ࡱ> `!eL7HF<:$* T ( 3xcdd``bd``beV dX,XĐ SRcgb кc@P5< %! `fjvF+B2sSRsnsi#y ŵp>o `uwl *77c |?0o&k͇0cMa`Hc)=BCb#ܤ3`=ep~.pZv0o(;+KRs:] VOa`!m2դb~I{@Rx5O; P Vb%v   iҥL]l-/k^1\-m?kCW3>]J?$o|pgFGQ~`x`_vNNHݫӄߌN m-Path Partition for Graphs With Special Blocks . $XoOpg September 2005,dd  UOutline  @Introduction Motivation Previous results Main result Future work A ? Graph G=(V,E)>  _ Subgraph H of a Graph G2    XPath  VPath-partition   YPath-partition   WPath-partition Number p(G) J  BPath-partition Number p(G)B  c#Cycle  [ Hamiltonian  v2Complete Graph Kn, &   C Motivation  RSkupieD 1974 Hamiltonian shortage SH(G) x*   $ D Motivation  E Motivation  F Motivation  G Previous Results  =Question  ; NP-Complete  \Previous Results  ] Connected  ^ Component  a! Cut-vertex  b"Block e%Complete Bipartite Graph z6Tree {7 Block Graph  H  Our Result   f'f-path partition  |f: V(G){0,1,2,3} f-path partition: P: collection of vertex-disjoint paths 1 f (v) `"3 in some path in P 2 f(v)=0 v itself is a path in P 3 f(v)=1 v end vertex of some path in P% @(@@@@@@@@ j*f-path partition number pf(G)X   p,Inductive Theorem  h(Inductive Theorem  i)Inductive Theorem  k+Inductive Theorem  ~:Basis  q-Basis  r.Basis  s/Basis  t0Basis  u1Basis  I  Our Result  J w3K |8Lx4M}9Ny5OPQRS?Time Complexity  The depth-first search to find all blocks and cut-vertices, and requires O(|E|) time Each subroutine requires O(|V(B)|+|E(B)|) operations. I@ @  T Future work  pPath partition Tree structure !small separator structure*98  @P!!b ` 33PP` 3333` ___MMM` 13` 333fpKNāvI` j@v۩ῑ΂H` Q_{>?" dd@,?n<d@ `7 `2@`7``2 n?" dd@   @@``PR    @ ` ` p>> 4 ,   (    <" BA    T d" BA    <L"U_ BA    Td">& BA    N"P BA    <"p BA    C x\?d?"bUv BA    6$ "U  P cN NN}/kGrjL#j_     0< "   0 cN NN}/kGr ,{Nd\ ,{ Nd\ ,{Vd\ ,{Nd\    6P  "@  Z*   6P "@`   \*   68 "`  \* B  s *޽h ? 3333  Blendsd      @ (  T +  "+bb P@ # "Dwoh  s *"PP  Bd" P@bb P 0  # "Nyh  s *"P    Bd"P 0 z   <" a*h   s *"    f?d?"+)   < ?"pP  P cN NN}/kGrjL#j_      0 " `    R cN NN}/kGroRjL#j_   6t "`p   ^*   6(  "`p   `*    6  "`  `*  B  s *޽h ? 3333  0x0(  xx x c $  `    x x c $t pP  H x 0޽h ? 33334   \(  x  c $ U     c $p  <$ 0  H  0޽h ? 3333c     | (  |x | c $  U   2 | # lh??` Ka  2 | # l??  Kb  2 | # l?? `  Kd  2 | # l(??  Kc  B |  `Do??` B |  `Do?? B  |  `Do??0 ` 0 B  |  `Do??  B  |  `Do??0   |  `| ?" 4 } V={a,b,c,d}B       V |  `x ?" e  E={ab,ac,ad,bc,cd}B    @      |  ` ??0 PvertexC    |  ` ??&B  NedgeC   H | 0޽h ? 3333   [(  r  S ޔ U   2  # lߔ??   Ka  2  # l?? @  Kb  2  # l?? p Kc  B   `Do??  B    `Do??     `D ?" 4  { V={a,b,c}B       3    ` ?"   E={ab,bc}B     &   `  c $A ??0 $ H  0޽h ? 3333    @(  x  c $<Ȕ U   2  # lɔ??` Ka  2  # l̔??  Kb  2  # l$Д?? `  Kd  2  # lӔ??  Kc  B   `Do??` B   `DԔ?? B    `DԔ??0 ` 0 B    `DԔ??  B    `Do??0    `ؔ ?" =  ^adcb   H  0޽h ? 3333  g_(  r  S  U   E  Tp ??^  kA path partition of a graph is a collection of vertex-disjoint paths that cover all vertices of the graph. llC k  H  0޽h ? 3333   ` X '(  x  c $X U   @B @ C ԔXA @B  C ԔA @B  C ԔX O @B @ C Ԕ O  :B  3    @B   C Ԕ :B   3  S@B  @ C Ԕ S2    `?"` P 2    `?"p2   `?"p ` 2   `?" p2   `?" 2   `?"` P2   `?" 2 # Z?"P @ 4B $ #   2 % Z?"  :B & 3 Ԕ  H  0޽h ? 3333[   p(  x  c $ U     T ??  _The path-partition problem is to find the path-partition number p(G) of a graph G, which is the minimum cardinality of a path partition of G. AC     C  :C  C   H  0޽h ? 3333d  b   ` (  x  c $\ U   F P  PBB B 3 ԔAX BB  3 ԔA BB  3 Ԕ XO BB B 3 Ԕ O  <B  #   BB   3 Ԕ  <B   #   SBB  B 3 Ԕ S2   Z?"` P 2   Z?"p2  Z?" p ` 2  Z?"  p2  Z?" 2  Z?" ` P2  Z?" 2  Z?"P @ 4B  #   2  Z?"  :B  3 Ԕ  ,   f ??P @0 p(G)=2T,,,,  H  0޽h ? 3333   ` X P(  r  S  U   @B @ C ԔXA @B  C ԔA @B  C ԔX O @B @ C Ԕ O  @B  C Ԕ   @B   C  :B   3  S@B  @ C  S2    `?"` P 2    `?"p2   `?"p ` 2   `?" p2   `?" 2   `?"` P2   `?" 2  Z?"P @ 4B  #   2  Z?"  :B  3   H  0޽h ? 3333n  @"(  x  c $@ U   d8   "p 2  # l??P Ka  2  # l ?? P  Kd  .@   ! 2  # l<??p Kb  2  # l??   Kc  B   `DԔ??P`p`B   `DԔ?? B    `DԔ??P  B    `DԔ?? B    `Do??  @B @ C  x@B  C  J @B  C x@B @ C  :B  3  J  @B  C  :B  3   @B @ C  2   `?"2   `?"  p 2   `?" 2   `?" 2   `?"@ P@0 2   `?"P@2   `?"P @H  0޽h ? 3333  0T6(  Tx T c $w U   :B T@ 3 XA :B T 3 A :B T 3 X O :B T@ 3  O  4B T #    2 T Z?"` P 2  T Z?"p2  T Z?"p ` 2  T Z?" p2  T Z?" :B  T@ 3  p :B T@ 3  :B T 3  :B T 3 @ :B T 3 @ p H T 0޽h ? 3333    0 (  (  x  c $lW U   x  c $(X       `\Y ?? y  Q Hamiltonian       `t] ??5 ; tmin p < AA  F       2  # l\c?? `  KG  B   `Do?? @ B    `Do?? @ B    `Do??` B    `Do??  2   # lg??   \Kp(     ` U   F       2  # l??? `  KG  B   `Do?? @ B   `Do?? @ B   `Do??` B   `Do??  2   # lDD??   ZK0(I      fH ?" t  G: Hamiltonian SH(G)=0 p(G)=1 C       C       H  0޽h ? 3333#   K(  x  c $0( U   2  # lx)??   \K2(     f, ?"= z  'G : K1 SH(G)=2 p(G)=1                  2  Z?" P@ B  ZD?" @0 B  ZD?"P @ @H  0޽h ? 3333k  (  x  c $ U   [   f ?"p !G: is not Hamiltonian and not K1 V" C    C !    Z| ?"u  SH(G)=p(G) (2      C        "  B CODEFg 33  vaM"6( 0 8BKWcr}m^PE:0& 2J`w #/:BHMOgh@ P2 2  BCdDEFg 55 (5DT e w &.6 @ K5XLe`su1Mf%-5<BHM-RCUVXh\x_`bbdddkl@ P"  B{C{DEFg 33 "/=O b u%-5=0HHS`^xm{(13K=cF{NV]bfknr-u=vMxYzeznzu{y{{gh@ H "  BNC8DEFg 33NLLK KG F/C=?M<`7s3,&(?Uks_K6 #(+/x2f3T5D767(788 8777gh@P =2   Z?"  2   Z?"` P 2   Z?"  2   Z?"pB   ZD?" 2  Z?"P @ B  ZD?" 0P    ` ?? SH(G) p(G)                  `L ?? @ =  SH(G) p(G)               H  0޽h ? 3333.  H  !V(  x  c $x U      `l ??p& = \Algorithmic viewC  2   f?? 0  B 2  # l,?? `@ B    `L ?? ftr  W NP-complete  C    !  ` ??0 @=  ePolynomial-time algorithmC  H  0޽h ? 3333  :2(  r  S  U     Z ??@ @Is the path-partition problem is NP-complete for general graphs?A 2AC A H  0޽h ? 3333  qix(  xr x S  U   O x  `H ??pC  eAs the Hamiltonian path problem is NP-complete for general graphs, so is the path-partition problem.ffC e  H x 0޽h ? 3333   (  x  c $H U   2  # l??p Ntree 2  # ll??P P  Rinterval    2   fxÒ??  Pplanar 2  # l4ǒ??P  V circular-arc    2  # l˒??   Oblock 2   f\??` p @ ichordal bipartite   2    ftҒ??P`  _chordal   2    f֒?? @p  vstrongly chordal*    2   # lْ??  _cograph  2   # lݒ??  gcocomparability  2   # lH?? P gbipartite distance hereditary 2   f?? 0  S bipartite    B  ZD?? `B  ZD??@@@@B  ZD?? @@pB  ZD?? pp B  ZD??`  B  ZD?? P  B @ ZD??0 P B  ZD?? ` B  ZD??P p B  ZD?? B  ZD??ppB  ZD?"0 00H  0޽h ? 3333   (  r  S p U   g   ` ??6y  IA graph G is connected if each pair of vertices in G belongs to a path.\JC  +C  C J 2  # lX??0  Ma 2  # lH??  Mb 2  # l@?? 0  Md 2  # l,??   Mc B   `Do??` ` B    `Do?? B    `Do?? B    `Do?? 00 H  0޽h ? 3333   P H  (  r  S  U   9   `Ĉ ??6E AThe components of a graph G are its maximal connected subgraphs.BBC *7    2  # lL?? P  Ma 2  # l䏒??`  `  Mb 2  # l쓒??@P` Md 2  # l??@@ ` Mc B    `Do?? @B    `Do?? p2   # l??p p  Ma B   `Do??P @P B   `Do??` @pH  0޽h ? 3333     ( (  r  S { U   >   `} ??uH  XA cut-vertex of a graph is a vertex whose deletion increases the number of components.$YXC Y 4B  # g }  4B @ # g   4B @ # g  } 4B  # g  4B @ # g } 4B   # g } 2   Z?"0 2   Z?"0 2   Z?"  2   Z?"@ 0 2  Z?"@0 0 4B @ # g Mo4B  # g Mo2  Z?"0 2  Z?"0 2  Z?"@0 2  Z?"@0 H  0޽h ? 33336   v(   r   S q U   H    `(s ??&5x  PA block of a graph G is a maximal connected subgraphs that has no cut-vertex. QQC *-    4B  @ # g o  4B   # g o / 4B  @ # g  / 2   Z?" p ` p2   Z?" p2   Z?"  H   0޽h ? 3333  p .(  r  S  k U   2  Z?"2  Z?" 2  Z?"  2  Z?" 2  Z?"P  @ :B  3 ` :B   3  :B  @ 3 P :B   3   :B  @ 3  :B  @ 3   0 H  0޽h ? 3333  ` d(  dr d S dY U   2 d Z?"@0 2 d Z?"p2 d Z?"P @ p2 d Z?"  2 d Z?"P @0 @ B  d@  `Do??PpB  d  `Do??0 B  d@  `Do??@ B  d  `Do??pPpP   d  `] ??   iH. A. Jung 1978C   , d  `q  ~,x: a specified vertex of a graph H in which f is a vertex labeling. i=0,1,2,3 f i :V(H){0,1,2,3} f i(y)=f(y) for all vertices y except f i(x)=i. C  C  C    C    C  C  C  C    C  C  C  C  C    C  C  C   H $ 0޽h ? 3333  rj((  (r ( S Hf U   P ( T ??LV  =pf0(B)-pf1(B) =pf1(B)-pf2(B) 1 f(x)=0 pf(G)=pf(A)+pf(B)-1. 2 f(x)=1 pf(G)=pf1-(A)+pf(B)-1.i                 C                  C                         C                            i H ( 0޽h ? 33338   0x(  0r 0 S Z U   & 0  `@b ??@ D 3 f(x) 2 and ==0 pf(G)=pf(A)+pf0(B)-1. 4 f(x)2 and =0 and =1 pf(G)=pf3(A)+pf(B). 5 f(x) 2 and =1 pf(G)=pf1- (A)+pf1+(B)-1. C                             C                             C                                 ` 0 c $A ??@ h  ` 0 c $A ??@ h  ` 0 c $A ??@ h  ` 0 c $A ??@ h  H 0 0޽h ? 3333  TLt(  tr t S lF U   2 t  `K ??f  2Pf (B)=pf (B-f -1(0)) +|f -1(0)|. WLOG f -1(0)=43                    C     #  2  H t 0޽h ? 3333  @(  @r @ S (+ U   h @  `D- ??Ff L  B: complete graph. f -1(1) `" or f -1(2)= pf(B)= |f -1(1)|/2 else pf(B)=1.T   C      # C     # C  C               C       C S  H @ 0޽h ? 3333N   D(  Dr D S P U    D  ` ??@\ L*B is a path x: end vertex of B with f(x)=3 pf(B)=pf(B-x). x: end vertex of B with f(x)=2 pf(B)=pf1(B). B: end vertex x and another vertex y with f(x)=f(y)=1 no vertex between x and y has a label 1 pf(B)=pf(B  )+1 B : path obtained from B by deleting [x, y]. C  C  C     C                C  C  C     C                C  C  C  C         C  C C  C  C            C  C  C  C      C   H D 0޽h ? 3333|  ,$H(  Hr H S h U     H  ` ??_  zB is a cycle. f -1(2)= pf(B)=|f -1(1)|/2. P: [x,y] in B such that f -1(1) )"P ={x,y} and f -1(2) )"P`" pf(B)=pf(B-P)+1.F} C     # C               C  C      C  C           C      C #                 C |  H H 0޽h ? 3333   H @  L (  Lr L S l U    L  `r ??  LB is a complete bipartite graph c1=d1=0 and c2 d2 and xC2 pf(B)=pf '(B) f ' is the same as f except f '(x)=1.0q C         C        C                     C  C     C p  *2 L # lLt??@ P C1 C2 C3V       L  `  ??l   KC   *2 L # lȕ??@ `   D1 D2 D3V        L  ` ??`  KD   2  L # l8??  rx f(x)=2:     L  ` ??`   \* C   2  L # lt??@ 0 rx f(x)=1:   H L 0޽h ? 3333j      P (  Pr P S 4* U    P  `/ ??0  LxC1. Also, either d21 with yD2, or else c1>d1 and d2=0<d3 with yD3. Then pf(B)=pf '(B-x), where f' is the same as f except f'(y)=1.     C      C      C        C        C      C                 C  C  C     C   2 P # l!??p @  rx f(x)=1:   2 P # lX??  P ry f(y)=2:    P  `T ?? Q  \* C   2  P # l(b??@  B 2  P # lHe??@ `0 ry f(y)=1:   H P 0޽h ? 3333M     pi (  x  c $ U     T,  ?"0P 7f-path partition: a collection of vertex-disjoint paths&8 7  8 (F 0p   @@` 2   f ??p  Ka     `  ??0p   I0 4L  # 2  # l?? Kd      ` ?? I3 L ; P [   #   2    f??; [  Kb      `d ??; m[  I1 B   ZD?? P 2  # l?? 0  Kc     f# ??p @  I2 B   `D??P P B   `D??P 0 P `  c $A ??0 8 H  0޽h ? 3333G!    P*,o (    6C U oExample,  z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Csz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Cz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@`` `CC z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@` `C    `$L ?"Ee K0    `tP ?"  K1     `S ?"E e  K2     `V ?"   K3    <Z+  D FB   S g  FB   S g E) F  < _"  D   <b` + F  D FB  S g FB  S g E FFB  S g ~ a FB  S g  ) FB  S g  ) FB  S g ~ | } a FB  S g Eh FB  S g }F FB  S g  FB  S g -. FB  S g ~ FB @ S g  0 FB @ S g } 2   `h?"  Kg  2   ` k?"  Kf  2   `(o?"@0  Kh  2    `r?"p @0 `  Ka  2 !  `v?"p `  Kb  2 "  `8z?"  Kc  2 #  `}?" Kj  2 $  `?"   Kl  2 %  `?" Ki  2 &  `<?"   Kk  2 '  `?"@0  Ke  2 (  `p?" @0  Kd   )  ` ?"  VPath Partition   + Z ?? Z V=pf0(B)-pf1(B)=2-1=1 =pf1(B)-pf2(B)=1-1=0|, 2                 C                   , H  0޽h ? 3333>  `X~(  X X Z ??p@w  ( f(x) 2 and =1 pf(G)=pf1- (A)+pf1+(B)-1 = pf1 (A)+1-1= pf1 (A)\ 2C                                               >6      X Z  ?? V=pf0(B)-pf1(B)=2-1=1 =pf1(B)-pf2(B)=1-1=0|, 2                 C                   , H X 0޽h ? 3333"  F">"p,.!(    6| U oExample,  z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Csz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Cz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@`` `CC z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@` `C    ` ?"Ee K0    ` ?"  K1     `H ?"E e  K2     ` ?"   K3    <d+  D FB   S g  FB   S g E) F  <` + F  D FB  S g FB  S g ~ a FB  S g  ) FB  S g  ) FB  S g ~ | } a FB  S g Eh FB  S g }F FB  S g  FB  S g -. FB  S g ~ FB @ S g  0 FB @ S g } 2   `?"  Kf  2   `L?"@0  Kh  2   `T?"p @0 `  Ka  2   ` ?"p `  Kb  2   `(?"  Kc  2    `d?" Kj  2 !  `?"   Kl  2 "  `?" Ki  2 #  `?"   Kk  2 $  `h?"@0  Ke  2 %  `"?" @0  Kd  VF    & 0  ' <! y  D NB ( S g   2 )  `$?"   Kg  2 *  `L,?"   Kf   +  `' ?"  VPath Partition   . Zt6 ?? * V=pf0(B)-pf1(B)=2-1=1 =pf1(B)-pf2(B)=1-1=0|, 2                 C                   , H  0޽h ? 3333>  l~(  l l Z6      l Ztk ?? V=pf0(B)-pf1(B)=2-1=1 =pf1(B)-pf2(B)=1-1=0|, 2                 C                   , H l 0޽h ? 3333$  e$]$/0#(    6 U oExample,  z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Csz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Cz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@`` `CC z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@` `C    ` ?"Ee K0    `@ ?"  K1     ` ?"E e  K2     `4 ?"   K3    <+  D FB   S g  FB   S g E) F  <` + F  D FB  S g ~ a FB  S g  ) FB  S g  ) FB  S g ~ | } a FB  S g Eh FB  S g }F FB  S g  FB  S g -. FB  S g ~ FB @ S g  0 FB @ S g } 2   `?"  Kf  2   `?"p @0 `  Ka  2   `$?"p `  Kb  2   `?"  Kc  2   `?" Kj  2   `4?"   Kl  2    `?" Ki  2 !  `?"   Kk  2 "  `?"@0  Ke  2 #  `8?" @0  Kd  VF    $ 0  % < y  D NB & S g   2 '  `?"   Kg  2 (  `?"   Kf  VF @0 j ) @0 j * <+ j D NB + S g 2 ,  `?"@0  Kh  2 -  `?"@`0 P Ke   .  ` ?"  VPath Partition   0 Z ??  V=pf0(B)-pf1(B)=2-2=0 =pf1(B)-pf2(B)=2-1=1|, 2                 C                   , H  0޽h ? 3333  NF\(  \ \ Z ??@ V=pf0(B)-pf1(B)=2-2=0 =pf1(B)-pf2(B)=2-1=1|, 2                 C                   ,  \ Z8  ??`  f(x)2 and =0 and =1 pf(G)=pf3(A)+pf(B) =pf3(A)+1Z 2                                     Z H \ 0޽h ? 3333f)  ((56((    6# U oExample,  z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Csz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Cz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@`` `CC z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@` `C    `, ?"Ee K0    ` 1 ?"  K1     `4 ?"E e  K2     `P7 ?"   K3    <:` + F  D FB   S g ~ a FB   S g  ) FB  S g  ) FB  S g ~ | } a FB  S g Eh FB  S g }F FB  S g  FB  S g -. FB  S g ~ FB @ S g } 2   `@?"p @0 `  Ka  2   `(C?"p `  Kb  2   `0G?"  Kc  2   `J?" Kj  2   `\&?"   Kl  2   `Q?" Ki  2   `U?"   Kk  2   `Y?" @0  Kd  tF @   ` j  <\+  D    <_+  D NB ! S g ) E F " <c"   D NB # S g NB $ S g  E F2 %  `e?"   Kg  2 &  `i?"   Kf  2 '  `m?"@0  Kh  2 (  `p?"@0  Ke   ) <st I Z  D FB * S g W h i :  + <0xt iZ  D FB , S g  h i u FB - S g   FB .@ S g 8 _ 2 /  ` {?"P p@  Kg  2 0  `?"P @  Kf  2 1  `@?"  Kh  2 2  `?"P @  Ke  2 3  `?"0  Kd   4  ` ?"  VPath Partition   6 ZĒ ?? * V=pf0(B)-pf1(B)=2-1=1 =pf1(B)-pf2(B)=1-1=0|, 2                 C                   , H  0޽h ? 3333  ZRp(  p p Z ??p@w  ( f(x) 2 and =1 pf(G)=pf1- (A)+pf1+(B)-1 = pf1 (A)+1-1= pf1 (A)\ 2C                                               >6     H p 0޽h ? 3333   h ` *+(    6  U oExample,  z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Csz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Cz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@`` `CC z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@` `C    `( ?"Ee K0    `x ?"  K1     ` ?"E e  K2     ` ?"   K3    <+  D FB   S g Eh FB   S g }F FB  S g  FB  S g -. FB  S g ~ FB @ S g } 2   `?" Kj  2   `?"   Kl  2   `?" Ki  2   ` ?"   Kk  2   ``?" @0  Kd  tF @   ` j  <@+  D   <+  D NB  S g ) E F  <T"   D NB  S g NB  S g  E F2   `H ?"   Kg  2   ` ?"   Kf  2    ` ?"@0  Kh  2 !  `8 ?"@0  Ke   " <    D FB # S g  8 9 FB $ S g 1 2 FB % S g   2 &  ` ?"  Ka  2 '  ` ?" p`  Kb  2 (  `8 ?" p`  Kc  2 )  `?"  Kd   *  `  ?"  VPath Partition  H  0޽h ? 3333|  ,$`(  ` ` ZX,  ??    xC1. d21 with yD2, Then pf(B)=pf'(B-x), where f' is the same as f except f'(y)=1.U 2     C      C      C                 C  C  C     C T  H ` 0޽h ? 3333a  (    6H  U oExample,  z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Csz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Cz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@`` `CC z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@` `C    `Q  ?"Ee K0    `$V  ?"  K1     `Y  ?"E e  K2     `h\  ?"   K3    <_ +  D FB   S g  FB   S g -. FB  S g ~ FB @ S g } 2   `c ?" Kj  2   `f ?"   Kl  2   `j ?" Ki  2   `Ln ?"   Kk     `q  ?"  VPath Partition  H  0޽h ? 3333  ld(    6y  U oExample,  z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Csz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Cz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@`` `CC z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@` `C    `̂  ?"Ee K0    `  ?"  K1     `  ?"E e  K2     ``  ?"   K3    < +  D FB   S g -. FB   S g ~ 2   `D ?" Kj  2   `l ?"   Kl  2   ` ?"   Kk     `l|  ?"  VPath Partition  H  0޽h ? 3333 H -%(    6x  U oExample,  z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Csz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Cz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@`` `CC z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@` `C    `tK  ?"Ee K0    `  ?"  K1     `  ?"E e  K2     `  ?"   K3    <X +  D FB   S g -. 2    `P ?" Kj  2   `@ ?"   Kl     `  ?"  VPath Partition  H  0޽h ? 3333  nf !!(    6d  U oExample,  z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Csz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@``Cz  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@`` `CC z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@` `C    `  ?"Ee K0    `  ?"  K1     `\  ?"E e  K2     `  ?"   K3 2    `P ?"   Kl  0F         < +  D   <    D   <  D   <   v  D   <  t  D NB  S g  NB  S g    NB  S g l m NB  S g 5 X NB  S g   NB  S g   NB  S g n  2   `8 ?"0  Ka  2   `T ?"   Kb  2   ` ?"   Kc  2   `( ?"  Kj  2   ` ?"  Kl  2   ` ?" p  Ki  2   ` ?" p  Kk  2    ` ?"0  Kd   !  `4  ?"  VPath Partition  H  0޽h ? 3333#  0!!K(    6  U oExample,    <|# +  D FB  S g  LB  c $pE) F  <' "  D   <P* ` + F  D LB  c $pLB   c $pE FLB   c $p~ a FB   S g  ) LB   c $p ) LB   c $p~ | } a LB  c $pEh FB  S g }F LB  c $p LB  c $p-. LB  c $p~ FB @ S g  0 FB @ S g } 2   `0 ?"  Kg  2   `,3 ?"  Kf  2   `47 ?"@0  Kh  2   `: ?"p @0 `  Ka  2   `? ?"p `  Kb  2   `DB ?"  Kc  2   `E ?" Kj  2   `H ?"   Kl  2   `K ?" Ki  2   `HO ?"   Kk  2   `R ?"@0  Ke  2    `|Q ?" @0  Kd   !  `T  ?"  VPath Partition  H  0޽h ? 3333   `$(  r  S   U   r  S       H  0޽h ? 3333    - % P (  x  c $Pp  U   x  c $ē         `  ?"D  S Future work    2   `??`  z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@`  2   fh5?? 2    f-^?? P z   RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@` P 3 2    f1)?? p 2    f"?? @`2    `?? 0z  RBCDEtF~g qY A++@ Xr +AYqrX@+ q<@@`  2   f??  H  0޽h ? 3333  @(    T_  ?? p@'  cThanks for your attentionC0  H  0޽h ? 3333xVkQv&hci|xЋ6bh* FlӤP !I<D2ⷬt2 "Lq qZvp[?a>-sH*c^}~i]2`LeAF>~)#v߭lxo{>Dp2Iρ'eOd_9߰ OSx.xq5IVK:eO.iyowȵL.L/w#\{|eˑE"bOw0E1ͪy/'J|{OLT C&{-b˔m#Hm%2xU1o@9- S!*Dea*AT@17Ejq+E,P&f`e`a`f&k6ݝ+9Yg߽{w{޽{{18ɳ` t!r% O0|7lql^Lϧ'Y89q G0ePP2xU1o@9- S!*Dea*AT@17Ejq+E,P&f`e`a`f&k6ݝ+9Yg߽{w{޽{{18ɳ` t!r% O0|7lql^Lϧ'Y89q G0ePP0xU=oA}|l#BET E#)0l(%E*rA_Z  !QS!݋, '>cW L6 Pvq1~rgm~B(tKq9/@SZL_O뮯~rSIM-ԆLÅ?YםBob3/5P|/uWk`Q?1k /BsK_j)n׃dC8B`ZN _nVFJ(Եʁ '_~/9vعu<9wgDz^Zb DZrA.d]Qhds'0g3߹UFέ`uXvZEǤR*F% d2Lo/ڙ\س^X'+NXko"72?:զO =\}!~m~ce* [7h/ӭIND;ðqjM2xU=l@9 S!*DJD+AMQ5qӸ2 LL $0!l޻;W&r ξ{|{O>b N6 Pvq sg4MQF_8r^zX{Wprۨę Ei;۱#1@)+m_Yo🯡%dǼ?7غf yd-RgbRw&swvNPr{6\ZB.Wd(8³v6~x䕶m 7d.ρ߇;g%ZUJo%%גvA%늪G+$>yν2EunSz5ܺRko;6Kt6!:}{ž`t/>)XvZ}i6}g[Vq;D`0V Ƃ2Co߃-kvV$ 'g"N?aTq0TPZr09 +:>0QEIB=wբuwj*oďYU ;Las. #aGKgs+^ j[aVv. e \/hc<A1HcZ6UZc]jvכ*\R1>7Q0K7*I,(   p@Equation Equation.30(Microsoft e z_}/hV 3.00`Equation Equation.30( Oh+'0# px $0 P \ h t.Path Partition for Graphs With Special BlocksweUserParLC:\Program Files\Microsoft Office\Templates\Presentation Designs\Blends.potUserogr177Microsoft PowerPointoso@V@@ RC@GP"g   6& &&#TNPP2OMii & TNPP &&TNPP    - "-- !-- "-&o& i=33--- !2. ---&G i=&--NN- $G Q Q=G=- $Q [ [=Q=- $[ e e=[=- $e i i=e=---&&&&+8yj--- !2,8,---&R8yi&--6- $R8\8\iRiߎ- $\8f8fi\i- $f8p8pifi- $p8y8yipi---&&&&0;\&--&&- $<0<:DD- $<:<Dkk- $<D"<N- $<N",<X- $<X,6<b- $<b6@<l- $<l@J<v- $<vJT<- $<T\< $<\\<&&& "- & $0\;\;0& ttb&-&& &&-&&]<0&&- $<0<:DD- $<:<Dkk- $<D"<N- $<N",<X- $<X,6<b- $<b6@<l- $<l@J<v- $<vJT<- $<T\< $<\\<&- --&&--- !oC---&!V\&--- $!V+V+\!\ - $+V5V5\+\###- $5V?V?\5\%%%- $?VIVI\?\(((- $IVSVS\I\+++- $SV]V]\S\...- $]VgVg\]\111- $gVqVq\g\333- $qV{V{\q\666- ${VV\{\:::- $VV\\===- $VV\\@@@- $VV\\CCC- $VV\\FFF- $VV\\JJJ- $VV\\MMM- $VV\\QQQ- $VV\\SSS- $VV\\WWW- $VV\\[[[- $VV\\^^^- $VV\\bbb- $VV\\eee- $VV\\iii- $VV\\mmm- $V%V%\\ppp- $%V/V/\%\ttt- $/V9V9\/\www- $9VCVC\9\{{{- $CVMVM\C\- $MVWVW\M\- $WVaVa\W\- $aVkVk\a\- $kVuVu\k\- $uVV\u\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $V V \\- $ VV\ \- $VV\\- $V)V)\\- $)V3V3\)\- $3V=V=\3\- $=VGVG\=\- $GVQVQ\G\- $QV[V[\Q\- $[VeVe\[\- $eVoVo\e\- $oVyVy\o\- $yVV\y\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $VV\\- $V#V#\\- $#V-V-\#\- $-V7V7\-\ $7VAVA\7\- $AVKVK\A\- $KVUVU\K\- $UV_V_\U\ $_ViVi\_\- $iVsVs\i\ $sV}V}\s\- $}VV\}\ $VV\\- $VV\\ $VV\\ $VV\\- $VV\\---&&&&g1& - &g& --Q1-- Aз- 33.2 TN++,.@"Tahoma- 33.2 ESeptember 2005# .--9h-- 33@"Tahoma- 33.62 rPath Partition for Graphs With !!!  ! '!!5 !. 33.2 'rSpecial Blocks!!  # .--"System-&TNPP &0lEquation Equation.30(Microsoft e z_}/hV 3.00mEquation Equation.30(Microsoft e z_}/hV 3.00nEquation Equation.30(Microsoft e z_}/hV 3.00oEquation Equation.30(Microsoft e z_}/hV 3.0/ 0DArialgsRobbb0bWo 0"De0}fԚgsRobbb0bWo 0 DTimes New Romanbb0bWo 00DTahomaew Romanbb0bWo 0"@DWingdingsRomanbb0bWo 0PDjwiԚdingsRomanbb0bWo 0A`DSymbolgsRomanbb0bWo 0 a.  @n?" dd@  @@`` @!m>FL FG ( ! ""$-/1 7 , ""&%( *#&' *)( )5'# "!   + &% %$!?!?2$L7HF<:$* m2$m2դb~I{m2$U`o-q6b 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g41d1d0bvppp@ <4!d!d 0bb<4dddd 0bb? %O =)>-Path Partition for Graphs With Special Blocks . $XoOpg September 2005,dd  UOutline  @Introduction Motivation Previous results Main result Future work A ? Graph G=(V,E)>  _ Subgraph H of a Graph G2    XPath  VPath-partition   YPath-partition   WPath-partition Number p(G) J  BPath-partition Number p(G)B  c#Cycle  [ Hamiltonian  v2Complete Graph Kn, &   C Motivation  RSkupieD 1974 Hamiltonian shortage SH(G) x*   $ D Motivation  E Motivation  F Motivation  G Previous Results   =Question  ; NP-Complete  \Previous Results   ] Connected  ^ Component  a! Cut-vertex  b"Block  e%Complete Bipartite Graph  z6Tree  {7 Block Graph  H  Our Result   A Our Results  f'f-path partition  |f: V(G){0,1,2,3} f-path partition: P: collection of vertex-disjoint paths 1 f (v) `"3 in some path in P 2 f(v)=0 v itself is a path in P 3 f(v)=1 v end vertex of some path in P%   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz|}~Root EntrydO)PicturesCurrent User,SummaryInformation({$PowerPoint Document(\DocumentSummaryInformation8Microsoft e z_}/hV 3.00lEquation Equation.30(Microsoft e z_}/hV 3.00mEquation Equation.30(Microsoft e z_}/hV 3.00nEquation Equation.30(Microsoft e z_}/hV 3.00oEquation Equation.30(Microsoft e z_}/hV 3.0/ 0DArialgsRobbb0bWo 0"De0}fԚgsRobbb0bWo 0 DTimes New Romanbb0bWo 00DTahomaew Romanbb0bWo 0"@DWingdingsRomanbb0bWo 0PDjwiԚdingsRomanbb0bWo 0A`DSymbolgsRomanbb0bWo 0 a.  @n?" dd@  @@`` 8m>FL FG ( ! ""$-/1 7 , ""&%( *#&' *)( )5'# "!   + &% %$>!?2$L7HF<:$* m2$m2դb~I{m2$U`o-q6b 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g41d1d0bvppp@ <4!d!d 0bb<4dddd 0bb? %O =;>-Path Partition for Graphs With Special Blocks . $XoOpg September 2005,dd  UOutline  @Introduction Motivation Previous results Main result Future work A ? Graph G=(V,E)>  _ Subgraph H of a Graph G2    XPath  VPath-partition   YPath-partition   WPath-partition Number p(G) J  BPath-partition Number p(G)B  c#Cycle  [ Hamiltonian  v2Complete Graph Kn, &   C Motivation  RSkupieD 1974 Hamiltonian shortage SH(G) x*   $ D Motivation  E Motivation  F Motivation  G Previous Results   =Question  ; NP-Complete  \Previous Results   ] Connected  ^ Component  a! Cut-vertex  b"Block  e%Complete Bipartite Graph  z6Tree  {7 Block Graph  H  Our Result   f'f-path partition  |f: V(G){0,1,2,3} f-path partition: P: collection of vertex-disjoint paths 1 f (v) `"3 in some path in P 2 f(v)=0 v itself is a path in P 3 f(v)=1 v end vertex of some path in P% @(@@@@@@@@  j*f-path partition number pf(G)X   p,Inductive Theorem  AInductive Theorem  h(Inductive Theorem  i)Inductive Theorem  k+Inductive Theorem  ~:Basis  q-Basis  r.Basis  s/Basis  t0Basis  u1Basis  I  Our Result  J w3K |8Lx4M}9Ny5OPQRS?Time Complexity  The depth-first search to find all blocks and cut-vertices, and requires O(|E|) time Each subroutine requires O(|V(B)|+|E(B)|) operations. I@ @  T Future work  pPath partition Tree structure !small separator structure*98  @P!!bp    (  r  S P U     < +  D FB  S g  FB  S g E) F  <"  D   <\` + F  D FB  S g FB   S g E FFB   S g ~ a FB   S g  ) FB   S g  ) FB   S g ~ | } a FB  S g Eh FB  S g }F FB  S g  FB  S g -. FB  S g ~ FB @ S g  0 FB @ S g } 2   `,?"  Kg  2   `?"  Kf  2   `4?"@0  Kh  2   `?"p @0 `  Ka  2   `(?"p `  Kb  2   `q ?"  Kc  2   ` ?" Kj  2   ` ?"   Kl  2   `D ?" Ki  2   ` ?"   Kk  2   ` ?"@0  Ke  2    ` ?" @0  Kd  H  0޽h ? 3333rAA$,(   p@Equation Equation.30(Microsoft e z_}/hV 3.00`Equation Equation.30(Microsoft e z_}/hV 3.0  ՜.+,0h     pùjpp\< EArial sөTimes New RomanTahoma WingdingsзSymbolBlendsMicrosoft {s边 3.0.Path Partition for Graphs With Special BlocksOutlineGraph G=(V,E)Subgraph H of a Graph GPathPath-partition Path-partition Path-partition Number p(G) Path-partition Number p(G)Cycle HamiltonianComplete Graph Kn Motivation Motivation Motivation MotivationPrevious Results Question NP-CompletePrevious Results Connected Component Cut-vertexBlockComplete Bipartite GraphTree Block Graph Our Result Our Resultsf-path partitionf-path partition number pf(G)Inductive TheoremInductive TheoremInductive TheoremInductive TheoremBasisBasisBasisBasisBasisBasis Our ResultPowerPoint ²PowerPoint ²PowerPoint ²PowerPoint ²PowerPoint ²PowerPoint ²PowerPoint ²PowerPoint ²PowerPoint ²PowerPoint ²PowerPoint ²PowerPoint ²PowerPoint ²PowerPoint ²PowerPoint ²Time Complexity Future workPowerPoint ²  ϥΦr ²]pdO OLE A{ vD<<_[UserUser@(@@@@@@@@  j*f-path partition number pf(G)X   p,Inductive Theorem  h(Inductive Theorem  i)Inductive Theorem  k+Inductive Theorem  ~:Basis  q-Basis  r.Basis  s/Basis  t0Basis  u1Basis  I  Our Result  J w3K |8Lx4M}9Ny5OPQRS?Time Complexity  The depth-first search to find all blocks and cut-vertices, and requires O(|E|) time Each subroutine requires O(|V(B)|+|E(B)|) operations. I@ @  T Future work  pPath partition Tree structure !small separator structure*98  @P!!b  0x0(  xx x c $  `    x x c $t pP  H x 0޽h ? 3333   qi(  x  c $H U   2  # l??p Ltree  2  # ll??P P  Pinterval    2   fxÒ??  Nplanar  2  # l4ǒ??P  T circular-arc    2  # l˒??   Mblock  2   f\??` p @ gchordal bipartite   2    ftҒ??P`  ]chordal  2    f֒?? @p  rstrongly chordal&    2   # lْ??  ]cograph  2   # lݒ??  ecocomparability  2   # lH?? P ebipartite distance hereditary  2   f?? 0  Q bipartite    B  ZD?? `B  ZD??@@@@B  ZD?? @@pB  ZD?? pp B  ZD??`  B  ZD?? P  B @ ZD??0 P B  ZD?? ` B  ZD??P p B  ZD?? B  ZD??ppB  ZD?"0 00H  0޽h ? 3333  { (  r  S p U   e   ` ??6y  IA graph G is connected if each pair of vertices in G belongs to a path.\JC  +C  C  J 2  # lX??0  Ka  2  # lH??  Kb  2  # l@?? 0  Kd  2  # l,??   Kc  B   `Do??` ` B    `Do?? B    `Do?? B    `Do?? 00 H  0޽h ? 3333     & (  r  S { U   <   `} ??uH  XA cut-vertex of a graph is a vertex whose deletion increases the number of components.$YXC  Y 4B  # g }  4B @ # g   4B @ # g  } 4B  # g  4B @ # g } 4B   # g } 2   Z?"0 2   Z?"0 2   Z?"  2   Z?"@ 0 2  Z?"@0 0 4B @ # g Mo4B  # g Mo2  Z?"0 2  Z?"0 2  Z?"@0 2  Z?"@0 H  0޽h ? 33332   r(   r   S q U   D    `(s ??&5x  PA block of a graph G is a maximal connected subgraphs that has no cut-vertex. QQC &-    4B  @ # g o  4B   # g o / 4B  @ # g  / 2   Z?" p ` p2   Z?" p2   Z?"  H   0޽h ? 3333  p .(  r  S  k U   2  Z?"2  Z?" 2  Z?"  2  Z?" 2  Z?"P  @ :B  3 ` :B   3  :B  @ 3 P :B   3   :B  @ 3  :B  @ 3   0 H  0޽h ? 3333  w` d(  dr d S dY U   2 d Z?"@0 2 d Z?"p2 d Z?"P @ p2 d Z?"  2 d Z?"P @0 @ B  d@  `Do??PpB  d  `Do??0 B  d@  `Do??@ B  d  `Do??pPpP   d  `] ??   eH. A. Jung 1978C   ( d  `?@ABCDEFGHIJKLMNOPQRSTU    S !    "p`PpH   0޽h ? 3333  wo ,(  ,r , S $ U   U ,  ` ??fR  f-path-partition problem is to determine the f-path-partition number pf(G) which is the minimum cardinality of an f-path partition of G. p(G)=pf(G) when f(v)=2 for all vertices v in G. -C  C       (C  C  C           C     C  C  C   H , 0޽h ? 3333  <A(  <r < S  U   2 < # l??p  KA  2 < # l?? 0  KB  2 <  f$Α?? P p  Mx    <  ` ??M KG    <  ` ??e v(  S end block  C   H < 0޽h ? 3333  0($(  $r $ S P U    $  `Α ??>q  ~,x: a specified vertex of a graph H in which f is a vertex labeling. i=0,1,2,3 f i :V(H){0,1,2,3} f i(y)=f(y) for all vertices y except f i(x)=i. C  C  C    C    C  C  C  C    C  C  C  C  C    C  C  C   H $ 0޽h ? 3333  rj((  (r ( S Hf U   P ( T ??LV  =pf0(B)-pf1(B) =pf1(B)-pf2(B) 1 f(x)=0 pf(G)=pf(A)+pf(B)-1. 2 f(x)=1 pf(G)=pf1-(A)+pf(B)-1.i                 C                  C                         C                            i H ( 0޽h ? 3333N   D(  Dr D S P U    D  ` ??@\ L*B is a path x: end vertex of B with f(x)=3 pf(B)=pf(B-x). x: end vertex of B with f(x)=2 pf(B)=pf1(B). B: end vertex x and another vertex y with f(x)=f(y)=1 no vertex between x and y has a label 1 pf(B)=pf(B  )+1 B : path obtained from B by deleting [x, y]. C  C  C     C                C  C  C     C                C  C  C  C         C  C C  C  C            C  C  C  C      C   H D 0޽h ? 3333j      P (  Pr P S 4* U    P  `/ ??0  LxC1. Also, either d21 with yD2, or else c1>d1 and d2=0<d3 with yD3. Then pf(B)=pf '(B-x), where f' is the same as f except f'(y)=1.     C      C      C        C        C      C                 C  C  C     C   2 P # l!??p @  rx f(x)=1:   2 P # lX??  P ry f(y)=2:    P  `T ?? Q  \* C   2  P # l(b??@  B 2  P # lHe??@ `0 ry f(y)=1:   H P 0޽h ? 3333M     pi (  x  c $ U     T,  ?"0P 7f-path partition: a collection of vertex-disjoint paths&8 7  8 (F 0p   @@` 2   f ??p  Ka     `  ??0p   I0 4L  # 2  # l?? Kd      ` ?? I3 L ; P [   #   2    f??; [  Kb      `d ??; m[  I1 B   ZD?? P 2  # l?? 0  Kc     f# ??p @  I2 B   `D??P P B   `D??P 0 P `  c $A ??0 8 H  0޽h ? 3333   `$(  r  S   U   r  S       H  0޽h ? 3333r}$>PH  \ Rfa coQye ~h0xprBuz X_AY$K,(   p@Equation Equation.30(Microsoft e z_}/hV 3.00`Equation Equation.30(Microsoft e z_}/hV 3.00lEquation Equation.30(Microsoft e z_}/hV 3.00mEquation Equation.30(Microsoft e z_}/hV 3.00nEquation Equation.30(Microsoft e z_}/hV 3.00oEquation Equation.30(Microsoft e z_}/hV 3.0/ 0DArialgsRobbb0bWo 0"De0}fԚgsRobbb0bWo 0 DTimes New Romanbb0bWo 00DTahomaew Romanbb0bWo 0"@DWingdingsRomanbb0bWo 0PDjwiԚdingsRomanbb0bWo 0A`DSymbolgsRomanbb0bWo 0 a.  @n?" dd@  @@`` @"n>FL FG ( ! ""$-/1 7 , ""&%( *#&' *)( )5'# "!   + &% %$!?"?2$L7HF<:$* m2$m2դb~I{m2$U`o-q6b 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g41d1d0bvppp@ <4!d!d 0bb<4dddd 0bb? %O =)>-Path Partition for Graphs With Special Blocks . $XoOpg September 2005,dd  UOutline  @Introduction Motivation Previous results Main result Future work A ? Graph G=(V,E)>  _ Subgraph H of a Graph G2    XPath  VPath-partition   YPath-partition   WPath-partition Number p(G) J  BPath-partition Number p(G)B  c#Cycle  [ Hamiltonian  v2Complete Graph Kn, &   C Motivation  RSkupieD 1974 Hamiltonian shortage SH(G) x*   $ D Motivation  E Motivation  F Motivation  G Previous Results   =Question  ; NP-Complete  \Previous Results   ] Connected  ^ Component  a! Cut-vertex  b"Block  e%Complete Bipartite Graph  z6Tree  {7 Block Graph  H  Our Result   A Our Results  f'f-path partition  |f: V(G){0,1,2,3} f-path partition: P: collection of vertex-disjoint paths 1 f (v) `"3 in some path in P 2 f(v)=0 v itself is a path in P 3 f(v)=1 v end vertex of some path in P% @(@@@@@@@@  j*f-path partition number pf(G)X   p,Inductive Theorem  h(Inductive Theorem  i)Inductive Theorem  k+Inductive Theorem  ~:Basis  q-Basis  r.Basis  s/Basis  t0Basis  u1Basis  I  Our Result  J w3K |8Lx4M}9Ny5OPQRS?Time Complexity  The depth-first search to find all blocks and cut-vertices, and requires O(|E|) time Each subroutine requires O(|V(B)|+|E(B)|) operations. I@ @  T Future work  pPath partition Tree structure !small separator structure*98  @P!!ba   !!(  x  c $\ܬ U     <,+  D FB  S g  FB  S g E) F  <"  D   <` + F  D FB  S g FB   S g E FFB   S g ~ a FB   S g  ) FB   S g  ) FB   S g ~ | } a FB  S g Eh FB  S g }F FB  S g  FB  S g -. FB  S g ~ FB @ S g  0 FB @ S g } 2   `|?"  Kg  2   `?"  Kf  2   `Ŭ?"@0  Kh  2   `Ǭ?"p @0 `  Ka  2   `Lì?"p `  Kb  2   `?"  Kc  2   `W?" Kj  2   `X?"   Kl  2   `h?" Ki  2   ``?"   Kk  2   `t4?"@0  Ke  2    `<9?" @0  Kd   !  `( ??e v(  S end block  C   H  0޽h ? 3333rA~/,(   p@Equation Equation.30(Microsoft e z_}/hV 3.00`Equation Equation.30(Microsoft e z_}/hV 3.00lEquation Equation.30(Microsoft e z_}/hV 3.00mEquation Equation.30(Microsoft e z_}/hV 3.00nEquation Equation.30(Microsoft e z_}/hV 3.00oEquation Equation.30(Microsoft e z_}/hV 3.0/ 0DArialgsRobbb0bWo 0"De0}fԚgsRobbb0bWo 0 DTimes New Romanbb0bWo 00DTahomaew Romanbb0bWo 0"@DWingdingsRomanbb0bWo 0PDjwiԚdingsRomanbb0bWo 0A`DSymbolgsRomanbb0bWo 0 a.  @n?" dd@  @@`` @"n>FL FG ( ! ""$-/1 7 , ""&%( *#&' *)( )5'# "!   + &% %$!?"?2$L7HF<:$* m2$m2դb~I{m2$U`o-q6b 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g41d1d0bvppp@ <4!d!d 0bb<4dddd 0bb? %O =)>-Path Partition for Graphs With Special Blocks . $XoOpg September 2005,dd  UOutline  @Introduction Motivation Previous results Main result Future work A ? Graph G=(V,E)>  _ Subgraph H of a Graph G2    XPath  VPath-partition   YPath-partition   WPath-partition Number p(G) J  BPath-partition Number p(G)B  c#Cycle  [ Hamiltonian  v2Complete Graph Kn, &   C Motivation  RSkupieD 1974 Hamiltonian shortage SH(G) x*   $ D Motivation  E Motivation  F Motivation  G Previous Results   =Question  ; NP-Complete  \Previous Results   ] Connected  ^ Component  a! Cut-vertex  b"Block  e%Complete Bipartite Graph  z6Tree  {7 Block Graph  H  Our Result   A Our Results  f'f-path partition  |f: V(G){0,1,2,3} f-path partition: P: collection of vertex-disjoint paths 1 f (v) `"3 in some path in P 2 f(v)=0 v itself is a path in P 3 f(v)=1 v end vertex of some path in P% @(@@@@@@@@  j*f-path partition number pf(G)X   p,Inductive Theorem  h(Inductive Theorem  i)Inductive Theorem  k+Inductive Theorem  ~:Basis  q-Basis  r.Basis  s/Basis  t0Basis  u1Basis  I  Our Result  J w3K |8Lx4M}9Ny5OPQRS?Time Complexity  The depth-first search to find all blocks and cut-vertices, and requires O(|E|) time Each subroutine requires O(|V(B)|+|E(B)|) operations. I@ @  T Future work  pPath partition Tree structure !small separator structure*98  @P!!br/A/[