −d−S −d0 −b0 −S −b .<!)&9 1 3 2 1 4 2 2#+1./<O'+:0(8)'.G2#+X73!)#N'+#"h-,=(8"%+0/Q,=" {a, b} +1.9M#)'+X-,=(8"%+02C?[,=:0(3. _ #p+73!)'. H 73+M9"%!(8)'2;#(89-,=" V (S ) ∪ V (S ) 3 4 8P« ¨ G; G/I `EPPRZUg¥ Z IC¦ Z`Uq¥ JI5` S \ {a, b} N (Z) ∩ C(T ) § r ##p.%!)'.b$&Z? (87+ (3.%9%+#).<!:0:5+9 u ∈ V (S) \ {a, b} 2',=)'. N (Z) ∩ C(T ) > ?[,=#" : .g73+H73+:0:5+ j > u ∈ A ∪ B ∪ C(T ∪ A ∪ B) no,=(3. u ∈ C(T ∪ A ∪ B) +1.%9 7I,XS !(3. a +9 b 5> !)'/ u ∈ A ∪ B (8:0p.%.%(8e#73+h/Q,=" ,='/#)d.%!:0:5+9H2#+ S )'+MAB!(89 r ( 2 8 = > !)'/ u ∈ A > u 2#!(89 9"%+X73+\AB!(3.(8)]2#+ a .#" S1 /Q,=" A (8)'2;#(89h#)'+0/78(3$&'+ (V (S1 ) \ {a}) ∪ V (S3 ) ∪ V (S2 ) (8)'2;#(89#)/%O'+:0(8)R2#+M73!)#N'+#"H(8:0-,=(8"%+ Pu 2#+ 8 = 4 #(3.%$&'+73+*AB!(3.(8)^2#+ u ∈ A AB+"%. b "<,=#p+73!)'.g$K'+ {a, b} +1.9q#)'+*-,=(8"%+ 2C?[,=:0(3. 8 = jAj (8:0#78(3$K'+ u .#" Pu )[email protected]+1.9 -,!.2',=)'. A /Q,=" A (8)'2;#(89#)'+X/78(3$&'+ > 73+L73+:0:5+ $&'+X73+\AB!(3.(8)a2#+ b .#" P +1.9G2',=)'. B nR,=(3.G/1+7I,/1!)K9"%+12;(89L7I,V:^, (8:^,=78(89%452#+ u Zr 2 8 b !#"[email protected]!"%2;"%+ <B ( u ∈ B > u 2#!(89 9"%+\73+\AB!(3.%(8)o2#+ b .#" S2 .( S2 +1.9h2#+ 73!)#N'+#"J,=:5!(8)'. 1 = !7f? #)c2#+ c0 , d0 8 .( S +1.9L2#+\73!)#N'+#" 0 = > :^,=(3.G2',=)'. 2 7f? #)!7f?[,=#9"%+h2#+1.*/Q,!. > #),="N#:5+)&9H.%(8:0(87I,=(8"%+ 7I,X.(89-,=9(3!) u ∈ A /1!)'2;#(89 #)'+h/1!)&9"<,!2;(3/9(3!) _ +1/(Z#"%!#AB+ 8N<= > ?[,=#" : .*73+h73+:0:5+ 1Ar ,=##78(3$K'4 Y S +9 T > !).<,=(89H$K'+ 81« ¨ G P ESTI5`F¥ ZG ¥ Za` `EP PRZUT` 0 ZUL¥ ZG ¥ Za`n`EPPRZUT` 0 0 0 `EI U a , c, d b ,c ,d ¥qJI5` C(T ) § 4 #(3.%$K'+ 8 = 8N<= +9 8W6q= !) a 6= a0 .%!#AB+)'!)'.*)'!'.*$K'+ b = b0 +1.9p.<.(8e#73+ > 2#+ Y p+#92#412;#(8"%+h73+1. 2#+ Sf,=(89.#(8A,=)&9%. 8« ^ S;K¦ GFI F¥ Z 0 Za`UM¥ JI5` ] JKEH `3S K Za`UGq¥ JI5` ZU K?Za` a , c, d N (Z) N (Z) \ C(T ) ¥ ZG JG/U H Za` `&EI U¥ JI5` C(T ) \ N (Z) § 8!«p^ S K¦ G/I¥ Z 0 0 Za`Uq¥ JI5` ] JK EHa`;STK Za`U¥ JI5` ] K%¦ JG/U H ZpZa`U c ,d N (Z) N (Z) \ C(T ) ¥qJI5` ]iZU 0 C(T ) \ N (Z) b ∈ C(T ) § - !9%!)'.H$&'+ 0 a ∈ / Z /Q,=".(8)'!)d7f? #)d2#+ c, d .%+"<,=(89H2',=)'. N (Z) ∩ C(T ) r ) 2#4sW')#(89#) /<O'+:0(8) Y .<!(89 00 73+.<!:0:5+92#+ P /1!:0:5+o.#(89 a N (Z) 73+ #78'.#"%F/%O'+o2#+ a .#" a − S − a > +9.%!(89 b00 73+o.%!:0:5+9T2#+ N (Z) 73+#78'. 1 1 1 #"%F/%O'+G2#+ b0 .#" S 8 73+1.H.%!:0:5+9%. a00 +9 b00 + (3.%9%+)&9 /Q,='.%+L2#+ a +9 b =-r !(3+)&9 2 zi ∈ V (Z) ∩ N (a00 ) +9 zj ∈ V (Z) ∩ N (b00 ) 9%+73.$K'+ zi −Z −zj +1.9H,='.<.(Z/1!#"9 $K'+L.%.(8e#73+ 8 )'!9%!)'.M$&'+ i < j > i = j ! i > j .%!)K9 .%.(8e#73+1. = 4 .%!)'. P = a1−S1−a00−zi−Z−zj−b00−S2−b0 )GAB4"(W-+$K'+ P +1.9Z#)G/<O'+:0(8)J2#+ G [email protected]+ 9"%4:0(89%41. q> : 1 ,=##78(3$K'4 a1 +9 b0 ?[,=#" .g73+73+:0:5+ H > P +9 {a0 } > (87p+ (3.%9%+#)b.<!:0:5+9 z 2#+ P $K#(#AB!(897f? #)b2#+ c ! d > +9!)^/<O'!(3.(89#)59%+7 z ,='.%.('#"%F/%O'+H$&'+*.%.(8e#73+ 2#+ a .#" P #(899%+ 41/%O-,=)#NB+" S +9 S > !)X.##p.<+$K'+ z AB!(89 c 9! ,H2#4sW')#(89(3!) 1 3 4 2#+1.`$&-,!.( 7 #"(3.%:5+1.`+)K9"<, 3)'+ z 6∈ V (a −S −a00 )∪V (S ) > +92#!)'/ z ∈ V (z −Z−z ) 1 1 2 i j /> > ?[,=#" : . 8Wq= !)V, c ∈ N (z) \ C(T ) +9 a0 , d ∈ C(T ) \ N (Z) !)'/ a0 )Z?[,h-,!.g2#+ > AB!(3.(8)'.2',=)'. V (P ) \ {a } > /Q,="#)^9%+7'AB!(3.(8)^2#+AF"<,=(89!e#78(8N&,=9%!(8"%+:5+)K9g.%+9"%!#AB+" 1 2',=)'. z −Z −z .%!'.h+(8)'+02#+5/1!)K9"%+12;(8"%+57I,2#4sW')#(89(3!)c2#+1.L$&-,!.( 7 #"(3.%:5+1. > /1+ i j $K#(Z+)&9"<, 3)'+"<,=(89 a0 ∈ N (Z) Y #)'+G/1!)K9"<,!2;(3/9(3!) 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V (P10 )∪V (P2 )∪V (P3 ) (8)'2;#(89H#)T#"(3.:5+L!#9(8:^,=72#+G/1!(8) a +9H2#+M9"(I,=)#N73+ {b , b , b } 1 1 2 3 k r ( :^,=)'$K'+ c a1 ,=73!"%.\73+VN"<,=#O'+ F 0 (8)'2;#(890-,=" V (H) ∪ V (b1 − P10 − c) +973+h9"(I,=)#N73+ {b , b , b } .<,=9(3.mSU!)&9H9%!#9%+1.73+1.*O % !9O : .%+1.H2;d73+:0:5+ 1)( > 1 2 3 /1+\$&#( :5!)&9"%+0$K'+ F 0 /1!)&9(3+)K9G#)#"(3.%:5+X!R#)'+X % "<,=:0(32#+ nR,=(3.M/1+7I, /1!)&9"%+12;([email protected]!#9(8:^,=78(89%452#+ F /Q,=" |F 0 | < |F | 8 (87q:^,=)'$&'+\73+1.h.<!:0:5+9%.h2#+ = P10 +)K9"%+ c +9 a1 k r ( AB!(89 7 7 c a1 +9 ,=X:5!(8)'.`#)L)'!) AB!(3.(8)02#+ a1 .#" H > ,=73!"%.`(87F+ (3.%9%+#)J/<O'+ 6 :0(8) Q 2#+ H 2#+73!)#N'+#"q,=L:5!(8)'. > +9`2#!)K973+1.`+ 9"%4:0(89%41.t.%!)K9 a +9#) 1 AB!(3.(8)b2#+ c nR,=(3.g,=73!"%.73+*N"<,=#O'+ F 0 (8)'2;#(89-,=" (V (H) ∪ V (P 0 )) \ V (Q∗ ) 1 +973+h9"(I,=)#N73+ {b , b , b } .<,=9(3.mSU!)&9H9%!#9%+1.73+1.*O % !9O : .%+1.H2;d73+:0:5+ 1)( > 1 2 3 /1+\$&#( :5!)&9"%+0$K'+ F 0 /1!)&9(3+)K9G#)#"(3.%:5+X!R#)'+X % "<,=:0(32#+ nR,=(3.M/1+7I, /1!)&9"%+12;([email protected]!#9(8:^,=78(89%452#+ F /Q,=" |F 0 | < |F | 8 (87q:^,=)'$&'+\73+1.h.<!:0:5+9%.h2#+ = Q∗ Qd RTMH #F r ( c AB!(8952#+ H #) ,=#9"%+T.<!:0:5+9^$&'+ a1 ,=73!"%.073+N"<,=#O'+ F 0 (8)'2;#(890-,=" % : V (H) ∪ V (P10 ) \ {a1 } +9g73+H9"(I,=)#N73+ {b1 , b2 , b3 } .<,=9(3.S !)K99%!#9%+1.g73+1.gO p!9O .<+1. 1 ) ( % 2;c73+:0:5+ > /1+b$&#(:5!)&9"%+V$&'+ F 0 /1!)&9(3+)&9X#)a#"(3.%:5+^!#)'+^ "<,=:0(32#+ Y no,=(3.b/1+7I,a/1!)&9"%+12;([email protected]!#9(8:^,=78(89%4o2#+ F /Q,=" |F 0 | < |F | (87:^,=)'$&'+ F 0 73+ > .%!:0:5+9 a !)'/ > c )'+bAB!(89J2#+ H $&'+^73+^.%!:0:5+9 a > + ,!/9%+:5+)&90/1+^[email protected]!) 1 1 AB!#7I,=(89 :5!)&9"%+" 2 KaaZ8 ^ ES U G/Ib¤H[JP YZ G1ZK EI G1Z ^_ESWZI U ] ] ]LU H[ES `hYZPS I1` G § P 1 P2 P3 ¥ Z M&EHP JI U;GFIRBHS `aPRZnEP U STP JK ¥ ZnU HSWJIK¤K?Za` ZU G F {a1 , a2 , a3 } {b1 , b2 , b3 } § "I `aGB E9`.ZEG/H G5Z JM5EGFHZ U H%PSTUka` Z&U i = 1, 2, 3 Pi ai bi § ^_ESWZI Ug© G/Ia K GF`R EG/HU5 YZPSTI¥FZ 9ZHa` ¥ EI U KZa` `EPPRZUT` STI UkHSWZG/Ha` P10 a1 b1 P JI G1ZI U ZU b2 b3 § / G a I K F G R ` E G/HU5 YZPSTIF¥ Z 9ZHa` ¥ EI U KZa` `EPPRZUT` STI UkHSWZG/Ha` P20 a1 b2 P JI G1ZI U ZU b1 b3 § / G a I K F G R ` E G/HU5 YZPSTIF¥ Z 9ZHa` ¥ EI U KZa` `EPPRZUT` STI Uk HSWZG/Ha` 0 P3 a1 b3 P JI G1ZI U ZU b1 b2 § ¨ KEHa`J© ª Js« Za`LYZP STI5` 0 ] ] M&EHP ZI ULGFI5BHS `P ZpEP U STP JKZ¥ Z ¥ ZL ES I ZU P1 P2 P3 G a1 ¥ Z U HaSWJIK¤K?Z {b1 , b2 , b3 } § ª Q « iG QSWZI STI ¥G/STU GFIBHaS `aPRZEs U S P JK¥ Z^ ESTI V (P ) ∪ V (P20 ) ∪ V (P3 ) a ZUXF¥ Z U HS JIK¤ K?Z 1 ] EG QSWZI ] ] M&[email protected]_Z\ \9H[JPSU¥ 1Z {b1 , b2 , b3 } P1 P20 P3 ZUgF¥ ZpU HSWJIK¤ K?Z EP U STPJKZX¥ ZX ESTI a1 {b1 , b2 , b3 } § ª £« iG QSWZI STI ¥ G/STU GFIB HaS `aPRZEs U S P JK¥ Z^ ESTI 0 V (P ) ∪ V (P2 ) ∪ V (P3 ) a ZUXF¥ Z U HS JIK¤ K?Z 1 ] EG QSWZI ] ] 0 M&[email protected]_Z\ \9H[JPSU¥ 1Z {b1 , b2 , b3 } P 1 P2 P3 ¥ ZL ES I Z&Ug¥ ZpU HS JIF¤ KZ EP U STPJKZX¥ Z G a1 {b1 , b2 , b3 } § & ')(*,+( )T#9(878(3.<+"<,\.<,=)'.:5+)&9(3!)+ #78(3/(89%+M73+SU,=(89*$K'+ (P ) +1.9,=V:5!(8)'. i ,='.%.%(C73!)#N^$&'+ (P 0 ) i ) /1!:0:5+)'/1+T -,="\#"%!#AB+"07I,R/1!)'/78'.(3!) 8 , = )6)'!9%+ H 73+V9"%! 2#+ G (8)'2;#(89-,=" V (P ) ∪ V (P ) r ( 73+1. .<!:0:5+9%. 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R #) ' G + 0 F p_ 8 = F /Q,=" |F 0 | < |F | +1/(Z#"%!#AB+h7I,0/1!)'/78'.(3!) , 4 "%!#AB!)'.H7I,0/1!)'/78'.%(3!) 8 e =p )d)'!9%+ H 73+G9"%!2#+ G (8)'2;#(89H-,=" V (P1 ) ∪ B_ !:0:5+ : ^ : = , ' ) & $ '+ ! \ ) P . = , 8 ( t 9 K $ '+ V (P3 ) a1 b2 > P20 .%. 2#+g2#+1.q.%!:0:5+9%.t(8)&9%4"(3+#"%. r !(89 c 73+0.%!:0:5+9G(8)K9%4"(3+#"J2#+ P20 $K#(qAB!(89 V (H) > +9G/%O'!(3.(,='.<.(q#"%K/<O'+5$K'+ .%.(8e#73+\2#+ b - !9%!)'.h$K'+ c + (3.%9%+X/Q,=" a ∈ H uH+1.9%+)K9 + ,=:0(8)'+"G$K-,=9"%+ 2 1 /Q,!./1!)'2;#(3.P,=)&9 #)'+h/1!)&9"<,!2;(3/9(3!)o! /1+G$KZ?[,1A,=)'/1+h73+h73+:0:5+ Y T % M Qr )5t% % 7 k r ( )'+MAB!(89H2#+ c H $K'+ a1 > ,=73!"%. P1 > P20 > P3 SU!":5+)&9 #)'+M "<,=:0(32#+L!#9( :^,=73+G2#+G/1!(8) +9H2#+M9"(I,=)#N73+ a {b , b , b } 1 2 3 k r ( )'+gAB!(892#+ 1 $K'+ c H a1 +9t#)\AB!(3.(8)52#+ a1 > ,=73!"%. V (P1 )∪V (P20 )∪V (P3 ) (8)'2;#(89H#)T#"(3.:5+L!#9(8:^,=72#+G/1!(8) +9H2#+M9"(I,=)#N73+ a {b , b , b } 1 1 2 3 k r ( :^,=)'$K'+ c a1 ,=73!"%.\73+VN"<,=#O'+ F 0 (8)'2;#(890-,=" V (H) ∪ V (b2 − P20 − c) +973+h9"(I,=)#N73+ {b , b , b } .<,=9(3.mSU!)&9H9%!#9%+1.73+1.*O % !9O : %. +1.H2;d73+:0:5+ 1)( > 1 2 3 /1+\$&#( :5!)&9"%+0$K'+ F 0 /1!)&9(3+)K9G#)#"(3.%:5+X!R#)'+X % "<,=:0(32#+ nR,=(3.M/1+7I, /1!)&9"%+12;([email protected]!#9(8:^,=78(89%452#+ F /Q,=" |F 0 | < |F | 8 (87q:^,=)'$&'+\73+1.h.<!:0:5+9%.h2#+ = P20 +)K9"%+ c +9 a1 7 k r ( AB!(89 7 c a1 +9 ,=X:5!(8)'.`#)L)'!) 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P1 > P2 > P3 > +9`2#+9"(I,=)#N73+1. {a1 , a2 , a3 } > {b1 , b2 , b3 } > 2#+h.%!"9%+M$&'+!#" i = 1, 2, 3 > 73+M/%O'+:0(8) Pi .%!(89*[email protected]+ 9"%4:0(89%41. ai +9 `r ` )a2;(89X,=73!"%.L$K'+073+ 7 ##73+9 bi !(89 mi 73+5.<!:0:5+9J,=a:0(878(3+a2; /%O'+:0(8) Pi 9 U N H J R P Z 1 + . * 9 I 7 , # 2 + (a , a , a , b , b , b , m , m , m ) F 1 2 3 1 2 3 1 2 3 KaaZ ^ ESTU /G I ¤ H[JP YZ G BHS `aPRZ 5 JS H ¥ Z VU JSTKTKZ P TS I1STP J KZ¥ Z RTMH #F U H[EG S P JSTH Z&U `ES U G/I F ^ GQ&E9`EI5` G5Z `ES Ua¥ ZcU HNJP Z G§ F Z&U M&EHP5\JH KZa ` YZPS I1` ] ] J9Z (a1 , a2 , a3 , b1 , b2 , b3ª , m1 , m2 , m3 ) P P P « ^_ESTU G/IcYZPSTc I ¥ Z ¥¦ Z U H%1PST2Uka` 3 ] ai , mi , bi ∈ V (Pi ) i = 1, 2, 3 § R G a 1 , m1 ¥qEI U K?Za` `&EPPRZUT` STI UkHSWZGFHa` P JI G5ZI U ZU ] ZH U ¥ Z KEKI ¤G1ZG/HP STI1S a2 , a 3 , b 2 b3 P JK?Z J9Z0 Z`*BH[EPBHSUkU ` § ¨ KEHa` ] G1Z K ¦ EI IBEU,Z Za`U GF] I YZP ST] I ¥ Z ]iZU ] ] a −R−m1 −P1 −b1 G R1 R1 P2 M&EHP 1ZI U G/b I BHaS `aPR Z JSTL H ¥ Z ¥FZpUVJS KTKZ PS I S P JK?Z P3 `JI1` § G % : - !9%!)'. r !(89 k 7I,T73!)#N'+#"X2#+ P1 > $K#(+1.9G-,=(8"%+^-,="LO !9O .%+ $K'+ |E(R)| ≤ k/2 )a)'!9%+ H 73+59"%!62#+ G (8)'2;#(89X-,=" V (P ) ∪ V (P ) qr ( 2 3 73+1..<!:0:5+9%.*(8)&9%4"(3+#"%. 2#+ R :^,=)'$K'+)&9 +)&9( : "%+:5+)&9 H > ,=73!"%..%!(89 R #)T#78'. 1 /1!#"9/<O'+:0(8)2#+ a AB+"%. b /1!)&9%+)& 2',=)'. V (R) ∪ V (m − P − b ) H ), 1 1 1 1 1 |E(R1 )| ≤ k +9*73+1.H/%O'+:0(8)'. R1 > P2 > P3 SU!":5+)&9#)T#"(3.%:5+ F 0 ,QAB+1/ |V (F 0 )| ≤ 8 4 |V (F )| #(3.<$&'+ G +1.9b.<,=)'.^9"%! (8:0-,=(8" > R1 +1.%9b2#+73!)#N'+#"-,=(8"%+ .(8)'!) = V (R1 ) ∪ V (P2 ) (8)'2;#(8"<,=(89#)T9"%!(8:0-,=(8" > +9 F 0 +1.9*#)T#"(3.%:5+M-,=(8"*:0(8)#(8:^,=7 ! ? (8)'4N&,=78(89%4d/( 7 2#+1.%.'.0+1.902#!)'/V#)'+T4N&,=78(89%4 /1+$&#(:5!)&9"%[email protected]+)6Sf,=(89 > R1 +1.9 Y 4N&,=7 a −R−m −P −b 7I,\/1!)'/78'.(3!)2;T73+:0:5+L+1.9H.<,=9(3.mSf,=(89%+ 1 1 1 )bp1+#92#!)'/H .##.%+"$KZ? (87-+ (3.9%+*#)b.<!:0:5+9(8)&9%4"(3+#" c 2#+ R , % ,=)&92#+1. AB!(3.(8)'.M2',=)'. H - !'.,=7873!)'.:5!)&9"%+"G$K'[email protected]+ (3.9%+)'/1+\2#+X/1+J.%!:0:5+9M+)&9"<, 3)'+ #)'+G/1!)K9"<,!2;(3/9(3!) )/%O'!(3.(89 c ,='.%.%(Z#"%K/<O'+G$K'+hp.<.(8e#73+h2#+ m - )A,XAB!(8" 1 $K'[email protected]+ (3.9%+)'/1+h2#+ c +)&9"<,=(8)'+h#)'+h/1!)K9"<,!2;(3/9(3!) pr !(89 S #)#78'.*/1!#"9/%O'+:0(8) 2#+ c AB+"%. b /1!)&9%+)&2',=)'. V (c−R−m ) ∪ V (m −P −b ) K )b, |E(S)| < k /Q,=" 1 > ?[,=#" : .H7I,^2#14sW')#(89(3!)R1 2#+ 1 ,=1'/#).%!:0:5+9M2#+ |E(R)| ≤ k/2 +9 c 6= a1 c> S \ b1 )Z?[,2#+\AB!(3.(8)2',=)'. H )2#4sW')#(89L2#+ .%!'. 7 /<O'+:0(8)'.L2#+ H Y H = H\a +9 2 3 4 H3 = H\a2 !#" i = 2, 3 > !))'!9%+ ci 73+5.%!:0:5+9L2#+ Hi AB!(3.(8)c2#+ c 73+5#78'. #"%F/%O'+G2#+ a i 8P« "I^5 ZGU `GQ& E9`&ZH G1Z Za`U GFI_Z JHU,Z\F¥ Z c2 c3 P2 § r ( % c2 = c3 ,=73!"%. V (S) ∪ V (P2 ) ∪ V (P3 ) (8)'2;#(89#)'+ "<,=:0(32#+]2#+R9"(I,=)#N73+ Y > !)'/ -r {b1 , b2 , b3 } +9 2#+h/1!(8) c2 #)'+h/1!)&9"<,!2;(3/9(3!) c2 6= c3 ( c2 :^,=)'$K'+ c3 > % ,=73!"%. V (S) ∪ V (P ) ∪ V (P ) /1!)&9(3+)K9#)'+ "<,=:0(32#+h2#+9"(I,=)#N73+ {b , b , b } +9 2 3 1 2 3 2#+X/1!(8) c Y #)'+X/1!)&9"<,!2;(3/9(3!) > !)'/ c c +1.9e#(3+)o#)'+X,=" 2 9%+\2#+ G > +9 > $K#(899%+ 2 3 41/<O-,=)#NB+" P +9 P > !)]+#9G.##.%+"G$&'+ c c +1.9h#)'+^,=" 2 9%+02#+ P _ +1/( 2 3 2 3 2 #"%!#AB+ 8N<= & ')(*,+( - > !)'/ V (S) ∪ V (P2 ) ∪ V (P3 ) (8)'2;#(89*#)T#"(3.%:5+ F 0 2#+M9"(I,=)#N73+1. {b1 , b2 , b3 } + 9 {c, c , c } ,QAB+1/ |V (F 0 )| < |V (F )| ##(3.%$K'+ |E(S)| < k > !)'/ F 0 1+ .95#) 2 3 #"(3.:5+b(8:0-,=(8" > /1+b$K#((8:0#78(3$K'+b$K'+ c − P − b +1.9J#)c/%O'+:0(8) 2#+^73!)#N'+#" 3 2 2 (8:0-,=(8"%+ > +9L2#!)'/ a −P −c #)a/<O'+:0(8)c2#+\73!)#N'+#"L-,=(8"%+ tr !(89 R0 #)#78'. 2 2 2 /1!#"9H/<O'+:0(8)d2#+ c AB+"%. a /1!)K9%+)&2',=)'. V (c−R−m ) ∪ V (m −P −a ) - ) 1 1 1 1 1 , |E(R0 )| < k /Q,=" |E(R)| ≤ k/2 > ?[,=#" : .h7I,2#4sW')#(89(3!)a2#+ c > ,='/#).%!:0:5+9 2#+ R0 \ a )Z?[,2#+0AB!(3.%(8)c2',=)'. H > !)'/ R0 +1.9L2#+573!)#N'+#"J-,=(8"%+b/Q,="J.(8)'!) > 1 > !)'/ V (R0 ) ∪ V (a2−P2−c2 ) (8)'2;#(8"<,=(89#)^9"%!^(8:0-,=(8" V (R0 ) ∪ V (P2 ) ∪ V (P3 ) (8)'2;#(89 #)L#"(3.:5+ F 00 2#+9"(I,=)#N73+1. {a , a , a } +9 {c, c , c } > +9 F 00 +1.9#)J#"(3.:5+ 1 2 3 2 3 T % M .j )5t% -,=(8"AB4"(W ,=)&9 |V (F 00 )| < |V (F )| /Q,=" |E(R0 )| < k ! _ 1+ /(F+1.9`#)'+/1!)K9"<,!2;(3/9(3!) 2 ) +#9o:^,=(8)&9%+)-,=)K9a2#!)#)'+"o#) ,=78NB!"(89O#:5+ # 2 + 2#49%+1/9(3!) 2#+1.R#"(3.:5+1. -,=(8"%. Y dZ ' ( l )TN"<,=#O'+ G .P,=)'.*9"%!(8:0-,=(8" f ( r ('73+HN"<,=#O'+H+1.9.<,=)'.#"(3.:5+H-,=(8" Y j 4 ,!.g2#+*#"(3.:5+-,=(8"ml ;r (8)'!) Y #)#"(3.:5+h-,=(8"H:0(8)#(8:^,=72#+ ' #" $ " * G k_ !)'.(32#4"%+"^.%'/1/1+1.%.(8AB+:5+)&9^9%!'.073+1. r$7 ##73+9%. (a1 > a2 > a3 > b1 > b2 > b3 > 9 m1 > m2 > m3 ) ∈ V (G) _ ,=73/#73+"p!#" i = 1, 2, 3 #) #78'./1!#"9/<O'+:0(8) Ri 2#+ ai AB+"%. mi 2#!)&9g73+1.g.%!:0:5+9%.(8)K9%4"(3+#"%.g:^,=)'$&'+)K973+H"%+1.9%+H2; r$7 ##73+9 > +9#)^#78'. /1!#"9/%O'+:0(8) S 2#+ m AB+"%. b 2#!)K973+1..%!:0:5+9%.(8)&9%4"(3+#"%.:^,=)'$K'+)&9 i i i 73+J"%+1.9%+\2; r$7 ##73+9 Zr ( [email protected]+)'.%+:Le#73+ ∪ (8)'2;#(89M#) i=1,2,3 (V (Ri ) ∪ V (Si )) #"(3.:5+h-,=(8" > 73+h.9%K/ B+"H+):54:5!(8"%+ kr (,='/#) r$7 ##73+9b)Z?[,a2#!)#)'42#+d#"(3.:5+-,=(8" j 4 ,!.V2#+#"(3.%:5+-,=(8"ml > r (8)'!) "%+9%!#")'+"`#)G#"(3.:5+t-,=(8"2#+q9<,=(87873+:0(8)#(8:^,=73+-,=":0(&/1+ .9%F/ B41. > +):54:5!(8"%+ & ')( " & *,+( - ( f O(n11 ) r ( G +1.9T.<,=)'.V#"(3.:5+R-,=(8" > ,=73!"%.(87M+1.9T/7I,=(8"$K'+7f?[,=78NB!"(89O#:5+ "%4p!)'2bj 4 !, . 2#+M#"(3.%:5+h-,=(8"ml r ( : : G .%. 2#+5#)a#"(3.:5+^-,=(8" > ,=73!"%. G .%. 2#+0#)c#"(3.:5+5-,=(8"J:0(8)#(8:^,=7 > $&'[email protected]!) )'!9%+ F i #)'+R/1+"9<,=(8)'+o49<,=+ > 7f?[,=78NB!"(89O#:5+o/1!)'.(32 : "%+R#) r$7 ##73+9 .! (a1 , a2 , a3 , b1 , b2 , b3 , m1 , m2 , m3 ) ∈ V (G)9 $&#(K+1.97I,9"<,=:5+2#+ F ?[,=78NB!"(89O#:5+ /Q,=73/#73+,=73!"%.2#+1.t#78'./1!#"9%.q/%O'+:0(8)'. R , S 4 ,=".( ,=##78(3/Q,=9(3!)'./1!)'.%41/#9(8AB+1. i i 2;]73+:0:5+ * > !)]AB!(89L$K'+ ∪ (8)'2;#(89L#)]#"(3.:5+0-,=(8" i=1,2,3 (V (Ri ) ∪ V (Si )) > !)'/R7f?[,=78NB!"(89O#:5+o"%+9%!#")'+]/1+o.%!'. 7 N"<,=#O'+ ! #).%!'. 7 N"<,=#O'+ BJ ¥ Y/QE 2#+ > : 2 :5+h9<,=(87873+ > ,=)'.73+#(8"%+M2#+1./Q,!. 9%+1.9%+"g9%!'.73+1. r$7 ##73+9%.)'41/1+1.<.(89%+ _ ,=7 7 ' O(n9 ) 49<,=+1. > /#73+"573+1.5#78'.5/1!#"9%.b/%O'+:0(8)'.V+95AB4"(W-+"b$K'+T73+.<!'. 7 N"<,=#O'+!e#9%+)& +1.95#) #"(3.:5+h!T#)'+h % "<,=:0(32#+h)'41/1+1.%.%(89%+ O(n2 ) 49<,=p+1. 5> !)'/ > ,=d#(8"%+ > 7f?[,=78NB!"(89O#:5+ .%+M9%+":0(8)'+G+) 11 49<,=+1. O(n ) "-! # 2 #" "- "+! K4 ! +1.t78(8)'+ 7 N"<,=#O'+1.2#+.#e2;(8AK(3.%(3!)'.`e#(8-,="9(3+1.2#+ K4 !'+)&9t#)\"!73+(8:0!"9<,=)&9 : 2',=)'.7I, #"%+#AB+]2;9O'41!" :5+2#+1.N"<,=#O'+1.T-,="mSf,=(89%. AB!(8"T-,="+ +:0#73+o73+ 9O'41!" : :5+ 6A6*> 49%+1/9%+"\/1+09 % +^2#+5.%!'. 7 N"<,=#O'+5+#9J2#!)'/ 2 9"%+^(8)&9%4"%+1.%.P,=)&9 i )'.#(89%+ +9h/1+7I,bS#9M)'!9"%+\:5!9(8A,=9(3!)](8)#(89(I,=73+ 7f?[,=78NB!"(89O#:5+52#!)#)'40/( 7 +)o.<!( p > > 2 +1.%.%!'.+1.9 +1.%.<+)&9(3+7tp!#"7I,52#49%+1/9(3!)2#+1. #"(3.:5+1.(8:0-,=(8"%. > +92#!)'/J!#" 7I, # "%+1/1!)#)-,=(3.%.<,=)'/1+G2#+1.HN"<,=#O'+1.H2C? "9%4:0(3.*-,=(8"%. .* vad RTMH vda #F vac vab Pad Pac vca Pab Pdc vcd vdc vdb vcb Pbd Pcb vba vbd vbc e"f g"hB*0k3! (8)'+ 7 N"<,=#O'+J2C? #)'+L.#e2;(8AK(3.%(3!)Te#(8-,="9(3+G2#+ K4 4 !#"*/1+H$K#( .#(89 !)^!#""<,L/1!)'.%#789%+"7I,GW'N#"%+ *#r ( 7 > F +1.%973+78(8)'+ N"<,=#O'+ 2 ? #)'+0.#e2;(8AF(3.(3!)]e#(8-,="9(3+ R 2#+ K > ,=73!"%.G!)/1!)&AF(3+)&9J2#+\)'!9%+" a > b > c +9 d C 4 73+1.$K-,=9"%+.<!:0:5+9%.g2; K /1!""%+1.!)'2',=)&9 - )T!"%2#!)#)'+h,="e#(89"<,=(8"%+:5+)&9 a > b > c 4 +9 d Y a < b < c < d _ /%O-,!$K'+0,=" 2 9%+ ij > i < j > 2#+ K > /1!""%+1.%p!)'2o2#!)'/\#) 4 /<O'+:0(8)V2#+ F $&[email protected]!)5)'!9%+ P K )^)'!9%+ v +9 v 73+1.+ 9"%4:0(89%41.2#+ P > 2#+ .%!"9%+ ij ij ji ij $K'+Mp!#" i = a, b, c, d [email protected]+)'.%+:Je#73+ T = {v ; j ∈ {a, b, c, d} > j 6= i} (8)'2;#(3.%+h#) i ij 9"(I,=)#N73+ _o (8)-,=73+:5+)&9 > !)!e#9(3+)K9h$&'+L73+1. .( /<O'+:0(8)'. P > P > P > P > P > ac ab ad bc bd Pcd > +9 73+1.2#! Q+L.%!:0:5+9%. vab > vac > vad > vba > vbc > vbd > vca > vcb > vcd > vda > vdb +9 Y v AB4"(W-+)K9 dc k ! +1./%O'+:0(8)'. Pab > Pac > Pad > Pbc > Pbd > Pcd .%!)K9H2;(3. !(8)&9%. k F =P ∪P ∪P ∪P ∪P ∪P ac ad bc bd cd k 4 !#" ab 7 i ∈ {a, b, c, d} > j ∈ {a, b, c, d} > i < j > 73+X/%O'+:0(8) Pij ,0!#" + 9"%4 :0(89%41. v +9 v ij ji k ! +1. +)' .<+:Le#73+1. {vab , vac , vad } > {vba , vbc , vbd } > {vca , vcb , vcd } +9 {v , vdb , vdc } .%!)K92#+1.9"(I,=)#N73+1. 2#+ F k ! +1da 2 ..%+#73+1.g,=" 9%+1.2#+ F .%!)K9/1+7873+1.2#+1./%O'+:0(8)'.+99"(I,=)#N73+1.#"%41/1412#+:5+)&9 2#41/"(89%. ) "%+:^,="%$K'+"<,\$&'+M/1+"9<,=(8)'.*2#+1.*/<O'+:0(8)'.*p+#AB+)K9 2 9"%+h2#+H73!)#N'+#"*)K#7873+ > /1+M$K#(p/1!""%+1.!)'2 #)'+h,=" 2 9%+M2#+ K )'!)V.%#ep2;(8AF(3.%41+ /4 ,="+ +:0#73+ > P !#" 7 4 ab "<,=(89 2 9"%+H2#+*73!)#N'+#")K#7873+ > ,='$&'+7 /Q,!.73+1..%!:0:5+9%. v +9 v .<+"<,=(3+)&9/1!);SU!) 7 ab ba 2;'. nF2;#73G73+/<O'!( (8)#(89(I,=7 2#+*[email protected]!"%2;"%+.#" {a, b, c, d} > 73+1..%!:0:5+9%.+9/<O'+:0(8)'. T % M <( )5t% / ( 7 2#+1.%.'..%!)&9*2#4sW')#(3.*.<,=)'.*,=:Je#(8N U9%4 1! +1.*.%!:0:5+9%. v .%!)&9,=#+7341.73+1. 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R = K +9V(87G+1.%9/7I,=(8"$K'+ L(R) )'+ 4 /1!)&9(3+)&9J-,!.G2#+\9"%!](8:0-,=(8"L+9G$K'+ R )[email protected]+1.%9G-,!.Me#(8-,="9( ` )+#9G2#!)'/0.# 7 p.%+" R 6= K 7 +1.9 /7I,=(8"M4N&,=73+:5+)&9h$K'+G73+1.2#+ 2#+")#( : "%+1.M.##p.%(89(3!)'.H)'+ 4 p+#AB+)K9 2 9"%+T.P,=9(3.mSU,=(89%+1.\.(8:J#789<,=)'4:5+)&9 > /Q,="\73+1.\78(8)'+ 7 N"<,=#O'+1.52#+VN"<,=#O'+1.\e#( 7 -,="9(3..%!)K92#+1.N"<,=#O'+1.g2#+ g+"NB+ F 7'"%+1.9%+ '.9%+ #"%!#AB+"[email protected]!)5)'+Hp+#9,1AB!(8" .(8:L#789<,=)'4:5+)K9 L(R) .<,=)'.9"%!T(8:0-,=(8"H+9 R )'!)Te#(8-,="9( -r ##p.%!)'.H2#!)'/ > +) AK'+h2C? #)'+G/1!)K9"<,!2;(3/9(3!) > $KZ? (87Z+)d.%!(89H,=(8)'.%( #+73!)'. a > b > c +9 d 73+1.`$K-,=9"%+.<!:0:5+9%. 2; K4 > +9 p!#" i < j ∈ {a, b, c, d} )'!9%!)'. C 7I,.#ep2;(8AF(3.(3!)L2#+7f?[,=" 2 9%+ ij <4 #(3.%$K'+ L(R) +1.9.<,=)'.9"%!L(8:0-,=(8"`+9 R ij )'!)^e#(8-,="9( > ,=73!"%. 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L(R) #)9"%!(8:0-,=(8" > +)'/1!"%+G#)'+G/1!)&9"<,!2;(3/9(3!) 2 i )d/1!:Le#(8)-,=)K9 73+1.2#+ 73+:0:5+1.H#"%41/1412#+)&9%. !)!e#9(3+)&9 Y > ¨ KEH `pEG QSWZI]© ^ ESTU [email protected]`G5Q%¥S <S `S EI G1ZK8 E I G5Z0¥FZ R K4 § Za`U U HS .SWJK?Za]g-¦ Za`U !m¥S H Z R R = K4 § EI U SWZI U G/IU HNEG S P JS H L(R) § EI U SWZI U G/b I BHS `P Z STH P 5JS H KaaZ# § L(R) 4 !#"2#+1.g"<,=(3.%!)'.*$K#(,=#-,="<,39"%!)K9H2',=)'.7I,G#"%+#AB+h2;V#"%F/%O-,=(8)T73+:0:5+ !) > , ep+1.%!(8)2#+G#"%+)'2;"%+X+)/1!:0#9%+J73+1.H.%!:0:5+9%.M,=d:0(878(3+R2#+1. /<O'+:0(8)'.2#+1.H78(8)'+ 7 5 N"<,=#O'+1. - "*/1+1.*/<O'+:0(8)'.*p+#AB+)K9 2 9"%+h2#+73!)#N'+#"*(8:0-,=(8"%+ > +9g)Z?[,QAB!(8"-,!.*2#+ /1+MSf,=(892#+h.%!:0:5+9H/1+)K9"<,=7 p )d+1.%92#!)'/L,=:5+)'4 7I,\2#4sW')#(89(3!)R.#(8A!,=)K9%+ Y r !(89 7 7 F 73+78(8)'+ N"<,=#O'+*2C? #)'+g.#e2;(8AK(3.(3!)\2#+ K4 ,1AB+1/g73+1.)'!9<,=9(3!)'.q/( 2#+1.<.'. r !(3+)K9 * r mab > mac > mad > mbc > mbd +9 mcd .( .<!:0:5+9%.T2#+ F ( !#"V9%!#9 3 7 i < j > +\.%!:0:5+9 mij +1.9M2',=)'. Pij +9AB4"(W-+ dPij (vij , mij ) − dPij (vji , mij ) ∈ .d$7 ##73+9 {−1, 0, 1} > ,=73!"%.!)T2;(89$K'+ 73+ (vab , vac , . . . , vcd , mab , . . . , mcd ) ∈ V 18 U N H J R P Z B ! +1.9#)'+ 2#+ F +1..%( .%!:0:5+9%. m > m > m > m > m +9 m .Q? (8) 7 ac ab ad bc bd cd 9%+"#" : 9%+)K9M/1!:0:5+X2#+1..%!:0:5+9%.Hjk#"%K/<O'+1.2#+1.H:0(878(3+ 2#+1./%O'+:0(8)'.ml P > P > ac ab !(3/(Z:^,=(8)K9%+)-,=)&973+M73+:0:5+h#"(8)'/(8-,=7 Y P > P > P +9 P ad bc bd cd KaaZ; ^ ESTU G/IL¤H[JP YZ `JI1`CB\HNJP SU¥ Z ^ ES U G/I`EGF`P ¤H[JP YZ S I ¥GFS U G F § ¥ Z "I `GQ&E9`&Z ¥FZ UVJS KTK?Z G /SZ`U FG I_Z `aG1Q¥S .S `aSWEI IBEI U HS .SWJK?Zo¥ Z G K4 § F PSTI1S P JK?ZRZ&U ¥ ZRU HNJP Z ^ ES U GFIdBK G `^ EG/HU (vab , vac , . . . , vcd , mab , . . . , mcd ) § P Y5ZPS ID¥ Z 9ZHa` ¥qEI U;KZ``EPPRZUT`S IBUkHaSVZG/H ` P JI G5ZI U;UVEG ` K?Za` v m vij JGU H Za` G1Z ab ¨ K EH `J© ab vab § J¦ ZI1`&ZPRQK?Z STI ¥ G/STUiG/IDK S I Z¤ H[JP YZb¥¦ GFI_Z `aG1Q¥S .S `aSWEIIBEI (F \vab Pab mab ) ∪ P U HS <S JK?ZX¥ Z ¥ Z UVJS [email protected] P JK?Z K § "I J G/I H `aG/4 K UVJU0`S PSTK JS H ZG EG/H3K?Za`T S I JGU H[Za`T EGB K?Za` Z&U;K?Za`3`aS# (vij , mij ) EGQB K?Za` (vji , mij ) § )0p!#""<,M/1!)'.#789%+"q7I, W'N#"%+ *A6 -,=NB+ .r& )0)'!9%+ 0 & ')(*,+( F = F \Pab r ( vab = mab > !\.( vab AB!(89 mab > ,=73!"%. 7I,H/1!)'/78'.(3!)02;J73+:0:5+g+1.9 9"(8AK(I,=73+:5+)K9 .<,=9(3.SU,=(89%+ ' )Vp+#9*2#!)'/M.##.%+"g$K'+ v +9 m .%!)K9*2#+1.g.<!:0:5+9%.g2;(3.%9(8)'/9%. ab ab +9)'!)d,!2 ,!/1+)&9%. > /1+h$K#(C(8:0#78(3$&'+J+)!#9"%+ m 6= v ab ba 8PR « ^ S K?Za` `EP PRZUT` STI Uk HSVZGFHa`X¥ Z P JI G5ZIBU ZI U S $H[ZPRZIBU 0 ] K JT EI K G `S EI P F ¥G KZPPRZZa`U `JU S ` M&JS U,Z § r !(89 % u 73+J.%!:0:5+9 2#+ vab −P −mab 73+G#78'. #"%K/<O'+J2#+ vab +9M, ,=)&9M2#+1. AB!(3.(8)'. & r 2',=)'. m −P −v !(89 u0 73+AB!(3.(8)^2#+ u 2',=)'. m −P −v 73+#78'.#"%K/<O'+2#+ ab ab ba 73!ab"%. ab ba > vab−P −u−u0−Pab−vba +1.9#)V/%O'+:0(8).<,=)'./1!"%2#+$&'+*[email protected]!)b)'!9%+ P 0 vba )R/1!)'.9<,=9%+L$K'+ 7 V (P 0 ) ∪ V (F 0 ) (8)'2;#(89H73+L78(8)'+ N"<,=#O'+J2C? #)'+J.#e2;(8AK(3.(3!)d)'!) T % M .r )5t% vad vac vab c2 Pad c3 e1 Pac e2 vda vca c1 Pab P vcd d1 vdc e3 vdb vcb e4 mab Pcd Pbd d2 d3 Pab vba Pbc vbd vbc eifTg"hq*A6;k 8 *.*q=q! 1+ ./<O'+:0(8)'.+)0N"<,!.+9q73+1.q9"(I,=)#N73+1.N"(3.%41. F +9 P #"%+#AB+H2#+ 7 /1!)'.9(89'+)&9 73+h.%!'. N"<,=#O'+h(8)'2;#(89 2#4sW')#(`,1A,=)&9 8V(= E .6 RTMH #F 9"(8AF(I,=73+2#+ K > !)'/ V (P 0 ) ∪ V (F 0 ) ,M#)'+9<,=(87873+.#4"(3+#"%+H!54N&,=73+ /1+7873+ 2#+ 4 1 / G + K $ #( Z 1 + . 9 % . . 8 ( # e 3 7 h + % . + #7 3 + 5 : + K ) 9 . ( Q / = , " 1 + % . * 9 #) T # 8 7 '. 1 / ! #" H 9 % / O'+:0(8) F> u = mab P 7 no,=(3.J2',=)'.\/1+b/Q,!. > (V (F ) \ V (v − P − m ) ∪ V (P ) (8)'2;#(89J73+b78(8)'+ N"<,=#O'+ ab ab ab 2C? #)'+J.#ep2;(8AF(3.(3!))'!)9"(8AK(I,=73+L2#+ K 2#+h9<,=(87873+G:0(8)#(8:^,=73+ > +97I,0/1!)'/78'.(3!)o2; 4 73+:0:5+G+1.%9e#(3+).<,=9(3.mSf,=(89%+ _ +1/(Z#"%!#AB+ 8N<= o ) +#9h2#!)'/0.##p.<+" [email protected]+ (3.9%+)'/1+02C? #)].%!:0:5+9 c 2#+ P , % ,=)K9L2#+1.AB!( 7 1 .(8)'.H2',=)'. F 0 > +9/%O'!(3.(t,='.%.(#"%F/%O'+L$K'+M.%.(8e#73+h2#+ v )2#4sW')#(89 2#+h: 2 :5+ ab % 7 d1 > .%!:0:5+9M2#+ P , ,=)K9G2#+1.HAB!(3.%(8)'.2',=)'. F 0 > +9/<O'!(3.(,='.%.(#"%F/%O'+\$K'+G. .(8e#73+02#+ m - !'.J,=7873!)'.hAB!(8"L$K'+\[email protected]+ (3.9%+)'/1+02#+0/1+1.L.%!:0:5+9%.L+)K9"<, 3)'+0#)'+ ab /1!)K9"<,!2;(3/9(3!) 81« Za`U [email protected],ZLF¥ Z 0 ZU Za`U [email protected],ZL¥FZ 0 N (c1 ) ∩ V (F 0 ) F N (d1 ) ∩ V (F 0 ) F )T.%+ ' )T.##p.<+2C?[,=e!"%2T$K'+ 73+M.%!:0:5+9 H = Pac ∪ Pbc ∪ Pbd ∪ Pad c1 , 2#+1.*AB!(3.%(8)'.H2',=)'.*73+h9"%! H )62#4sW')#(89\2#+ /%O'+:0(8)'.02#+ Y t ) H Hac = H \ vad +9 Had = H \ vac p 2#4sW')#(89 c 73+L.%!:0:5+9H2#+ H 73+M#78'.#"%F/%O'+J2#+ v +9AB!(3.(8)2#+ c )d2#4sW')#(89 2 ac ac *1r ( 3 7 ] + % . ! 0 : 5 : + 9 # 2 + 3 7 o + # 8 7 '. T # % " F % / ' O + # 2 + + T 9 B A ! 3 ( . 8 ( ) # 2 + c3 Had vad c1 c2 = c 3 > ,=73!"%.^[email protected]+)'.%+:Le#73+ V (H) ∪ V (c − c − P − v ) (8)'2;#(89b#)'+T % "<,=:0(32#+2#+d/1!(8) 2 1 ab Y ;> !)'/ Hr c2 +92#+o9"(I,=)#N73+ {vab , vac , vad } #)'+]/1!)K9"<,!2;(3/9(3!) c2 6= c3 ( c2 :^,=)'$K'+ c > ,=73!"%.G!).##.%+\.<,=)'.Mp+"9%+02#+\NB4)'4"<,=78(89%4^$&'+ v > c > c +9 v 3 ac 2 3 ad ,=#-,="<,=(3.%.<+)&92',=)'.t/1+9q!"%2;"%+g.%#" H i 9t!)\AB!(89 $K'+73+1.`9"%!(3.q/<O'+:0(8)'. c −P−v > 1 ab % c1 −c2 −Hac −vac +9 c1 −c3 −Had −vad (8)'2;#(3.%+)K9g#)'+ "<,=:0(32#+M2#+ /1!(8) c1 +92#+ 9"(I,=)#N73+ {v , v , v } Y #)'+0/1!)K9"<,!2;(3/9(3!) > !)'/\73+0.%!:0:5+9 c ,b!#"G.%+#73. 1 ab ac ad AB!(3.(8)'.J2',=)'. H 73+1.L2#+ .%!:0:5+9%. c +9 c $K#(.%!)&9J+)c!#9"%+^,!2 ,!/1+)&9%. t ) 2 3 .##.%+h.<,=)'.p+"9%+G2#+MNB4)'4"<,=78(89%4J$&'+ c c +1.9*#)'+L,=" 2 9%+L2#+ P ∪ P 2 3 ac bc r ( .%. : 2#+ +)]#78'.G2#+0.%+1.MAB!(3.(8)'.L.#" c H > 2#+1.MAB!(3.(8)'.L.#" Pcd > ,=73!"%.G!) > )'!9%+ c 73+ AB!(3.%(8)2#+ c .#" P 73+#78'.#"%K/<O'+h2#+ v ; )AB!(89*$&'+73+1.*/<O'+:0(8)'. 4 1 cd dc % c1 −P −vab > c1 −c4 −Pcd ∪ Pad +9 c1 −c2 −Hac −vac (8)'2;#(3.%+)K9G#)'+X "<,=:0(32#+ 2#+G/1!(8) c +92#+G9"(I,=)#N73+ {v , v , v } Y #)'+G/1!)K9"<,!2;(3/9(3!) B> !)'/ c :^,=)'$K'+ Pcd 1 ab ac 1 ad o (8)-,=73+:5+)K9 .( > c1 AB!(89\2#+1.X.%!:0:5+9%.\2#+ H ,=73!"%.J(87)'+^AB!(89X2#+ F 0 $KZ? #)'+ 2 =, " %9 + > ,=(8)'.(*$K'+V73+b#"%4AB!(89 8W6q= )6+#9\2#!)'/.##.%+"0$K'+ c :^,=)'$&'+ H 1 > ,=)'./1+G/Q,!. 2C?[,=#" : . 8N<= !)T+#9H.##.%+"$K'+ ,\2#+1.HAB!(3.(8)'.2',=)'. ∗ c1 Pcd > > r ( Z ) [ ? ] , K $ Z? #) % . + #7 B A ! 3 ( % . 8 ( ) = , 3 7 ! % " 5 . 3 7 1 + V . % / ' O + 0 : 8 ( ' ) . c1 c ∈ Pcd > c − c1 − P − vab > % c−Pcd −vcd ∪ Pac +9 c−Pcd −vdc ∪ Pad (8)'2;#(3.%+)K9*#)'+ "<,=:0(32#+M2#+M/1!(8) c +9*2#+ 9"(I,=)#N73+ {v , v , v } Y #)'+\/1!)K9"<,!2;(3/9(3!) > !)'/ c ,b#78'.(3+#"%.AB!(3.(8)'.h2',=)'. 1 ab ac ad ∗ B )0)'!9%+ c 73+AB!(3.(8)52#+ c 2',=)'. P ∗ 73+#78'.t#"%F/%O'+2#+ v +9 c 73+AB!(3.(8)52#+ Pcd 2 1 3 cd cd ∗ 73+M#78'.#"%F/%O'+L2#+ v -r ( c +9 c )'+h.%!)K9-,!.H,!2 ,!/1+)&9%. > ,=73!"%.*73+1. c1 2',=)'. Pcd 2 3 dc /<O'+:0(8)'. c −P−v > c −c −P −v ∪P +9 c −c −P −v ∪P (8)'2;#(3.%+)K9q#)'+ 1 ac 1 3 ab 1 2 cd cd cd dc ad % "<,=:0(32#+b2#+0/1!(8) c +9J2#+\9"(I,=)#N73+ {v , v , v } Y #)'+0/1!)&9"<,!2;(3/9(3!) > !)'/ 1 ab ac ad 73+1..%!:0:5+9%. c +9 c .%+/1!:0!"9%+)&9+)'/1!"%+ ,=(8)'.(-$K'+7f?[,=)#)'!)'/1+ 8W6q= ! ,G#"%+#AB+ 2 3 !#" d +1.9 .(8:0(87I,=(8"%+ _ +1/(Z#"%!#AB+ 8W6q= 1 )L)'!9%+2#!)'/ c , c 73+1.2#+ AB!(3.(8)'.`2#+ c 2',=)'. F 0 > +9 d , d 73+1.`2#+ AB!(3.(8)'. 2 3 1 2 3 T % M .6 )5t% 2 + d 2',=)'. F 0 A!,=)K92#+Lp!#"%.#(8AF"%+ > (87Sf,=#9HAB4"(W-+"h$K'+G73+1. 78(8)'+ 7 N"<,=#O'+1.h$&'+ # 1 )'!'.H,=7873!)'.9"%!#AB+")'+L2#4NB4) : "%+)&9H-,!.H+) L(K ) Y 4 8«^ ES U G/I &` EGF`P¤HNJPBYZ\¥ Z 0 GFS"Za`U K ZpK S I Z¤H[JP Y Z\¥`¦ G/I Z a` G1Q¥S .S `aSWEI J F ∪P ¥ Z ^ 0 S % K ¦ / G I ¥ a Z n ` U a H W S J K I ¤ K Za`¥ Z EIBU SWZIBU ] JKEHa` Za`U IBEI $&'+73/1!)'$K'+ K4 § J c1 J U HaS .SWJK § i 6 ) NB4)'4"<,=7 > !#"XAB4"(W-+"5$&Z? #) 78(8)'+ 7 N"<,=#O'+ J 2C? #)'+.#e2;(8AK(3.%(3!) 2#+ K +1.9 4 )'!)R9"(8AF(I,=7 > (87q.59h2#+JAB4"(W-+"G$K'+ J .%. : 2#+X#)]/1!(8)o$K#(t)Z?[,=#-,="9(3+)&9L$KZ? #)a.%+#79"(I,=)#N73+V2#+5e-,!.<+52#+ J _ [email protected]+1.9Je#(3+)a73+b/Q,!.G(3/( Y J )'+5+#9L-,!. 2 9"%+573+ 78(8)'+ 7 N"<,=#O'+ 2#+ K /Q,=" c )'++#9g,=#-,="9%+)#(8"$&Z? #)b.%+#7-9"(I,=)#N73+ 2#+ J 5_ +1/( 4 1 #"%!#AB+ 8Wq= 8!« i Ic5ZG/U `aGBE`&ZH G5Z Z`U GFI Z JHU,Zo¥FZ ZU G/I Z JHU,Zo¥ Z c2 c3 Pac d2 d3 Pcb ∪ Pbd § 2 r ( c2 c3 +1.9`#)'+g,=" 9%+g2#+ Pcd > ,=73!"%. [email protected]+)'.%+:Le#73+ V (c1−P−vab )∪V (Pac )∪V (Pad )∪ 7 7 V (Pcd ) (8)'2;#(89*73+78(8)'+ N"<,=#O'+G2C? #)'+M.#ep2;(8AF(3.(3!)e#(8-,="9(3+h2#+ K4 2#+9<,=(87873+h.9"(3/ 7 : W 8 q h = Y 7 9%+:5+)&9(8);S 4"(3+#"%+ /1+7873+X2#+ F > $&#( +1.9)'!) 9"(8AF(I,=73+X2C?[,=#" . #)'+J/1!)K9"<, 2;(3/9(3!) -r ( c c +1.9#)'+M,=" 2 9%+h2#+ P > ,=73!"%.*[email protected]+)'.%+:Le#73+ V (c −P −v ) ∪ V (F 0 ) 2 3 1 bc ab (8)'2;#(89h73+\78(8)'+ 7 N"<,=#O'+^2C? #)'+5.#e2;(8AF(3.(3!)oe#(8-,="9(3+52#+ K 2#+\9<,=(87873+0.9"(3/9%+:5+)K9 4 (8);S 4"(3+#"%+ /1+7873+L2#+ F > $&#(Z+1.%9*)'!) 7 9"(8AK(I,=73+J2C?[,=#" : . 8Wq=MY #)'+G/1!)K9"<,!2;(3/9(3!) > !)'/ 2 2 c2 c3 +1.9#)'+\,=" 9%+L2#+ Pac ∪ Pad )d:5!)K9"%+J2#+L: :5+J$K'+ d2 d3 +1.9 2 4 % #)'+,=" 9%+2#+ P ∪ P ,="b. :549"(3+T+95.<,=)'.0+"9%+2#+NB4)'4"<,=78(89%4 > !) p+#9 bc bd .##p.<+"$&'+ c c +1.9*#)'+L,=" 2 9%+G2#+ P d_ +1/(Z#"%!#AB+ 8TF= 2 3 ac )a.##.%+0.<,=)'.h+"9%+52#+0NB4)'4"<,=78(89%4b$&'+ vac > c2 > c3 +9 vad =, #-,="<,=(3.%.<+)&9 2',=)'.H/1+9 !"%2;"%+J.#" H ).%##p.%+L4N&,=73+:5+)&9h$&'+ d > d + 9 v =, #-,="<,=(3.%.<+)&9 2 3 ad 2',=)'./1+9H!"%2;"%+G.#" H 8s« `&EI Ug¥ Za`@`EP PRZUT`L¥S `U STI &UT`pZU IBEIJB¥&JB ZIBUT` d1 § ` r 4 ,=" 8W6q= +9 8TF= !) .<,=(89G$&'+ ( c AB!(89 d > ,=73!"%[email protected]+) 7 > c1 +9 d1 .%!)K9J2;(3.9(8)'/9%. 1 1 .%+:Le#73+ {c , d } ∪ V (H 0 ) (8)'2;#(89 #)d78(8)'+ 7 N"<,=#O'+L2C? #)'+J.#e2;(8AK(3.(3!)de#(8-,="9(3+L2#+ 1 1 7 : 8Wq=Y #)'+ K4 2#+M9<,=(87873+h.9"(3/9%+:5+)&9(8);SU4"(3+#"%+ F > $&#(C+1.9)'!) 9"(8AF(I,=73+G2C?[,=#" . C _ W 8 q j = /1!)&9"<,!2;(3/9(3!) ).<,=(89H2#!)'/G$K'+ c :^,=)'$&'+ d +1/(Z#"%!#AB+ c1 ZU 1 1 )T)'!9%+G,=73!"%. % e1 73+h.<!:0:5+92#+ c1 −P −vab , ,=)&9H2#+1.*AB!(3.%(8)'.2',=)'. (mab − Pab −vab )∗ > +9M/%O'!(3.(,='.%.(`#"%K/<O'+\$&'+Jp.<.(8e#73+L2#+ c1 )R)'!9%+ e4 73+X.%!:0:5+9 % 2#+ d −P −m , ,=)K9g2#+1.AB!(3.(8)'.2',=)'. (m −P −v )∗ > +9/<O'!(3.(p,='.<.('#"%F/%O'+ 1 ab ab ab ab $&'+Lp.%.%(8e#73+J2#+ d C )R/<O'!(3.(89M,=73!"%. e 2',=)'. (m −P −v )∗ > AB!(3.(8)R2#+ e > 1 2 1 ab ab ab +9 e 2',=)'. (m −P −v )∗ > AB!(3.(8)2#+ d 5! +1.H.%!:0:5+9%. e +9 e .%!)&9H/<O'!(3.(3. 3 1 2 3 ab ab ab ,='.%.(C#"%F/%O'+1.H$K'+M.%.(8e#73+M7f? #)d2#+M7f?[,=#9"%+ 8P« e1 = 6 vab § r (8)'!) 37 +1.M/<O'+:0(8)'. vab −P −c1 > vab −vac −Pac −c2 +9 vab −Pab −e3 −e4 −P − > S ! " :5+)K9*#)'+ % "<,=0 : (32#+h2#+/1!(8) v +92#+9"(I,=)#N73+ {c , c , c } d1 −d2 −Hac −c3 1 2 3 ab _ +1/(Z#"%!#AB+ 8W*q= .66 RTMH #F ! +1. / "(89 : "%+1. p!#"H73+G/%O'!( 2#+1. .%!:0:5+9%. /( 7 =, #" : . (8:0#78(3$K'41. 37 +1.H#"%!#"(349%41. > 8Wjq= +9 8W*q= +973+*Sf,=(89 .(8N)-,=734+)^2#4e##9g2#+#"%+#AB+ K$ '+ + 9 > > m v .<! )&92;(3.%9(8)'/9%. > ab ba :5!)K9"%+)&9M$&'+ c −P −e −e −P −e −e −P −d +1.%9#)d/%O'+:0(8)R2#+ G 2#!)K9 1 1 2 3 4 1 ab 73+1..%!:0:5+9%.(8)K9%4"(3+#"%.:^,=)'$K'+)&9 F 0 K )^)'!9%+/1+H/%O'+:0(8) P 0 F )^/1!)'.9<,=9%+H$K'+ [email protected]+)'.%+:Je#73+ E = V (P 0 ) ∪ V (F 0 ) (8)'2;#(89G73+078(8)'+ 7 N"<,=#O'+b2C? #)'+5.#e2;(8AK(3.%(3!)])'!) 9"(8AF(I,=73+G2#+ K 8 )'!)d9"(8AK(I,=73+h-,=" 8Wq=[= 4 8m« JRUVJS KTKZJ¥ Z Za`U `U HS . U,ZPRZI U S I.MQ HSWZGFH Z RKJ3UVJS KTK?ZL¥ Z E F§ 7.592#+G:5!)&9"%+"M$K'+G73+1.H.%( /<O'+:0(8)'.M2#+he-,!.%+L2#+ ! K ) 9 . 9"(3/9%+:5+)K9 :5!(8)'. E 2 2C?[,=" 9%+1. $&'+h73+1.H.( /<O'+:0(8)'. 2#+Ge-,!.%+h2#+ F )T)'!9%+ α 73+h)'!:Je#"%+L2C?[,=" 2 9%+1.2; /<O'+:0(8) v −P −m )d)'!9%+ β 73+G)'!:Je#"%+J2C?[,=" 2 9%+1.2#+ v −P −m ) ab ab ab ba ab ab )'!9%+ δ 73+M)'!:Le#"%+J2#+1. ,=" 2 9%+1. e 2#+ F 0 ,=#-,="9%+)-,=)K9 2#+1./<O'+:0(8)'. 2#+Me-,!.<+ F ! +G)'!:Le#"%+J2C?[,=" 2 9%+1. 2#+1. /%O'+:0(8)'.2#+ E +1.%9(8);S 4"(3+#" !d4N&,=7 δ + 2α − 3 _ +99%+Xe!")'+X/1!""%+1.%p!)'2],=o#(8"%+\2#+1./Q,!. /1+78#(q > e4 +9 mab .%!)&9h/1!);SU!)'2;'. > 5(87p+ (3.%9%+*#)5#)#(3$K'+.%!:0:5+92#+ P +)&9"%+ c +9 d > e AB!(89 v > e AB!(89 1 1 ab 2 > ,=1)'./1+/Q,!ab. 73+)'!2:Je#"%+ vab +9g P +9 vab −Pab −mab .%!)&92#+ : :5+H73!)#N'+#" > 2C?[,=" 2 9%+1. 2;d/%O'+:0(8)R2#+ E ,=787I,=)&9 2#+ c 1 + . * 9 # e 3 ( + ) 4 & N = , 7 d1 2α − 3 ! +)'!:Le#"%+*2C?[,=" 2 9%+1.q2#+1.q/%O'+:0(8)'.2#1+ F +1.9t4N&,=7 α+β +δ = 2α−ε+δ ,1AB+1/ 'i : ε ∈ {0, 1} /Q,=" > 2C?[,=#" .H7I,^2#4sW')#(89(3!)2#+ mab > (87`+ (3.9%+ ε 9%+7 $&'+ α = β + ε 9 2 2#!)'/H73+)'!:Le#"%+2C?[,=" 9%+1.g2#+1./<O'+:0(8)'.2#+He-,!.%+2#+ E +1.9.%9"(3/9%+:5+)&9g(8);S 4"(3+#" /1+78#(p2#+1./%O'+:0(8)'.g2#+*e-,!.%+H2#+ F q> !)'/*7I,M9<,=(87873+ 2#+ E +1.%9.9"(3/9%+:5+)&9g(8);SU4"(3+#"%+ /1+7873+G2#+ F d_ +1/(Z#"%!#AB+ 8V(= - !'.H,=""(8AB!)'. 2#!)'/ #)'+h/1!)&9"<,!2;(3/9(3!) 2 )+#9:^,=(8)&9%+)-,=)&92#!)#)'+"*#),=78NB!"(89O#:5+h2#+2#49%+1/9(3!)2#+1.78(8)'+ 7 N"<,=#O'+1. 2#+.#e2;(8AK(3.%(3!) )'!) 9"(8AF(I,=73+2#+ K 2',=)'.\73+1.\N"<,=#O'+1.0.<,=)'.\ % "<,=:0(32#+ q 7Sf,=#9 4 )'!9%+"h$K'+X/1+99%+X"%+1.9"(3/9(3!)a,= .#ep2;(8AF(3.(3!)'. 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