On-line vertex-covering

Texte intégral

(1)Laboratoire d'Analyse et Modélisation de Systèmes pour l'Aide à la Décision CNRS UMR 7024. CAHIER DU LAMSADE 183 Septembre 2001. On-line vertex-covering Marc Demange, Vangelis Th. Paschos.

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(69) p R f"Yo T 1t]u=Z=ZH\kt]X P 6⊆ P NJ"V=XlR TWV=Xl[]X X&^_PtmTdtwY*TxhaX(YtmTx\R=XikXl[mTWX&^ p ∈ P \ P YIR<Y*TwhaX(Yt Tw\R=X ikXl[mTWX&^ p ∈ P \ Pj N#"V=iXZY*TWV p , v , v , p _PtxYIu=q`;XlR9TW_aR=qb _rTWV []Xjt]ZHXo TjTWj\ Mio&\R9TW[dY}_Po TW_aR=qt]\ TWV=Xwf"Yo TiTWVY*T Mi _Ptj`Y*^_aù=`N i i j j QSRTWV=X \TWV=Xl[VYIR<Et]u=Z=ZH\ktmX =u}T N"V=XlR _rTWVTWV=XtdYI`;XYI[]qu=`;XlR9Tdt Hf2\[jYIR9g T \ikXl[mTW_Po&Xt p YIR< p _aR P ∩ PPj}TW⊂V=XwPZi Y*TWV p|P,jv| >, v 1, p _PtYIu=q`;XlR9TW_aR=q _rTWV|[]Xt]ZHXo TTW\ M N \R<tm_P}Xl[R=i\( cTWV=Xwjf2\haha\( _ajR=q;q[diYIZ=V B(N, E i ) o&i\R<jt TW[]u<j o TWX Ytf2\haha\( t 0 B 2 f \ w [ & X k i l X m [ g X } q ; X \ f X } d [ l Y b Y k i l X m [ W T & X < ^ 5haX&T N HX;TWV=XtmX&T \fikXl[mTW_Po&Xt tm\6}[dYl Rg XYIhPt]\ • M o&\R<t]_P}Xl[ikXl[mTW_Po&Xt\f P YtikXl[mTW_Po&Xt\f N g=_aR\TWV=XlM[ \[W=t N = N ∪ P g M a h & X T H 6 X W T = V 6 X k i l X m [ W T & X ^ \ f Y ] t ] t \ & o : _ * Y W T X e r _ W T V g a _ j f W T V=Xl[]XDX&^_PtmTdtYIR X}qXDha_aRGF_aR=q • Xl_rTWV=nXl[ ij v \[ v _rTWVDY ikXl[mTWNX&^ Mp ∈ P }TWV=XlR n p v∈ivjE ∈N M i j ij k B k gX&^Z=[]XtWtm_a\R mOIYIR<bTWV=Xw}_PtWo&u<t]t]_a\RDYIH\(ikXHYIR9g n ∈ N _Ptha_aRGX _rTWV6Y*T`;\kt T ∆ − 1 ikXl[mTW_Po&Xt \f P 2TWV=Xx`Y*^_aù=`v}Xlq[]XlXw\f G _Pt ∆ YIR<\R=XwX}qX ij TWV=XwM`Y*TdodV=_aR=q \R=X ha_aRGt v _rTWV v BN"V=_Pt i j []Xl`YI_aR<tTW[]u=X_af P = ∅ N\R<t]X

(70) 9u=XlR9TWhrg p 6 m(∆ − 1) N j QSRTWV=X\TWV=Xl[VYIR<_af o&\R9TBYI_aR<tSY t]X&T I \f ι _Ptm\h:Y*TWXikXl[mTW_Po&XtMTWV=XlR|TWV=XYI[]qu=`;XlR9TS}X&ikXlha\ZHX YIH\(ikXw[]Xl`YI_aR<tiYIha_P|\RTWV=G Xq[dYIZ=V G0(V \ I, E)

(71) N XN N tF_aR<o&X n = 2m + p YIR<=9gbhaXl`;`bYO' p 6 m(∆ − 1) 5[]Xt]ZN! p − ι 6 m(∆ − 1) }\R=XwX(Yt]_ahrgbqX&Tdt 5[]Xt]ZN! n − ι 6 m(∆ + 1)YIR<|[]X(YodV=XtTWV=Xxf2\haha\( _aR=qhaXl`;`YN n 6 m(∆ + 1) ' 7 ) 3 +& S&) wWSn ) ,NUY%&$|,.pM$'*!S'UV/mZ'/MWXUY,}v/Xp m > n/(∆ + 1) a +.1UYE/#wWSnHbSp ι ,.pM$*S'UV/m Z'/MWXUY,}v/Xp UYE/M& m > (n − ι)/(∆ + 1) a L

(72) 7F/H'*7 , ( $WAS&G w'WTSn UYE/JWTSUY,}$H$Q1UYE/Jp, / $QdS&)i ,.&),.iRS'*Z'/MWMUV/Tj(Qv$Z'/XWJUV$|UYE/dp, I/A$Q UYE/JZ'/MWXUV/j(Qv$Z'/MW#,.&4mk)v/mHqX G ,.pdq$kE&4m/m\SxqT$Z'/ qX a k. l. l. l. k. k. l. . k. . .

(73)

(74) . ∆/2. ¤ *' //E¥ {St]t]u=`;XJ=[WtmTSTWVY*T G VYtR=\_Pt]\h:Y*TWXDikXl[mTW_Po&XtM}XlR=\TWX9g C Y;`;_aR=_a`YIh

(75) ikXl[mTWX&^6o&\(ikXl[j\f G YIR<D9g TWV=Xtm_!lX\fTWV=X _aR<}XlZHXlR<}XlR9TtmX&TxYtWt]\o&_:Y*TWX6 _rTWV H_N XN! N SXlR=\TWX 9g TWV=X t]_!lX\fYs `Y*^_aù=` `Y*TdodV=_aR=q M \f G N g6_rTWXl` y\f¡f"Yo T Cyx

(76) YIhaq\[]|V_rTWV=\` C|x5 = sE5 "!$#%_aR<}u<mo&XtwY ikXl[mTWX&^Fnpo&\(ikXl[\ft]_!lX 2m NXoYIhahe=RYIhahrgTWVY*T <9g_rTWXl` Oj\ff"Yo Tjyx P = V \ X(M ) _Pt_aR<}XlZHXlR<}XlR9TN tF_aR<o&X C _Ptt]u=Z=ZH\ktmX `;_aR=_a`bYIh}1TWV=X_aR<}XlZHXlR<}XlR9T t]X&T V \ C _Ptw`Y*^_a`YIhN|\R<tmX

(77) 9u=XlR9TWhrg u<tm_aR=q L; M=N} XxqX&T s > n/(∆ + 1) ~y |C| = n − s 6 (∆ + 1)s − s = ∆s XbR=\( }_Pt TW_aR=qu=_Pt]V TWV=Xf2\haha\( _aR=q T \ oYtmXt }XlZHXlR<}_aR=q\R TWV=XiYIhau=Xt\f s YIR< m 0:5_. s 6 m 5_a_. s > m N. <\[oYtmX s 6 m =u<t]_aR=q;X&^Z=[]XtWtm_a\R ~y= XxqX&T 0. <\[oYt]X. s>m. =u<t]_aR=q;haXl`;`YO'} XxqX&T 0. M. |C| ∆s ∆ 6 6 2m 2m 2. . |C| n−s 2m + p − m m + m(∆ − 1) ∆ = 6 6 = 2m 2m 2m 2 2. \`=_aR=_aR=q X&^Z=[]XtWt]_a\R<t MYIR< \R=XxZ=[]\(ikXtTWV=XjTWV=Xl\[]Xl`N \R<tm_P}Xl[R=\( TWVY*T o&\R9TBYI_aR<t YtmX&T I \f_Pt]\h:Y*TWXikXl[mTW_Po&Xt&NR"V=XlR C HXl_aR=q|`;_aR=_a`bYIh}_rT}\FXt R=\T o&\R9TBYI_aRYIR9g _Pt]\h:Y*TWXG ikXl[mTWX&ÊNR <u=[mTWV=Xl[]`;\[]X MAX MATCHING(G[V \ I]) = MAX MATCHING(G) N sXlR<o&XTWV=XSYIRYIhrg}t]_PtZHXl[]f2\[]`X1P u<t TYIH\(ikX[]Xl`YI_aR<t¡iYIha_PbYIR<o&\R<o&hau<}XtTWV=XZ=[]\F\f\f TWV=XTWV=Xl\[]Xl`N ~Tj_Pt Xlhahrn?FR=\( R L; M=NTWVY*T f2\[jYIR9gq[dYIZ=V G YIR<Df2\[jYIR9g`bY*^_a`YIh

(78) `Y*TdodV=_aR=q M 5\f oYI[W}_aRYIhrn _rTg m \f G ~ τ (G) > m \R<tmX

(79) 9u=XlR9TWhrgH9gTWV=Xl\[]Xl` O'=TWV=Xwf2\haha\( _aR=qo&\[]\hah:YI[mgV=\hP=t_a`;`X}_:Y*TWXlhrgkN . . .

(80) ¢ $n/UY,.'*i/ A5S'A * )<'$ &4/J, ,. pdqT( $'$'kEW#&mrS/&Gm\JSrwqW$SZ'n/A qX UYE/WTa SUY,}$ $Q@UYb/p, I/1$QS'&G iA,.&G,.iRS*Z'/XWMUV/j(Qv$Z'/MWUV$ UYb/p, I/1$QyUYb/ . . ∆. # . / .@BA ! . 7 ) A5J /'I!2%(L ¥/H';%*L

(81) 7 +)l7 t = n XYI[]Xw[]X(YgR=\( cTW\;t]ZHXo&_af:g6YIR<6YIRYIhrgElXwTWV=Xxf2\haha\( _aR=q;\R}n~ha_aR=XYIhaq\[]_rTWV=`f2\[{SSN # %!$ > :3. # . /.. C ← ∅ t ← 1 Gt ← ({vt }, ∅)

(82) "! t<n v ← vt+1 t ← t+1 Et ← Et ∪ !"$#$ #%'&() Vt Vt ← Vt ∪ {vt+1 } Gt ← (Vt , Et ) ! * ∃u 6= v : uv ∈ E ∧ u ∈/ C # C ← C ∪ {v} *! t , + "-' + . . ¤ '*/H /E!:%*!2/ .c, 10 E/v$i@nG/ UY,NUY,.ZI,NUYJWTS'UY,}$A$Q@S'*Dwr$W,NUYri32 465 dS wrS,.&)pU S&`$nUY,.iS'*4$87 (+*-,.&4/S'*Dwr$W,NUYri M$W:96;c,.pqT$'kE&mr/mASrqT$'Z/1qX ∆ ) ,N*!/@,NUYp@v$inb/ UY,NUY,.ZI,NUYdWTS'UY,}$ S wrS,.&)pU =<+SppkEi ,.&Ew UYES'U #W/MUYkbW&)pqT$'UY|SJiRSIj,.iAkEi6iRS'UVvM,.&Ew M S&4m X(M )> ,.pqT$'kE&mr/mASxqT$Z'/1qX ∆/2 a@?y$'UY qT$'kE&m'pJSWT/dUY,!wxUVa ¤ '*//E¥ <\haha\( _aR=q YIhaq\[]_rTWV=À G ∈ C ∃uv ∈ E t]u<odVTWVY*T u ∈/ C NsXlR<o&XTWV=XikXl[mTWX&^Fnpo&\(ikXl[ C o&\`;Z=u}TWX_Pt`;_aR=_a`YIhN "V=XlRYIZ=Z=ha_Po∀v Y*TW_a\R<t\fo&\[]\hah:YI[mgOxYIR<b\f

(83) TWV=Xl\[]Xl` O'}[]Xt]ZHXo TW_rikXlhrgHo&\R<o&hau<}X TWV=Xw[dY*TW_a\kto&h:YI_a`;X N. _r^R=\( Y =o&\R<tm_P}Xl[Yt TBYI[ \R ikXl[mTW_Po&XtlNQ9i_a\u<t]hrg (S ) = 1 N tFu=Z=ZH\ktmX TWVY*T_rTdt o&XlR9TWXl[ ∆_Pt ∈TWV=NX=[WtmT¡ikXl[mTWX&^;[]X&ikX(YIhaSX∆+1 gFTWV=X[]X∆t T+\fH1ikXl[mTW_Po&XtoYIRHX[]X&ikX(YIhaτX _a∆+1 RYIR9g \[W}Xl[lN "V=XlR YIhaq\[]_rTWV=B ` _ahah9R=\T_aR<o&hau<}X TWV=Xt TBYI[mnpo&XlR9TWXl[_aR I V=_ahaX _rT

(84) _ahah9_aR<o&hau<}XYIhah9TWV=X¡[]Xl`bYI_aR=_aR=qikXl[mTW_Po&Xt \f S N|"V=Xl[]Xlf2\[]XTWV=Xo&\`ZHX&TW_rTW_ri_rTg[dY*TW_a\DYodV=C_aX&ikX _aRTWV=_PtoYt]X_PtwX

(85) 9uYIh TW\ ∆ N| <u=[mTWV=Xl[]`;\[]X t]_aR<o&X ∆+1 }TWV=Xo&\`;ZHX&TW_rTW_ri_rTg[dY*TW_a\ ∆/2 YIq9YI_aR<t T x5 x5 ! #%;_PtYIhPt]\YodV=_aX&ikX N τ (S )=m=1 .

(86) . . .

(87)

(88) . ∆+1. > : > ' / K 7F' $ /H . 0 / . %*L

(89) 7 +/H 79%*!:%*!:C

(90) !:% /E¥) . ) A2J /H'I!:%(L ¥~/'%*L

(91) 7D+)l7 t = n tFu=Z=ZH\kt]XTWVY*T ikXl[mTW_Po&XtYI[]XRFu=`HXl[]X_aR;TWV=X\[W}Xl[ TWV=X&gYI[][]_rikXg_aRbt TWXlZ i kikXl[mTWX&^ v _Pt []X&ikX(YIhaX N¡{hPtm\ o&\R<t]_P}Xl[TWVY*T _aRtmTWXlZ i {v , . . . , v } = C ∪ S V=Xl[]X C }[dYl twTWV=X;ikXl[mTWX&^Fnpt]X&iT_aR<o&hau<}X_aRTWV=X i i ikXl[mTWX&^ o&\(ikXl[ u=R<}Xl[ o&\R<tmTW[]u<o TW1_a\RbYIR< i S = {v Ni "V=X7=RYIh<q[dYIZ=V _Pt¡}XlR=\TWXYt u<t]uYIhahrg i 1 , . . . , vi }\Ci 9g G(V, E) YIR< _rTdtj`bY*^_aù=` }Xlq[]XlX;9g ∆ NA"V=X Z=u=[]ZH\ktmX;\f TWV=_Ptwt]Xo TW_a\R_PtjTW\|Z=[]\(i_P}X ha_a`_rTdtj_aR TWV=X;o&\`;ZHX&TW_rTW_ri_rTg YIq9YI_aR<tmTYIR\Z}TW_a`YIh \9Hn~ha_aR=XYIhaq\[]_rTWV=`\S\fYIR9g \R}n~ha_aR=XYIhaq\[]_rTWV=` tm\hri_aR=q6{S _rTWV t = n 5\(ikXl[jYIhah

(92) TWV=X Ylg}tTWV=X _aR=Z=u}T]n~q[dYIZ=V _Pt[]X&ikX(YIhaXBN1X&Tu<to&\R<t]_P}Xl[TWV=X t]\hau}TW_a\RD\f¡{S YtYT \n~Z=h:YlgkXl[Wt q9YI`X V=Xl[]XTWV=X|=[WtmT\R=X5Z=h:YlgkXl[OIx[]X&ikX(YIhPtTWV=X_aR<t TBYIR<o&XYIR<TWV=XbtmXo&\R<\R=X 5Z=h:YlgkXl[jyo&\R<tmTW[]u<o TdtTWV=Xtm\hau}TW_a\RN "V=XlR= XxZ=[]\(ikXwTWV=Xxf2\haha\( _aR=q;TWV=Xl\[]Xl`N L

(93) 7F/H'*7 3 [Da C1$%S* wr$WX,NUYi vS&cSvM,}/MZ'/Rv$i@nG/ UY,NUY,.Z ,NUYHWTS'UY,}$\pUYWX,}vMU}*-`qT/ U}UV/MWdUYES& ∆ /MZ'/X&B, S&B,.p $iR$W+n,}v $Q G ,.F p EI&$) &2,.& SmZ'S&4v/Ia . .

(94) _EaDC1$HS'* wr$WX,NUYi vS& SvM,}/MZ'/Av$'i@nG/MUY,NUY,.ZI,NUY|WTSUY,}$RpUYWX,}v U}*3%qT/ U}UV/MWJUYES& ∆ − 2 /MZ'/M&:, G ,.pdSUYW// S&4m n ,.D p EI&4$) &2,.& SmZ'S&4v/Ia ¤ '*//E¥ /E¥!:%(7 , "V=X_Pt]\`;\[]Z=V=_Po|\f G []X&ikX(YIhaX _aReYiYIR<o&X6o&\R<t]_Pt Tdt;\fSY }_PtNP \_aR9Tbo&\hahaXo TW_a\R \f p tmTBYI[WtMX(YodV6\f\[W}Xl[ ∆ + 1 YIR<\f ∆ − 1 _Ptm\h:Y*TWXikXl[mTW_Po&XtM V=Xl[]X ∆ YIR< p YI[]X1}^X_aR9TWXlqXl[Wt&N QS9i_a\u<tmhrgFTWV=X}Xlq[]XlX\f G _Pt ∆ YIR<;_rTdt¡\[W}Xl[ n = p(∆ + 1) + ∆ − 1 N{St]t]u=`;XTWVY*T Z=h:YlgkXl[SO[]X&ikX(YIhPt TWV=Xwq[dYIZ=V| _rTWV|[]Xt]ZHXo TTW\ TWV=Xwf2\haha\( _aR=q;[]u=haXt 0 ! _af C o&\R9TBYI_aR<t ∆ _Pt]\h:Y*TWXikXl[mTW_Po&Xt5f2\[ TWV=Xq[dYIZ=V|YIha[]X(Yg[]X&ikX(YIhaXTWV=XlR v _Ptha_aRGX;TW\YIhah i+1 TWV=Xt]iXxikXl[mTW_Po&XtMg !5! _af v ∈ S 5_aR\TWV=Xl[ \[W=t v VYtR=\THXlXlRTBYXlRb_aRTWV=XStm\hau}TW_a\RYIR< v _Pt R=\Tha_aRGXTW\YIR9g ikXl[mTWi X&^ vi j < i HYIR<_af i 6 in − ∆ }TWV=XlR|ikXl[mTW_Po&Xt v , v , . . . , v f2\[]` i Y tmTBYI[[]\F\TWX|_aR v g j i i+1 i i+∆ !5!5! _af p tmTBYI[WtVYlikXwHXlXlR[]X&ikX(YIhaX<TWV=Xw[]XtmT\fTWV=XjikXl[mTW_Po&Xt[]X&ikX(YIhaXDYI[]Xw_Ptm\h:Y*TWXg !:C _af[]u=haXt ! YIR< !5! oYIR=R=\THXYIZ=Z=ha_aX YIR< i 6 n − 1 =TWV=XlRikXl[mTWX&^ v _Pt_Ptm\h:Y*TWX _rTWV6[]XtmZHXo T i+1 TW\ TWV=Xxq[dYIZ=VDYIha[]X(Yg|[]X&ikX(YIhaX N {Z=Z=ha_PoY*TW_a\R<t\f[]u=haXt ! TW\ !2C YIH\(ikX_a`;Z=ha_aXtjTWVY*TZ=h:YlgkXl[yoYIR=R=\T}\|HX&T]TWXl[wTWVYIR o&\(ikXl[]_aR=q6X}qXt \f YIR9gtmTBYI[9g_rTdthaX(YlikXtM V=_ahaX\Z}TW_a`YIh

(95) \9Hn~ha_aR=Xt]\hau}TW_a\RDo&\R<t]_PtmTdt\fTWV=X tmTBYI[mnpo&XlR9TWXl[Wt&N"V=Xl[]Xlf2\[]XY [dY*TW_a\;\f ∆ _PtYodV=_aX&ikXDY*THXt TSYIR<TWV=_Pto&\`;Z=haX&TWXtTWV=XwZ=[]\F\f\f_rTWXl` Oj\fTWV=XxTWV=Xl\[]Xl`N ¤ '*//E¥/E¥S!:%(7 3 1X&T ∆ HXYIRD_aR9TWXlqXl[Sq[]X(Y*TWXl[TWVYIR \[X

(96) 9uYIhTW\G MYIR<Dt]X&T n = ∆(∆ + 1) + 1 N \R<tm_P}Xl[TWVY*TZ=h:YlgkXl[Oj[]X&ikX(YIhPtTWV=Xxq[dYIZ=V6f2\haha\( _aR=q;TWV=Xw[]u=haXtHXlha\( J0 ! _af C o&\R9TBYI_aR<t ∆ _Ptm\h:Y*TWXikXl[mTW_Po&Xty2 _rTWV []XtmZHXo T TW\STWV=Xq[dYIZ=V;YIha[]X(Yg[]X&ikX(YIhaX YIR< i 6 n−2 TWV=XlR i v _Ptha_aRGX|TW\YIhahTWV=XtmX_Ptm\h:Y*TWX|ikXl[mTW_Po&Xt g i+1 !5! _af v ∈ S 5_aR\TWV=Xl[ \[W=t v VYtR=\THXlXlRTBYXlRb_aRTWV=XStm\hau}TW_a\RYIR< v _Pt R=\Tha_aRGXTW\YIR9g ikXl[mTWi X&^ vi j < i

(97) YIR<D_af i 6i n − ∆ − 1 TWV=XlR ikXl[mTW_Po&Xt v , v , . . . , v i f2\[]` YbtmTBYI[j[]\F\TWX i i+1 i+∆ _aR v g j i !5!5! o&\R<t]_P}Xl[ v ∈ S v _Pt]\h:Y*TWXD _rTWV []XtmZHXo TxTW\TWV=X q[dYIZ=VYIha[]X(Yg []X&ikX(YIhaXYIR< n − 2 > i > g t]X&T i A =i {vi : j < i, v ∈ C , ∀k 6 i, v v ∈/ E} 5_N XN! A _PtxTWV=XtmX&T \fTWV=X_Ptm\h:Y*TWX n−∆ ikXl[mTW_Po&Xt HY*T_aR<tmTBYIR9T jt =TBYXlR_aR jC YIR<i B = {v j k, . . . , v } g}TWV=XlR0 i i+2 n−1 Y v _Ptha_aRGXTW\ v YIR<TW\YIR9gXlhaXl`XlR9T\ftmX&T A g i+1 i 5TWV=XwXlhaXl`;XlR9Tdt\f B f2\[]` YIR_aR<}XlZHXlR<}XlR9TjtmX&TSYIR<6YI[]Xwha_aRGXTW\ v i !:C _af []u=haXt ! YIR< !5! }\|R=\TYIZ=Z=hrgYIR< _af i 6 n − ∆ TWV=XlR ikXl[mTWX&^ v _Ptj_Pt]\h:Y*TWX _rTWV TWV=X i+1 q[dYIZ=VDYIha[]X(Yg|[]X&ikX(YIhaXg C v _Ptxha_aRGXTW\ ∆ ikXl[mTW_Po&Xt\f}Xlq[]XlX OZ=_PoX_aRTWV=XtmX&ikXl[dYIho&\R=R=Xo TWX o&\`;ZH\R=XlR9Tdt\fTWV=X qn[dYIZ=V[]X&ikX(YIhaXu=R9TW_ah

(98) tmTWXlZ n − 1 N pR =qu=[]X OY q[dYIZ=V }T]TW_aR=q[]u=haXt ! TW\ C _Pt;t]V=\( Rf2\[ YIR< N {¡Xl[mTW_Po&XtYI[]X RFu=`HXl[]X6_aRTWV=X\[W}XlG[TWV=X&gVYlikX HXlXlRD[]X&ikX(YIhaX NysXl[]XH[]u=haX !5n!5! =_Pt31YIZ=Z=ha_aXD∆f2\[=ikXl5[mTWX&^ yFN"V=XlR YIR< N"V=Xxo&_a[Wo&haXSikXl[mTW_Po&Xt[]XlZ=[]XtmXlR9TTWV=Xx{Snpt]\hau}TW_a\R} V=_ahaXSTWV=XxtL

(99) 9uYI[]X A = {19, 20} \R=Xt[]XlZ=[]XtmXlR9T TWV=XbB_aR<=}Xl{29, ZHXlR<30} }XlR9Tt]X&T YtWtm\o&_:Y*TWX _rTWV _rTN { q[dYIZ=V }T]TW_aR=q _rTWXl` O\fTWV=XbTWV=Xl\[]Xl` o&\u=hPbHXxYtTWV=XS\R=XS\fe=qu=[]X O_aR<}u<o&Xb9g;TWV=XxtmX&T\f1ikXl[mTW_Po&Xt {1, . . . , 27} Z=hau<t\R=XS_Pt]\h:Y*TWXtT

(100) 9uYI[]X ikXl[mTWX&ÊN pf[]u=haX !"!5! _PtR=\T YIZ=Z=ha_aXkTWV=XlR _aR tmTWXlZ TWV=Xq[dYIZ=V;o&\R9TBYI_aR<t tmTBYI[WtMkTWV=Xl_a[ikXl[mTW_Po&Xt \fH}Xlq[]XlXwO `YF_aR=q|ZYI[mTw\f TWV=X;t]\hau}TW_a\R o&\R<tmTW[]u<o TWXn−1 9g Z=h:YlgkXl[yFNApR TWV=_Ptwo∆ Yt]X τ (G) = ∆ + 1 2TWV=X []\F\Tdtx\f F.

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(117) ` ` ` `_` _$ #"! `_` _$ #"! `_`_$# baba h gfeh gfedcdch gfe h gfeh gfedcdch gfe h gfeh gfedcdch gfe hghgd cd chg d cd c d cd c dcdc '(' 2 1 )*) 2 1 4 3 21 +,+ 43 8 765 -.- 8 765 87 /0/ _` _ _ _ _ _ _ _ _. `. `. `. `. `. ` _` _ _ _ _ _ _ _ _. `. `. `. `. `. `. `. ` __ __ __ __ __ __ __ __ __ ` ` ` ` ` ` ` ` ` __ `` _ ` __ `` _ `__``_ baabba dch gfech gfeddc dch gfech gfeddc dch gfech gfeddc d chgc hgdd c d cc dd c d cc dd c dccddc `` ` ` ` ` ` ` ` ` _` _ _ _ _ _ _ _ _. `. `. `. `. `. `. `. ` h gfeh gfedcdch gfe h gfeh gfedcdch gfe h gfeh gfedcdch gfe hghgd cd chg d cd c d cd c dcdc _` _ _ _ _ _ _ _ _ _ _ _ a. `. `. `. `. `. `. `. `. `. ` ` b _` _ _ _ _ _ _ _ _ _ _ _ a. `. `. `. `. `. `. `. `. `. ` ` b __ __ __ __ __ __ __ __ __ ` ` ` ` ` ` ` ` ` __ `` _ ` __ `` _ `__``_ baab^ba dch gfech gfeddc dch gfech gfeddc dch gfech gfeddc d chgc hgdd c d cc dd c d cc dd c dccddc `` ` ` ` ` ` ` ` ` _` _ _ _ _ _ _ _ _. `. `. `. `. `. `. `. ` _ ` _ ` _ ` _ ` _ ` _ ` _ ` _ ` _ ` _ ` _ `_ ]^ba] h gfedc h gfedc h gfedc hgd c d c d c dc 18. 1. 28. 10. 7. 8. 17 19. 9. 16. 29. 30. 21. 20. 27. 15. 2. 3. 4. 5. 6. 11 12 13 14. 22 23 24 25 26. 31. _aqu=[]XO'0 { q[dYIZ=V}T]TW_aR=q[]u=haXt ! TW\ C _rTWV. n = 31. YIR<. ∆=5. N. TWV=X t TBYI[WtZ=hau<tikXl[mTWX&^ v < V=_ahaXwTWV=X t]\hau}TW_a\R o&\R<t TW[]u<o TWXD_Pt\f t]_!lX ∆(∆ + 1) N~"V=Xo&\`;ZHX&TW_rTW_ri_rTg [dY*TW_a\;_Pt_aR|TWV=_PtoYt]X ∆ N n tFu=Z=ZH\kt]XR=\( TWVY*Tj[]u=haX !5!"! _PtSYIZ=Z=ha_aX 5[]XoYIhahTWVY*TTWV=_Pt_PtTWV=XoYt]X f2\[ikXl[mTWX&^ y _aR`=qu=[]XOIBN "V=XlR;_aRtmTWXlZ i 9TWV=Xq[dYIZ=Vo&\R<t]_Pt Tdt\f k tmTBYI[Wt@2TWV=Xl_a[haX(YlikXt `YF_aR=qZYI[mT\fETWV=Xtm\hau}TW_a\Rbo&\R<tmTW[]u<o TWX 9gZ=h:YlgkXl[y Z=hau<t;TWV=XDikXl[mTW_Po&Xt\f A ∪ {i} N pRTWV=_PtoYtmXTWV=X6TW\TBYIhRFu=`HXl[\fjikXl[mTW_Po&XtikXl[]_!=Xt H _rTWV |A| < ∆ 5_af¡R=\T []u=haX ! \u=hPDHX YIZ=Z=ha_aX n = ∆(∆ + 1) + 1 = k(∆ + 1) + |A| + |B| + 3 \R=X`;\[]X;TW_a`;X>wYIR< |B| 6 ∆ − 2 5HXoYIu<tmX i > n − ∆ BNHXb}X}u<o&X k = ∆ − 1 NpR TWV=_Pt oYtmX V=_ahaXTWV=X t]\hau}TW_a\Rh=RYIhahrg o&\R<t TW[]u<o TWX 9gZ=h:YlgkXl[|y VYtbY*ThaX(YtmT τ (G) = ∆ − 1 + 3 = ∆ + 2 k i l X m [ W T P _ & o X 1 t 2 W T V=XjhaX(YlikXt\fTWV=X ∆ − 1 t TBYI[WtZ=hau<tt]X&T A Z=hau<tt]X&T B Z=hau<t x BN"V=X (∆ − 1)∆ + ∆ = ∆2 o&\`;ZHX&TW_rTW_ri_rTg[dY*TW_a\;_a`;Z=ha_aX_PtTWV=XlRDY*ThaX(YtmT ∆ − 2 NpR%=qu=[]XO' τ (G) = 7 2TWV=Xx\Z}TW_a`bYIh1tmi+1 \hau}TW_a\R_Pt TWV=X t]X&T {1, 10, 18, 21, 27, 28, 31} = V=_ahaXwTWV=Xtm\hau}TW_a\R o&\R<tmTW[]u<o TWX69gZ=h:YlgkXl[jy2TWV=X t]X&T\fTWV=X o&_a[Wo&haX ikXl[mTW_Po&Xt\f =qu=[]XOI_Pt\foYI[W}_aRYIha_rTgDyFN pRb\[W}Xl[TW\o&\R<o&hau<}X=haX&Tu<tR=\TWXSTWVY*TTWV=Xjq[dYIZ=V|Y*Tt TWXlZ o&\R<t]_PtmTdt\f Yo g}o&ha_Pojo&\R=R=Xo TWX o&\`;ZH\R=XlR9Tdt X(YodV\fETWV=Xl` o&\R9TBYI_aR=_aR=q Y*ThaX(YtmT\R=XikXl[mTWX&^;\f1n}Xl−q[]1XlXON "V=_Pt `∆ YXt TWVY*T[]u=haX C _Pt f2X(Yt]_a=haXjYIR< quYI[dYIR9TWXlXtTWVY*T¡TWV=Xy=RYIhHq[dYIZ=V_PtYjTW[]XlXNftF\G9TWV=XZ=[]\F\f5t \f _rTWXl` yxYIR<;\fETWV=XTWV=Xl\[]Xl` YI[]Xo&\`;Z=haX&TWXN. u. lKp

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(120) x{ x { x { x { x { x { x

(121) { #. . . *%. . t i←2 C ← C ∪ X(MAX MATCHING(Gi )) * u ∈ Vi \ X(MAX MATCHING(Gi )) !!* ∃v ∈ (∪ 16j6i−1 Vj ) \ C : uv ∈ E . #. C ← C ∪ {u}. *"!. ,+"-'+ " . QS9i_a\u<tmhrgTWV=XDt]X&T =RYIhahrgeo&\`;Z=u}TWX9g ' _PtbY ikXl[mTWX&^Fnpo&\(ikXl[ SYIhrTWV=\u=qV R=\TbR=Xo&Xt]tdYI[]_ahrg ` _aR=_a`YIhNtF\G1Z=[]\ZH\kCt]_rTW_a\R O }\FXtwR=\Tw[]XlZ=[]Xt]Xl R9TTWV=X \[WtmToYt]X;f2\[x_rTdtwo&\`;ZHX&TW_rTW_ri_rTg [dY*TW_a\=N \TWX ; TWVY*T|f2\[bTWV=XoYt]X V=Xl[]X o&hau<tmTWXl[Wt6YI[]XYtWt]u=`;X _rTWV=\u}T6YIR9g []XtmTW[]_Po TW_a\Rjt]X&T]TW_aR=q |C| 6 n − |I| 2 V=Xl[]X I }XlR=\TWXt;TWV=Xt]X&T\fS_Pt]\h:Y*TWX ikXl[mTW_Po&XtM_afjYIR9g);YIR< u<t]_aR=qhaXl`;`bYyxo&\`;ZHX&TW_rTW_ri_rTg [dY*TW_a\ _Pt_a`;`;X}_:Y*TWXlhrg}X}u<o&X N ∆+1 L

(122) 7F/H'*7 > /MU λ q/2UYE/:&GkbiqT/MW$LUYb/:,.p $'*SUV/m Z'/MWXUY,}v/Xp2$Q G ,.&UYWT$Imk)v/m ,.& C S'&m p /MU Pt S'&m ρ =i λ/|A| a 0 E/M& UYE/Bv$i@nG/ UY,NUY,.ZI,NUY WTS'UY,}$ $Q2S'*Dwi$'W,NUYi 2,4 5 S wrS,.&)pURS& λ = λ i $nUY,.iS'* i=1 96;KS'*Dw$'W,NUYi ,.p qT$kE&4m/m\SrqT$'Z/ q a . . 2 + (∆ − 2)/(2 − ρ). ¤ '*//E¥ SXlR=\TWX69g TWV=X|ikXl[mTWX&^Fnpt]X&Tdt_aR9TW[]\}u<o&Xe_aR C }u=[]_aR=qTWV=X6X&^Xo&u}TW_a\Re\f t ah _aR=Xt x { TW\ x { \fYIhaAq\i []_riTWV==` 2, "... , ; YIR<|t]X&T A = ∪t A NQS<tmXl[mikXjTWVY*TikXl[mTWX&^Fnpt]X&T A _PtX&^ZH\kt]X _rTWV[]Xt]ZHXo T TW\STWV=X15R=\R}n~`bY*^_aù=`\

(123) `Y*TdodV=_aR=q ∪t Mi=2g9`;i\[]Xl\(ikXl[ k_rT¡}\FXt R=\T¡o&\R9TBYI_aRYIR9g _Pt]\h:Y*TWX ikXl[mTWX&ÊNQS<tmXl[mikXjYIhPt]\TWVY*TYIR9g;_Ptm\h:Y*TWXikXl[mTWX&^;_Pt X&î=1 ZH\ktmXi; _rTWV[]Xt]ZHXo TTW\YIR9g;`bY*TdodV=_aR=q\f G gFV=XlR<o&X }F λ 6 |A| ⇐⇒ ρ 6 1 SXlR=\TWX9g M i = 1, . . . , t Y;`Y*^_aù=``bY*TdodV=_aR=q\f G Et]X&T m = |M | i = 2, . . . , t N@"V=XlRHTWV=X oYI[W}_aRYIha_rTg|\fiTWV=Xw\R}n~ha_aR=Xt]\hau}TW_a\R C o&\`;Z=u}TWX|9g ' i oYIR|Hi Xx []_rT]TWiXlRDYtf2\haha\( tM0 |C| = 2. t X i=1. ~. mi + |A|. 1 X&T E 0 HXTWV=XtmX&TS\f X}qXtTWVY*TSVYlikX XlR9TBYI_ahaXD_aR9TW[]\}u<o TW_a\R \fTWV=XikXl[mTW_Po&XtS\f A _aR C YIR<D}XlR=\TWX 9g B(A ∪ (V \ X(M )), E 0 ) TWV=XxZYI[mTW_:YIh

(124) tmu==q[dYIZ=V6\f G _aR<}u<o&X|9g A ∪ (V \ X(M )) YIR<9g E 0 N {hPtm\G }XlR=\TWX|1 9g M 1 YD`Y*^_aù=` `Y*TdodV=_aR=q \f B YIR< 9g m TWV=XoYI[W}_aR1YIha_rTg \f 1M N Xl`YI[L YIhPt]\|TWVY*T M ∪t BM _PtY`Y*TdodV=_aR=q\f G

(125) R=\TR=XtWt]XoYI[]_ahrg`BY*^_aù=``

(126) YIR< TWVY*TTWV=X Bq[dYIZ=V B _Pt =_aZYI[mTW_rTWX _rTWBVbo&\i=1 ha\[mnpo&h:iYt]t]Xt A YIR< V \ X(M ) N ^Z=[]XtWtm_a\RB~ oYIRf2\[¡TWV=XZ=u=[]ZH\ktmXt\f\u=[ Z=[]\F\f+ 1 1 HXw[]X& []_rT]TWXlR Yt ! t X }O τ (G) > m +m i. B. YIR<<u<t]_aR=q;X&^Z=[]XtWtm_a\R<t ~YIR<c}O=TWV=Xo&\`;ZHX&TW_rTW_ri_rTg|[dY*TW_a\;\f ' ;HXo&\`;Xt i=1. ct . OLVC. =. |C| |C| 6 t P τ (G) mi + mB. _aRYIhahrg<u<t]_aR=q;haXl``Y yx} XxqX&T. i=1. mB >. t P 2 mi + |A| |A| − 2mB i=1 = =2+ t t P P mi + mB mi + mB i=1. |A| |A ∪ (V1 \ X(M1 ))| > ∆ ∆. O. }C. i=1. mO.

(127) Q RTWV=X\TWV=Xl[¡VYIR<9haX&T i ∈ {1, . . . , t} YIR< haX&T I HXTWV=Xt]X&T¡\fTWV=X_Pt]\h:Y*TWXikXl[mTW_Po&Xt¡\fEo&hau<tmTWXl[ G N X S i TWV=XlR;VYlikX |I| = λ +(|I|−|A ∩I|) k\[ |A |−λ = |A |+(|I|−|A ∩I|)−|I| 6 p −|I| 6 m (∆−1) i i i i i i i i V=Xl[]X6TWV=X:=[Wt T_aR=X

(128) 9uYIha_rTgeV=\hP=tHXoYIu<t]X A ⊆ P YIR<Yt; X6VYlikX YIha[]X(Yg `;XlR9TW_a\R=Xe_aReTWV=X HXlq_aR=R=_aR=q\fSTWV=X6Z=[]\F\f+TWV=X6t]X&T I \ (A ∩ I)i HXl_aR=i q _Pt]\h:Y*TWXe_aR G _rT_Pt;X&^ZH\ktmX _rTWVe[]XtmZHXo T TW\DYIR9g `bY*TdodV=_aR=q\f G gTWV=Xt]Xo&\R<_aR=X

(129) 9i uYIha_rTgV=\hP=txTWVYIRGtwTW\haXl`;i`Y ONHtFu=`;`;_aR=q|_aR=X

(130) 9uYIha_rTW_aXt i f2\[ i = 1, . . . , t } Xx\}TBYI_aR |A | − λ 6 m (∆ − 1) i. i. i. t X. mi >. mOOI. |A| − λ ∆−1. Stm_aR=q;X&^Z=[]Xt]t]_a\R<t }F mOYIR<cmOOI<X&^Z=[]XtWtm_a\R }CHXo&\`Xt i=1. ct . OLVC. 6 2+ 6 2+. |A| − 2mB t P mi + mB. i=1 ∆−2 ∆ ∆(1−ρ)+∆−1 ∆(∆−1). 6 2+. = 2+. 2|A| ∆ |A| |A|−λ + ∆−1 ∆. |A| −. ∆−2 (∆ − 2)(∆ − 1) 6 2+ ∆(2 − ρ) − 1 2−ρ. mO(y. YIR<TWV=XwZ=[]\F\f\fTWV=XjTWV=Xl\[]Xl`_Pto&\`;Z=haX&TWXN XoYIhahTWVY*T 5X&^Z=[]Xt]t]_a\R }FLBNf <u=[mTWV=Xl[]`;\[]X<X&^Z=[]XtWtm_a\R mO(y_Pt_aR<o&[]X(Yt]_aR=q _rTWV ρ g=V=XlR<o&X t]X&T]TW_aR=q ρ = 1 TWV=ρX6f2\ha1ha\( _aR=q;[]Xt]u=hrT_Pt_a``;X}_:Y*TWXlhrgb\}TBYI_aR=X N )<' 3 0 E/Av$i@nG/ UY,NUY,.Z ,NUY|WTS'UY,}$\$LJS'*Dwr$WX,NUYi 2,4 5 S wrS,.&)pU S& $nUY,.iRS'* 9 ;hS'*Dwr$W,NUYri ,.p ¢ qT$'/kE'*&/ mrA5/A m\ Srq$Z'/ q ∆ a QSR6TWV=X \TWV=Xl[SVYIR<o&\R<tm_P}Xl[STWVY*Txo&hau<t TWXl[WtxYI[][]_rikX _rTWV=\u}Tj_Pt]\h:Y*TWX6ikXl[mTW_Po&Xt&N#pR6TWV=_PtSoYt]XEf2\[xYIR9g <t]\G ρ = 0 YIR<<u<t]_aR=q;X&^Z=[]XtWtm_a\R O(yx}TWV=Xwf2\haha\( _aR=q;V=\hP=tlN i = 1, . . . , t λi = 0 /MWp`SWXW,.Z'/ ) ,NUYE$kbUd,.pM$'*!S'UV/m Z'/MWMUY,}v/Mp UYE/v$inb/ UY,NUY,.ZI,NUY WTS'UY,}$ $Q ¢ S'*Dwr/$'*W/ ,NA5UYA r)<i ' > 2, 4 5 S wrES/M,.&4&)/XpZ'U /MSW& v $*3kEnpXUY,.UViR S* 96;KS'* wr$WX,NUYi ,.pdq$kE&4m/mHSrqT$Z'/AqX (∆ + 2)/2 a \TWXTWVY*T TWV=X6tm\hau}TW_a\R o&\`;Z=u}TWX 9g YIhaq\[]_rTWV=` ' _PtR=\T;R=Xo&XtWtdYI[]_ahrg `;_aR=_a`YIhNe\R<t]X&n

(131) 9u=XlR9TWhrg<TWV=Xx[]Xt]u=hrT\fo&\[]C\hah:YI[mgDyoYIR=R=\THX}Xl[]_rikX|9 g|}_a[]Xo TjYIZ=Z=ha_PoY*TW_a\R6\fo&\[]\hah:YI[mgON "V=XH\u=R< \fo&\[]\hah:YI[mgMoYIRHXt]ha_aqV9TWhrg_a`Z=[]\(ikX9gxTWV=Xf2\haha\( _aR=qS YlgkN tF_aR<o&Xf2\[ i = 1, . . . , t TWV=XxikXl[mTW_Po&Xt\f A YI[]XX&^ZH\ktmX| _rTWV|[]XtmZHXo TTW\ M =u<t]_aR=qhaXl`;`YbO A ∩ I = ∅ } XxqX&T i i i mOM |Ai | 6 mi (∆ − 1) QSRDTWV=X\TWV=Xl[jVYIR<

(132) tm_aR<o&X R=\o&hau<tmTWXl[wo&\R9TBYI_aR<tx_Ptm\h:Y*TWX ikXl[mTW_Po&Xt TWV=X =_aZYI[mTW_rTWX q[dYIZ=V B(A ∪ (V \ o&\R<t]_P}Xl[]Xb_aRTWV=XZ=[]\F\f\f TWV=Xl\[]Xl` MwVYt `Y*^_aù=` }Xlq[]XlXjH\u=R<}XYIH\(ikX9g ∆ −1 1 X(M1 )), E 0 ) Y*ThaX(Yt T\R=X X}qX;ZHXl[xikXl[mTWX&^ _aR A ∪ (V \ X(M )) 1ha_aRGtx_rTxTW\ikXl[mTW_Po&Xt\f X(M ) i = 1, . . . , t BN 1 1 i \R<tmX

(133) 9u=XlR9TWhrg mO |A| 6 (∆ − 1) |V1 \ X(M1 )| 6 (∆ − 1)mB V=Xl[]XTWV=X6h:YtmT_aR=X

(134) 9uYIha_rTg V=\hP=tHXoYIu<t]X_aRUY=_aZYI[mTW_rTWXDq[dYIZ=VTWV=XDoYI[W}_aRYIha_rTge\fwY`Y*^_aù=` `Y*TdodV=_aR=q_Pttm`bYIhahaXl[TWVYIRTWV=X oYI[W}_aRYIha_rTg6\fTWV=X tm`YIhahaXtmTjYI`;\R=q TWV=X o&\ha\[mnpo&h:YtWtmXtlNytF\GEo&\`=_aR=_aR=q X&^Z=[]XtWtm_a\R<t mOMYIR<cmO <X&^Z=[]XtWtm_a\R }CHXo&\`;Xt 0. . . . ct . OLVC. 62+ . |A| |A| − 2 ∆−1 |A| − 2mB ∆−3 ∆+1 6 2 + =2+ = 2|A| t P 2 2 ∆−1 mi + mB. mO(. YIR<|X&^Z=[]XtWtm_a\R m O(haX(Y=t_a`;`X}_:Y*TWXlhrgTW\ TWV=Xxf2\haha\( _aR=qR=RYIh

(135) o&\[]\hah:YI[mgkN /MWp`SWXW,.Z'/ ) ,NUYE$kbUd,.pM$'*!S'UV/m Z'/MWMUY,}v/Mp UYE/v$inb/ UY,NUY,.ZI,NUY WTS'UY,}$ $Q ¢ S'*Dwr/$'*W/ ,NA5UYA r)<i ' 2, 4 5 S wrES/M,.&4&)/XpZ'U /MSW& v $*3kEnpXUY,.UViR S* 96;KS'* wr$WX,NUYi ,.pdq$kE&4m/mHSrqT$Z'/AqX (∆ + 1)/2 a . i=1. . C.

(136) : 3 ?;.@BA ! . 7DCE7F'(%79G @ +/CH7F'I! .

(137) J K !:%*L t = 2 tFu=Z=ZH\kt]XbR=\( TWVY*TTWV=Xb_aR=Z=u}Tq[dYIZ=V _Pt []X&ikX(YIhaX _rTWV=_aR T \ o&hau<tmTWXl[Wt YIR< N {St]t]u=`;XYIhPtm\|TWVY*T n 1TWV=X\[W}Xl[\fTWV=XR=RYIh q[dYIZ=V_Pt1FR=\( RY*TwTWV=X;HGXl1q(V _aR=R=1,_aR=EqD1)\fTWV=X;Gq92YI(V `;2X,NRE2)X []XoYIhahTWVY*T 1f2\haha\( _aR=q\u=[wYtWtmu=`Z}TW_a\R<tM1\R=X VYtSTW\|}Xo&_P}X V=_PodV ikXl[mTW_Po&Xtx\f¡TWV=X =[Wt To&hau<t TWXl[x _ahah HXlha\R=q;TW\TWV=X1=RYIh

(138) t]\hau}TW_a\R6HXlf2\[]XxTWV=X YI[][]_riYIh1\fTWV=Xt]Xo&\R<o&hau<t TWXl[lN :3 5, G L

(139) ) . /!5l/ A )}%(70CE7F'(%(!"+7 pR TWV=_Ptt]Xo TW_a\R Xt]u=Z=ZH\ktmXTWVY*TR=\xY=}_rTW_a\RYIh=V9gZH\TWV=XtmXt¡YI[]XY}`;_rT]TWX \R TWV=Xf2\[]`t\f=TWV=Xo&hau<tmTWXl[Wt YIR<6YIRYIhrgElXxTWV=X o&\`;ZHX&TW_rTW_ri_rTg|[dY*TW_a\;\f

(140) TWV=Xwf2\haha\( _aR=qYIhaq\[]_rTWV=`N # " %!$. wv && # v $ G xzyw{ 5 V 6 n/2 C ← V x{ %" x { x { V > n/2 C ← X(MAX MATCHING(G )) x { %" x { x { wv v && # $ G x { ,+"-'+ C ← C ∪ X(MAX MATCHING(G )) x { ,+"-'+ C←C∪V . 1. 1. 1. 1. 1. 2. 2. #. . " . /.. 2. . L

(141) 7F/H'*7 +. ESpR&$:,.p $'*SUV/m`Z'/MWMUY,}v/Mp UYE/M& UYE/%v$inb/ UY,NUY,.ZI,NUY2WTS'UY,}$$L|S* wr$WX,NUYi (2 465 S wrS,.&)pU7S&c$nUY,.iRS'* G96;hS* wr$WX,NUYi Z'/XWX, /Xp c 6 (∆ + 5)/2 a 0 ,.pWTS'UY,}$,.p#UY,!w'rUVa 2 OLVC ¤ '*//E¥ SXlR=\TWXf2\[ i = 1, 2 9g M TWV=X`Y*TdodV=_aR=qkto&\`Z=u}TWX9gYIhaq\[]_rTWV=` x5 E5 "!$#%;\R G i Y*Tha_aR=Xt x { YIR< x { YIR<|9g m TWV=iXl_a[tm_!lXtlN i tFu=Z=ZH\kt]Xxha_aR=Xt xzyw{ YIR< x { \fYIhaq\[]_rTWV=` " ;YI[]XxX&^Xo&u}TWX Nf"V=XlR}TWV=Xxt]\hau}TW_a\R[]X&TWu=[]R=X_Pt _rTWV N\`=_aR=_aR=qX&^Z=[]XtWt]_a\Rf2\[ _rTWV X&^Z=[]Xt]t]_a\R<~jYIR< C = V1 ∪ X(M2 ) TBYF_aR=q;_aR9TW\Yolo&\u=R9ThaXl`;|C| `bY 6yYIn/2 R<TW+V=Xw2m f"Yo2TTWVY*T m 6 m =TWV=Xxf2\haha\( _aR=q;CV=\hP=t 0. . .

(142). . . 2. + 2m2 n ∆+1 ∆+5 mO F 6 +26 +2= m 2m 2 2 tFu=Z=ZH\kt]XR=\( eTWVY*Tha_aR=Xt \f1YIhaq\[]_rTWV=` " YI[]XX&^Xo&u}TWX_aR<t TWX(Y N"V=XlR |V | 6 n/2 YIR< TWV=Xbt]\hau}TW_a\R []X&TWu=[]R=X _PtwTWV=Xb\R=X\fha_aR=X _N XN! C = X(M ) ∪ V N:pR TWV=_PtoYt]X|2YIhPtm\DTWV=X 1 2 c2 . =. x{. OLVC. |C| 6 τ (G) YIR< . x{. n 2. x{. IY []qu=`;XlR9TdtZ=[]X&i_a\u<t]hrg }X&ikXlha\ZHX V=\hP N sXlR<o&XX&^Z=[]Xt]t]_a\R mO F;YIhr Ylg}tq_rikXt;TWV=XDo&\`ZHX&TW_rTW_ri_rTg [dY*TW_a\YodV=_aX&ikX N 1X&Tu<tR=\( t]V=\( TWVY*TTWV=XYIRYIhrg}t]_PtjYIH\(ikX _PtSYtmg`;Z}TW\TW_PoYIhahrgTW_aqV9TNj\R<t]_P}Xl[jY;q[dYIZ=V G(V, E) o&\hahaXo TW_a\R \f R t TBYI[Wt HX(YodV6\f `Y*^_aù=`}Xlq[]XlX ∆ N\R<tm_P}Xl[TWV=X t]u==q[dYIZ=V G \f G o&\R<t]_Pt TW_aR=qb\f Y|tmX&T\f n/2 X&^ZH\kt]XikXl[mTW_Po&Xtx _rTWV[]Xt]ZHXo TTW\6Y`Y*^_aù=` `Y*TdodV=_aR=q M \f G1 N Xl`YI[L6TWVY*T M o&\R9TBYI_aR<t \R=XX}qXZHXl[ t TBYI[YIR<TWVY*T V (G ) _PtYxtmX&T\fE_Ptm\h:Y*TWX;ikXl[mTW_Po&Xt \f t]_!lXR=\Th:YI[]qXl[¡TWVYIR n/2 N tFX&T G = G[V \ V (G )] YIR<YtWtmu=`XjTWVY*T 1 G _Pt[]X&ikX(YIhaXZHXl[o&hau<t TWXl[Wt G YIR< G N"V=XlR 2. 1. 1. |C| = τ (G) = |C| τ (G). =. "V=_Pto&\`;Z=haX&TWXtTWV=XxZ=[]\F\f\f

(143) TWV=XxTWV=Xl\[]Xl`N. n n 2n + 2R = + 2 2 ∆+1 n = R ∆+1 ∆+5 2. O. 2. mO( mO O mO C.

(144) : 3 P3 ¢ A &%(7F'I;L

(145) )9CE7 . /!"l/ A )=%(70CH7F'(%*!5+7 tFu=Z=ZH\kt]XjR=\( TWVY*To&hau<tmTWXl[Wt G YI[][]_rikXS _rTWVR=\_Ptm\h:Y*TWXikXl[mTW_Po&XtlN 1X&T n HXTWV=XS\[W}Xl[\f G N SXlR=\TWX i 9g M i = 1, 2 Y;`Y*^_aù=` `Y*TdodV=_aR=qb\f G H9g P TWV=Xt]X&TS\fTWV=X X&î ZH\kt]X6ikXl[mTW_Po&Xt _rTWiVD[]XtmZHXo T i \f M =9g p _rTdtoYI[W}_aRYIha_rTg6YIR<o&\R<tm_P}Xl[TWV=Xwi f2\haha\( i_aR=q;\R}n~ha_aR=XYIhaq\[]_rTWV=`N i i # x n, { %!$. zx yw{ % v > 1 x { C ← ∅ wv v && # $ G x { M ← MAX MATCHING(G ) x { ! * n 6 n/ # C ← C ∪ V x { C ← C ∪ X(M ) wv v && # $ G x { M ← MAX MATCHING(G ) x { C ← C ∪ X(M ) x

(146) { x { A,+"←-'+{v C∈←V C\∪X(MA ) : ∃u ∈ V \ C, uv ∈ E} { x # n, /. 1. 1. 1. 1. 1. 1. 2. 2. 2. 2. . 2. 2. 2. 1. 2. .

(147) L 7F/H'*7 &4m/MWUYb/rnG$'UYb/Xp,.p@UYES'Ufv *3kbpUV/MWpSWWX,.Z'/) ,NUYR&$A,.pM$'*!S'UV/m Z'/MWXUY,}v/Xp UYb/XW/1/j,.pXUYp UYE/%*!SW?wr/MpU1S'iR$&EwBUYE/HWT$I$'UYpH$Q|UYE/ nG$'*-'&$iA,}S'* 2 − 3 + 1 pk)vM UYbS'U#UYE/`v$inb/ UY,NUY,.ZI,NUYWTS'UY,}$c0$Q S'*Dwr$W,NUYri 2 465 S wrS,.&)pUfS&:$nUY,.iS'/* 96; S'*Dwr$W,NUYri6,.p#qT$'kE&mr/m Srq$Z'/1qX 2 + ((∆ + 1)/ ) a. . . . . 0. 0. ¤ '*//E¥ tFX&T f2\[ i = 1, 2 m = |M | YIR<R=\TWXTWV=Xf2\haha\( _aR=qf"Yo TTWVY*ToYIRbHX_a`;`;X}_:Y*TWXlhrg}X}u<o&X i 2f []\` YIhaq\[]_rTWV=` " =N i 1)+% > UV=XlR=X&ikXl[ ha_aR=X x { _PtX&^Xo&u}TWXTWV=XlR ha_aR=X x

(148) { o&\`;Z=u}TWXt A = ∅ gTWV=Xl[]Xlf2\[]XTWV=X=RYIh 2 o&\(ikXl[]_aR=q C o&\`;Z=u}TWX|_aRha_aR=X x { tdY*TW_PtQ=Xt C = V ∪ X(M ) N 1 2 1X&TSu<t@=[WtmTjt]u=Z=ZH\ktmXTWVY*TSha_aR=X x { _PtX&^Xo&u}TWX N"V=XlREu<t]_aR=qbf"Yo TxMEEX&^Z=[]XtWtm_a\R ~YIR<6haXl`;`bYyx XxqX&T 0 n n n + 2m2 |C| n1 + 2m2 ∆+1 ~y = 6 6 +26 +2= +2 1 X&Tu<t =R \( tmu=Z=ZH\kt]XTWVY*Tbha_aR=X x { _Pt;X&^Xo&u}TWXe_aR<t TWX(Y N "V=XlRTWV=XDt]X&T ha_aR=X x { 9 gYIhaq\[]_rTWV=` ki Xl[]_!=XtM0 τ (G). m. m. m. n ∆+1. . C. =RYIhahrgeo&\`;Z=u}TWX_aR. ~yOI SXlR=\TWX 9g Q ⊆ V \ X(M ) TWV=X tmX&T\fjikXl[mTW_Po&Xtb\f V TWVY*TbVYtbXlR9TBYI_ahaXeTWV=XD_aR9TW[]\}u<o TW_a\R \f t]X&T A _aR C <YI1R<b9g 1B(Q , A1, E ) TWV=Xxt]u==q[dYIZ=V|\f G _a1R<}u<o&X9g Q ∪ A NftF_aR<o&XjTWV=X&gbYI[]XjH\TWV _aR<}XlZH2XlR<}XlR9T1"tmu=<t]X&Tdt \f P1 YIR<2 PB F[]XtmZHXo TW_rikXlhrg) B _Pt¡=_aZYI[mTW_rTWXN SXlR=\1TWXjYIhP2tm\ 9g M Yx`Y*^_aù=` `Y*TdodV=_aR=q;\f B YIR<tmX&T m 1 = |M 2 | N tF_aR<o&X M ∪ M ∪ M _PtY`Y*TdodV=_aR=q;f2\[ G B 1 2 B B B ~yy τ (G) > m1 + m2 + mB \R<tm_P}Xl[¡tmX&T gk\9i_a\u<t]hrg |X(M )∩Q | = m N tF_aR<o&X G _Ptt]u=Z=ZH\kt]X _rTWV=\u}T _Pt]\h:Y*TWX 1 ikXl[mTW_Po&Xt YIR9g;X(M ikXl[mTWX&B^)∩Q \f X(M VYtY*TB`;\kt T ∆1 − 1 R=BXl_aqVFH\[Wt_aR 1 A NQRTWV=XS\TWV=Xl[VYIR< M B ) ∩ Q1 HXl_aR=q`bY*^_aù=` f2\[ B <YIR9gikXl[mTWX&^b\f A []Xo&Xl_rikXtX}qXtf2[]\`Y*ThaX(Yt T\2R=XikXl[mTWX&^b\f X(M ) ∩ QBN 2 1 B tF\G ~yM |A | 6 m (∆ − 1) |C| = |X (M1 )| + |X (M2 )| + |A2 | = 2 (m1 + m2 ) + |A2 |. 2. B. OO.

(149) {hPtm\G=t]_aR<o&X. G1. YIR<. G2. YI[]XSH\TWVYt]t]u=`;X _rTWV=\u}T_Ptm\h:Y*TWXbikXl[mTW_Po&XtMYIZ=Z=ha_PoY*TW_a\R|\f

(150) haXl`;`Y yq_rikXtM0 m1 > m2 >. ~y . n1 ∆+1 n2 ∆+1. \`=_aR=_aR=qDX&^Z=[]Xt]t]_a\R<t%~yOI~yyy~yMYIR<K~y ZHXl[]f2\[]`;_aR=q tm\`X;ha_rT]TWhaX|YIR< X(YtmgYIhaqXl=[dY YIR< TBYF_aR=q;_aR9TW\Yolo&\u=R9T n + n = n =\R=XwqX&Tdt 0 1. 2. |C| 2 (m1 + m2 ) + |A2 | |A2 | − 2mB 6 =2+ 62+ τ (G) m1 + m2 + mB m1 + m2 + mB. XoYIhahTWVY*T XYI[]Xwo&u=[][]XlR9TWhrg6o&\R<tm_P}Xl[]_aR=qboYtmX. n1 > n/. n2 6 n −. =_N XN!. ~y. ∆−3 ∆−1 |A2 | |A2 | n ∆+1 + ∆−1. ~y'F. n −1 =n . tm_aR=qX&^Z=[]Xt]t]_a\R ~y f2\[ m }XlR=\TW_aR=q 9g p TWV=XRFu=`HXl[;\fTWV=X|X&^ZH\kt]X ikXl[mTW_Po&Xt;\f S []Xt]ZHXo TTW\ M HYIR<u<t]_aR=q;X&^Z=2[]XtWtm_a\R ~y'F= Xx\2}TBYI_aR. V2. _rTWV. 2. |A2 | 6 p2 = n2 − 2m2 6 n2 −. 2n2 ∆−1 (∆ − 1)( − 1) = n2 6 n ∆+1 ∆+1 (∆ + 1). Xl`YI[LwYIhPtm\TWVY*TX&^Z=[]XtWtm_a\R%~y _Pt

(151) _aR<o&[]X(Yt]_aR=q _rTWV |C| 62+ τ (G). (∆−3)(−1) (∆+1) n (−1)n n ∆+1 + (∆+1). |A2 |. ~y. N tF\Go&\`=_aR=_aR=qX&^Z=[]XtWtm_a\R<t7~y

(152) YIR<|~y0. =2+. (∆ − 3)( − 1) 2 − 1. ~y'O. \TWXTWVY*T Ff2\[Y}^X − 3)( − 1)/(2 − 1) 6 (∆ + 1)( − 1)/(2 − 1) YIR< TWVY*TX&^Z=[]XtWtm_a\R~y _Ptx}Xo&[]X(Yt]_aR=q| _rTWV =V (∆ _ahaX X&^Z=[]XtWtm_a\Rh~y'O_Ptx_aR<o&[]X(Yt]_aR=q=N "V=XtmXT \X&^Z=[]XtWtm_a\R<tYt g`Z}TW\TW_PoYIhahrg o&\_aR<o&_P}Xw V=XlR ∆+1 (∆ + 1)( − 1) >1 +2=2+ ⇐⇒ 2 − 3 + 1 = 0 =⇒ 0 w 2.62 2 − 1. tFX&T]TW_aR=q = 2.62 * X qX&T X

(153) 9uYIhTW\ 0∆/2.62 N. cC2 . OLVC. 6 (∆+6.24)/2.62. ~y'C. N"V=_Pt If2\[

(154) h:YI[]qX¡iYIhau=Xt\f ∆ I_PtYtmg`;Z}TW\TW_PoYIhahrg. :> ' / K 7F' $ /H . 0;¥~/'%*L

(155) 7D+k/H¡79%*!:%(!2C

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(157) 9u=_rTWXxTWXodV=R=_PoYIh}_rT_Ptq_rikXlR_aRDYIZ=ZHXlR<}_rÊN L