On the Existence of Paths and Cycles

Diss. ETH No. 15897 

Michael Hoffmann 

On the Existence 

of Paths and Cycles 

2005

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Zusammenfassung 

Y 

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%'D$(' 

0 #¡ );' 7$(' 0 0 ] );- P 5' 0c¦ ¨ 0 -=1 P +.-=1 E ' 0 3 ':),m < %' E +U5 %'*),2d' M-=1 @¢@K < r 

1 

E ' 0 # ':3 E ' 3g':+,) 7$,C.5' 0 i ) 2©

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# 1e %'*),+&' ' %< E );- P 5 7$ + E ' 0 -R1 # ':) i @ - # $('D

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Contents 

Abstract 

Zusammenfassung 

Acknowledgments 

v 

vii 

ix 

1 Introduction 1 

R¦ 

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l ¡ 

7$,$('OCN+ 0 ¦ ¦ ¦o¦ ¦o¦ ¦ ¦o¦o¦X¦o¦o¦o¦X¦o¦o¦X¦o¦o¦ ¦ ¦o¦ ¦o¦ ¦ 

V¦£¢ 

p - 00 7C:-R< 7$,$('DC*+ A0 $ ¦o¦o¦X¦o¦o¦X¦o¦o¦ ¦ ¦o¦ ¦o¦ ¦ a 

V¦£¢¦7 

¢ 

¢ 

-=2d':

5 %$ +&5'O$ 7$ 0 Y!'D$ + E -/+&'O$ ¡ 1'O$ + A0 $ )&'D

1¥+ 0E $(1C.5 -g3 0 %3 13 + 1) 7$¢-g5©-/) # +&5-/+ 7$ ¡ / r 5-/) # P ) 2©

6 ':

@ 4+ '*l %$ +.$ C:- 0 2^'C 3 P 1¥+&' # 0 < 0 'O-/) + %3g' @ +&5' ) # '*) A@ +&5'9$,' E 3g' 0 +;$ 

 

0E +;5' C 0 Y!':l 5V1

Figure ¥ 

 

3: 

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Q 

0 'S$(' E 3g' 0 +;$B+&5-/+ -/);'LC 0 Y!':lV< F 0 # ' P ' 0 # ' 0 + ¦B 'LC 0 ¢¤'OCN+&1¥);' # +;5-/+ -[C 4) r 

k 0 13 2^':) @ ' # E 'D$ 0 +&5'fY %$ ?2 %

Figure 6: 

"3¥5ED¢¡ 93:70$ 50 B IK(8(

5 7$ -[$ %3 P

6 5 %C.5 +&' 0 # $+ 0 C:

G 7$- P -/+&5 @K ) 6 5 %C.5 0 + 6 Y!':),+ 7C*'D$-=)&' 

u, v, w A@ - E );- P 5 6 5'*+&5'*) +&5'*);' ':l 7$ +;$ 

(8 ( 7 D 59( K 5 3(0 

5 

5' 3 + 4Y-/+ 0 7$o+;5' @K 3 # 'D

0¥@ ),+;1 0 -/+;':< F -R$ @ -=)[-$M- 0 '¡ gC 4' 0 + 43 P

G /IHKJ ¦ 

(u, w) r P -/+&5 0 - # E ).- P 5 %$¢-=

M 'O$ - E );- P 5 C 0 +;- 0 -bC.5 ) # ':Y!' 0= $ %e:' 

> 3 

) 

1 E 5b- $ P 'DC ¨' # Y!'*),+&':l ¦ 

+&5¥) 

any ) #

) 

) 

k = g(k) 

 

¥ 

f(k)p(|x|) 

Chapter 1. Introduction 

Da 

Parameterized Complexity. 

C P ' 6 4+&5g+&5'9- PP -/)&' 0 +JC 3 P 1+;-/+ 0 -=< 0sr 

%

-R$ P )&'D$,' 0 +&' # -=+ G ¡ ¥ ¡ ¢ ¢ - 0 # +&5' 0 0 Y ?+;' # + -k$ P 'OC 7-=< 7$,$(1' A@ 

6 

+; 4 ¤ 5E 5 79$£¢¥¤ ?=5E79$ :D 

 

@ B ; ( ¡ A ¢n¦ j ' -=

k +&5¥) 1 E 5 - 

5'D$('k)&'D$(1

5 %$ C.5- P +&':) # 'E¨ 0 'D$ +&5' 2I¢¤'OCN+;$ A@ 1¥) 0 +;'*)&'O$ +- 0 # # ':) ?Y!'O$ $ 3g' A@ +&5' 4) 

 

) P ':),+ %'D$ ¦ ¡ +X-=

G = (V, E) - 0 # H = (W, F) 7$+&5' E ).- P 5 

G H := (V W, E F) ¦ 

 

Chapter 2. Basics and Notation 

 

E P 8 )[- 5 );- 

E ¦ E # 2 +&5'q$(':+ A@ %C:'D$[- # F 0 A@ ' # 'D$ " );- 5 

$,'*+ P 

0 

§ 

A@ 

5 

 

P # 

- P 5 

§ 

B 5D( & 7E ? +&5' A@ 

¦ ";?@( 

( 4 4 ¡ A@ - 6 -=

D %$ - $(' ¡ 1' 0 C:' 

W = (v 1 , . . . , v k ) @K ) 

l 7$f- 0 ' r # %3g' 0 $ 0 -=

l 7$ # %$,C 00 'DC*+&' # 0 + + 6 C 3 P A0 ' 0 +;$ 2 F )&'D3 Y-=< A@ - 0F 

p, q ¢ 

s = pq # '8¨ 0 ' 4+;$ F58=&K? -$ ||s|| := ||p − q|| ¦ 

 

2.3. Geometry 

l # 7$,C A00 'DC*+;$ ¨ 2 

l 

l ¦ 5' 4) )&'D$ P 'OCN+ 4Y!' C:< $(1¥);' ) ' ¡ 1 ?Y r 

§S':3 Y-R< A@ - 0F $ 0E

ε(p) := {q ¢ 

p = (p x , p y ) q = (q x , q y ) - 0 # r = (r x , r y ) 6 ' $,- F 

p %$S+ +;5' 79 &? +&5' );- F −⇀ qr 

@ - 0 # A0 < F @ # ':+ 

¡¡¡¡¡¡ 

¡¡¡¡¡¡ 

s p 

+ 

s r 

-R< 0E 

(p, q, r) < 0 

1(q) 0 

0 - 0E

¡ - 0 # 

X -/)&' P ' 0 

¡ - 0 # +&5' 0 +;'*);$('OCN+ 0bA@ - ¨ 0 4+&' @ -R3 4< F A@ P ' 0 $(':+;$ %$ P ' 0¦ 

X -R$[- 0 ':

A 0 (X, O) ¦ L +;' +&5-=+k+&5'h$,'*+;$ 0 O A 

0 P -/),+ 7C*1

or 7 5¦'5879$ ε > 0 ?=587855 3(0( δ ε ( B ? ? ; δ(p) \ P ( K 

c @ - 0 # 0 < F @ +;5' F CN) $,$S-=+ c ¦ 

¦ 

2Y 1$,< F! - 0F $(1¥2 E );- P 5 A@ - .¡ ¢ .¢ %$[- .¡ ¢ .¢ -R$ 6 ':

¤ 

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¡ 

¨ 

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¨ 

¨ 

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6 4+&5 

5D&@59( ?=5 K ?=587 4 4 ¡ 9$ D K 5D 3 59( 

¡ 

2.5. Geometric Straight Line Graphs 

R 

Proof. 

¢ 

⇒ U W G = (V, E) - 0 # 0 - ¢ .¡ +&5-=+ 

C*) $,$o-=+X- ¤@¤pJ0 0 + 

c $ ∂U ∂W c / 

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# V c 

+;5' 0 2 F # 'E¨ 0 4+ 0 @ ¢ .¡ 

'*)M+ 6 6 -=

@ 

¡ 

¡ 


 

¢ 

⇐ U W -R

%< 57 %$ 8 :D 5D 2©$(+;-RC: 0 

B ?? ; 7 4 "

γ 

γ([x, y]) ¥ 

(0, 1) 

¤ 2©$('*)&Y!' +&5-=+ -UY %$ ?2 %

K n 

- 0 # 5' 0 C:' 00sr P 4 ¦ 

¢ .¢ ¤ 8 ) P 0 +;$ 0 E ' 0 ':);-=

? ) (0 $&7E ? !6 5'*)&'¢Y!'*)&+ %C:'D$C ),)&'O$ P 0 # + $,' E 3g' 0 +;$ 

&7E 

0 # - 0 ' # E 'oC A00 'DC*+;$ + 6 Y!'*),+ 7C*'O$ @ - 0 # 0 < F @ $ 3g' P 0 +;$ A@ +&5'¢+ 6 

- 

E 3g' 0 +;$L-/)&'X3 1¥+&1-R

0 ?+;' (< F5 # $&@:4C:1¥),Y!'O$ ¦`¡>0 ) # '*)q+ H@ )&3U-=

¦ 


a 

W 7$X-h$ %3 P

¢ 

¤ ?=50785 B 85 '587 5 

T C 0 +.- 0 $q-HC*< $(' # $,1¥2+,);- 4

¨ 

2.7. Polygons 

! 


2 F 

0 ' +&' 

H := G[V(C)] D H +;-=+ 0 1©C*' 

T v -R$ # 'O$,CN) ?2d' # -=2 Y!' ¦ pJ0 $ # '*)- Y!':),+&':l A@ - 0 # +&5'fC F C*

P 1 

+&5-/+[$ +;-=),+;$ 6 ?+&5 +&5'

p n p 1 p 2 

¦ 5'f+ 1¥);$ 

α := (p i−1 , p i , p i+1 ) - 0 # β := (p j−1 , p i , p j+1 ) 0 T Y!':);$(1$ 

a(T ) < a(T) 0 

T 7$- 0 ¨ 1

¦ 

1

¤ 2©$('*)&Y!'b+&5-=+ 

0 

G 2 

( 

E := E {{x, v c } | c ¢ C - 0 # x ¢ V(c)} 

2.7. Polygons 

¢© 

+ 1)k+ +,).-:Y!':);$(' 'D-C;5 C*1+ r ' # E ' + 6 7C*' ¦ W9$ 0E +&5' @

£ 

P 2 

%$L2 1 0 # ' # 2 F - # ?)&'OCN+&' # 00sr 

Proposition 2.15 ∂P ( ?=5 78¥507 # 8& B 4 8 :D@70$ ¡ ©£ (P) 

¥ 


¢ 

¢ "¥&¡ ¢¨§©¥ ¢ ¡ £ "¥& ¡¡ £ )2! ©$' ¡ 

$%)$ 2 ¨$ ¨$% £#)# 

¢¤£¦¥ 

) ¥£¢ ¡ £ "¥&¡¢ © $' $%)¡ )2$ !&' ¡! 

¢ 

Figure 

¦ 4 59( ¡ 3 # $&@ ( 

15: 

P 2 

0 # 1©C*' # 2 F +&5' ¨ 1

£ 

∂P 2d'DCD-=1$('g-=

¢ ¢ 

Chapter 2. Basics and Notation

1 %$ 

 

Chapter 3 

Hamiltonian Polygons 

5 %$cC.5- P +;'*) 7$ # '*Y +&' # + +&5'$ +;1 # FfA@s -R3 4

2Y 1©$(< F +;5' '*l %$ +;' 0 C:' @ - -R3 4

¢ 

0 + %< 6 ' ¨ 0 -=

Chapter 3. Hamiltonian Polygons 

¢! 

E 3g' 0 + 0 +&5' 0 +&':) ) A@ $(1C.5 - P < FsE0k6 5 7C.5©¢¤1$(+ $ 43 P 

¥ ? K( 59( 3 ?=5 B (87 B "#%$&@ 

3.2. Frame Polygons 

¢ 

y 

z 

y 

z 

¢ § ¥£¢ £ ! £ $'¡( 

¢¤£ ¥¡ $%$ £#&%$' £ $'¡( 

y 

z 

y 

z 

¢ ¥£¢$%"&'$ ¨) £ $'¡( 

¢*) ¥£¤§$%)2$'¡( 

y 

z 

y 

z 

Figure 17: 

¢ ¥£ !)$'¡( 

¢ ¥ $ $%

¡¢¡ ¡ '507 65 

¡¤£ ¡ '507 65 

single vertex P 

¤ 2©$(':),Y!'+&5©-/+ C A0 Y 

¦ 

£ 

v ¢ V @ "57 ( 4 B 5 

?=50 ?=5 =& 7 

v ¢ P¢ 

(∂S) %$ -=< 6 - F $J- @ );-=3g' @K ) S ¦i ) P '*)&+ 4'O$ ¡§£ - 0 # 

¡¦¥ 3U- F < mh-q2 4+9$ +,).- 0E ' -=+ ¨);$ +9$ E 5+ 21¥+S+;5' F -=

¥ ¢ ¡ £ £#1 . $'&£ ¢¡ ¥ 

¢* 

£ ! §¥) 

α > π 

u $' 

£ 

v P¡ 

¦ # 59( 7 ¡ 7E 50( 

 

3.2. Frame Polygons 

V 

p j+1 

p j−1 

p i−1 

v 

α 

p i+1 

s 

¢¤£¦¥ ¢ £ 1¡ 

¢ £#1 .$'&£ ¢¡ £¢ ¥ § ) £ ! 

¢¨§©¥ ¡¡£ 

¤ V(s) (P) s ¨ 

¡ 

v 

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v 

¢ ¥ ¢ ¡ £ £#1 . $'&£ ¢¥ ©¨ ¥ £# ./ 

£ ""¥ £ ¨0$%1 $ 

P 

 

 

p j+1 

p j−1 

v 

α 

p i−1 

p i+1 

u 

v 

Figure 18: 

¢ ¥ ¢ ¡ £ £# .$'& £ ¢¡ © ¥ £#./

d, e ¢ 

P 

¦JpJ0 $ # '*)9- # 1¥2

Figure 19: 

D £ 5 '507 65 @ 579( B 5 K( 785¤£ 5 '507 65 

'587 5 

P ¦ 5'k$(' E 3g' 0 +;$¢+&5©-/+

¢ 

(P , u) ¦ 

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@

u +1 ¦ 

%3 P < F_r C 00 'DCN+;' # ¦ 

7$$ 

¡ £ - 0 # ¡ ¡ @K

¨ ¢9- @ );-=3g' 

(P, u) ¦ 

P - 0 # - 0b ) %' 0 +;-=+ 0 u(P) ¦ 

q ¦ ¤ 

¦ 

p i ¢ V(P) 

¦ 


a 

5' £ -=+&1¥);-=+ A0Ui 5-R$,' %$$(13g3U-/) %e:' # 0 +&5' P ':);-/+ 0 2^'D< =6X¦B¡ +\C 0 $ 7$ +;$ 

P 

4+&'*).-/+&' # - PP 

Figure 21: 3 

(P, u, p j ) 

(P, u, p i ) 

; 7E;65 $ B 785;65 B 5D&@50(

Definition 3.9 

 

u(p i 1 ) = −1¦ 

¡ ¡ 7 

alternation u P §7E 5 ( I37 

(p < 7950¥+; ( 

i , p i 1 ) 

B ? ? ; 

P u(p i +1¦ ) = < 

v ( "7 45070:; 3 u :D u(v) = +1 

r 6 -R$S-R

¥ 

P 

-R

¢ 

¥ 

¡ %$M$,-/+ 7$ ¨' # -R$ 6 'D

¥ 

¥ ;¦ 5 7$ C:- 0 2d' -C;5 4':Y!' # 'D-$ %< F @6 'h$ P 

¥ 

p i 

+&5-=+9$ P 

1 %$ 

¢ 

p i 

0 

(P, u,¨ 

P 

¦ j 'U5-DY!' # 'D$&CN) 42d' # 

, p i , r, ⇀ s ) 

¥ 

P 8 Er 

(q, t) ¦ p

¥ 

Figure 24: ¥ 

¦ 

587 5 

$? ©'5 D 5 &785E5 7 E507 ¦ 50:D¡ 5 £ 5 

(q, t) %$ +&5' 0 < F A0 ' P );'D$(' 0 + 

¡ ¤ ?=5 0587 785 £ 5 4 3 ( B 5D '507©4 "#%$&'¥( 78 

¡ 

( ; (0 5 ¨ 

¢ 

P 

2©1¥+ @ ) +&5'J3 3g' 0 + 6 ' 

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a ¢ 


x 

t 

p i 

r 

t 

+ 3U- F 2d'f)&' ¦'*l 0 +&5'f)&'D$,1

)&' P ) 2

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aa 

q 

q 

x 

⇀ s 

p i 

r 

p i 

t 

t 

r 

Figure 26: 

658 D¡ 5¤£ 5 $ B 785E;5A $ B 5D&@50(4 & B ; %< 5E7 

¦ 

5+& 50¥( E ' 

( 

(r, p i , x, t) 

P 

P 

0 # -=

¥ 

¡ -R$¢+&5'q'*l¥-=3 P ?=5 ( 7070 :D3=& 7E 5 ¥ 

3.4. Dissection 

a! 

Y!':),+&':l @ - E ' # 'O$ 7C ¦ ¤ 

t 7$S- $ 0E

(P , u,¨ 

p k p l 

7$ - C 3g3 0 

) ¦ 

Figure 28: ¥ 

(P, u,¨ 

P - 0 # 

, p i , ⇀ s , p k p l ) 

D p k 

7$ - C A0 Y!'*l 


a 

Operation u,¨ 

4 (Drag-Edge(P, , p i , ⇀

Proof. ¢ 

¥ 

p l 

 

p k p l 

0 

3.4. Dissection 

a 

P -=),+ @ ) 3 +&5'q1©$(1-=

0 < F¦ 

p l 

¤ -=) E 1'X-$ @ ) 

p k 

A@ 

p k 

- PP 'D-/).$o-R$ A@ +;' 0 0 

p k 

¦ 

p i 

A@ $ 3g' # 7$,$('OCN+ 0 P < FVE0 

P 

- 0 # 

P - 0 # +&5' # 7$,$('OCN+ 0 P < FVE0 

D 1 0 # ':) 

 

¥ 

¡ & 

D - 0 # 2 +&5 ' # E 'D$ 0 C # ' 0 + + p i 

-/)&' 

D ¦X¡ @ );- ERr ¨ # E ' 7$ 0 +[- PP 

# E '[+&5-=+ 6 -R$L5 ?+ 0 +&5' P )&'*Y 1©$ 4+&':);-/+ 0 @ - 0_F £¥$('D'X8 E 1¥)&' >2 

+;5'[' 

)- 0 '*l¥-=3 P ?=5 7E 5 

4 

3.4.5 A First Dissection Algorithm 

'*+f1$f$,13g3g-=) %e:'q+&5' -=< EA ) 4+&53 @ )[+&5' 7$,$('DC*+ A0 i 5©-R$('k-$ # 7$,C*1©$,$(' # 

-=2 Y!' ¦h +&' +&5-/+ +&5 7$ 7$ ¢¤1$(+ - P )&'D< %3 0 -=) F # 'O$,CN) P + 0¦ 

-=+&'*) +&5'g-=< r 

E ) 4+&53 

6 % =78 B 5D 785> 3¥+3 ( 7E 5 P 7 S 

u :D C4 (8 B 5D (E( 5 B ¨ P 

P ← C 0 Y (∂S) u +1 - 0 # ¨ ← {P} - 0 # );' P 'D-=+k+&5' 

4F=& 4 ? 79 58¥; 

£ +;-=),+ 6 4+&5 

@ 

p i 

-/);'LC 3 3 0 ' # E 'D$ A@ 

D -R< 0E ⇀ s 

- 0 # 1 P # -/+&' ¨ -C:C ) # 0E < F¦ 

P - 0 # 

D ' 0 +&'o+&5' # 7$,$('OCN+ 0 P < FsEA0 3 # ¨' #b# 1¥) 0E +;5 %$ ?+&':);-/+ 0¦ 

# 

@ +;5' @ );-R3 ' 7$¢3 # ¨' # 0 +&5 %$ 4+&'*);-=+ A0 +&5' 0 ' ?+;5'*) ¨ l+;' 0 # r §L' ¦'*l 

¡ 

 

¥ 

& ¡ & ¡ ; £ & - 0 # ¥ ;¦ 

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7$- 0b 2VY 1$LC 0 $,' ¡ 1' 0 C:' A@ +&5'oC:- 00 7C:-=< # 7$,$('OCN+ 0 2d'DC:-R1$(' 

£ 

¥ 

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§L'DCD-=

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) 

< FVE0 - 0F 3 )&'k2d'DC:-R1$('g+&5' 0 ':l+X);- F %$ $(5 + @ ) 3 +;5 %$oY!'*),+;'*l ¦U 5 7$ 

P 

 

 

¢ 

¥ 

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\0 +&'[+&5-=+-=

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P 

A@ - # 1¥2©

q i 

 

1 i k 3k1$ +o2d' )&' ¦':l 0 P 

¦ 8 E 1¥)&' $(5 /6 $ - 0 

P

K ¡ 

¢ 

P 

-$ - 6 ' # E ' ¦ " 

t +;5- 0 + r -R$ P -/),+ A@ - 6 ' # E ' ¦ 

¥ 

£ 

¡ 6 5 %C.5 ' 0 $(1)&'D$\-[C 3g3 A0 ' # E ' @ 'D-C.5 

3.5. Simplification 

A 

P 

¡ £ 5 < # $2^'OC:-R1$('©£ 

(P ) 

(P)

p i 

-R$ P -/),+ A@ - 6 ' # E ' ¦ 

p i 

6 '9

P 

¦ 

p i 

%$X$ +,) 7CN+;< F C 0 Y!'*l 0 

p i 

0 

P 

+&5' 0 0hE ' # 'D$ 7C 6 %

0 

E ' # 'D$ %C ¦ A@ 

F ¡¦¥ Z 

0 

r A@ p i 

0 

¦ 

6 ¤ %$,' 

¦ 

+&5':) 

r Q 7$L)&' ¦©'*l 0 

r 

P );'D$(' 0 + 0 +;5' 0 

P F # -$k-R

u(s) = −1 ¦ Z1+M+&5' 0 s - 0 # 4+;$ P )&' # 'DC:'D$,$ ) 0 +&5' @ );-R3g' 

p i 

 

¦ 

r q - 0 # t - 0 #

5 £ 5 

t - PP 'D-=);$ + 6 7C*' 0 

¦ 

y 0 

(t, x, p i , r) +&5' 0 +&5'`);- F ⇀ s 

CD- 0 2d' $,< E 5_+&< F 

¦ 

 


! 

+,) %- 0E

q 

q t ¡ D¡ 

Proposition 3.28 

6587 ?=5 78K; D 59( B 7085D 8 45 ?=5 78$ 

∡ D p i 

3¥6 4 (8 79 B %$ B 45 =&;F50( 

t - PP 'D-/).$ + 6 %C:' 0 

 

p i 

%$ - C 3g3 A0 )&' ¦'*l $ 0E

£ 

3 

¢ 

3 

a 2d' - Y!'*),+;'*l +;5-/+ 7$SY %$ ?+&' # 2 F Z1 4< # r p - P 

ρ(a) @ ) 3 a + p i 

-R$ @

Figure 35: 3 

q 

ρ(a) 


 

q 

t 

q 

t 

p i 

p i 

£ ! £ $ $%)$ 

¢¨§©¥¡¢ 

 

(p i , x, q) 

¢ £¦¥ ¡ ¢ £ ! £ ¡ 

q 

t 

q 

t 

a 

ρ(a) 

p i 

p i 

¢*) ¥¡¢ £ ! £ $%11¡01 

¢ ©¥ £#"&%0" £ ¨ 

; 7E;65 D 50( K 785 (

¢ 

Figure 36: ¥ 

587 5 

t $ 85 785 ( 5D D 793=& 3 

¥ 

¥ 

; 7E;5 

3 

; 7E;65 


 

q 

x 

q 

⇀ s 

p i 

p i 

t 

t 

r 

r 

¢ § ¥ ¡ ¢ £ ! £ # 

¢¤£ ¥ ¡ ¡ © 

Corollary 3.31 

¨ ¦ 50:D¡ 5 £ 5 B 785;59( ( 5 ¥ B ?=50 ?' ( 

B 3( K 785 3( 5ED D 703=& ?=5 # 4 3=& B 43: 


q ) t 0 < F¦ ¡ @ 4+ C*)&'D-=+&'D$ - 0 - 0 + r C:- P -/+ 

- 0 1 0 $&- @ 'h- 0 + r CD- P -=+ 0 ' @ 

" $ # 7$,C*1©$,$(' # 0 i ) P $ ?+ 0 s¦ as¨ lV+&' 0 # r §S' ¦':l C:- 0 C*)&'D-=+&' 

q q 7$ 0 + )&'*Y 7$ ?+;' # 0 +&5' @

0 

0 

0 

p l 

¦ 

p i 

0 

P 

%$ 0 +M3 # ¨' # - 0 # +&5V1$)&'D3U- 0 $9- C:- P 

p i 

%$q

3 

1 E 5`+&5' -R< E ) 4+&53 0 P -/),+ %C:1

Proposition 3.34 ?=5 4 7E 5 ( 7950¥65D 0$ :D 

u y 

{y, z} ¨ {y, z} 3 ( ?=50 

3 

< ¢ B ; ¥ ? K(£50( 

q ¢ P¢ 

7C*'O$ A@ +&5' @ );-=3g'M+&5¥) 1 E 5 1¥+ +;5' £ -=+&1¥);-=+ A0 - 0 # %$&$('DC*+ 0bi 5-R$('O$ ¦ 

Y!':),+ 

2©$(':),Y!'+&5©-/+ ¨ lV+&' 0 # r §S' ¦':l 0 < F )&' P

¢*) ¥ 

Figure 38: 

65879:; ( 3 ?=5 70 58 +; $ 5@D ( 5 

3.6. Preparations for Bridging 

 

¢¤£ ¥ ¢¨§©¥ 

a 

¢ ¥ 

a 3 ¡ & 705 ¡ D¢¡ ¡ 3 3 

¥ B 

4507 5 

; 7E;65

¡ ¡ ?=50785 3( ; (8 5 ( 5 785 £ 5 4 3 :D!; (8A 5 

?=5 # 4 3=& 

¥ 

£ &¦ 

¥ 

¥ 

¡ & 

¥ 

¡ 43 P 

Figure 39: 3 

¢* ¥ 

p i 

¢ ¥ 

5 B > # $K&' ? ; ? K(© &' D 5D&@5 ¢ 4 B 5D&@50( 

( 

?=5 7E 5!:D2?=5 (E? @D 5D 705 & < ¡

¨ ¢9- @ ).-=3g' 

¢ 

3 

P - 0 # - 0b ) %' 0 +;-/+ 0 u(P) = u y (P) @ )L$ 3g' y ¢ V(P) ¦ 

x # ' 0 +&'f+;5' P )&' # 'DC:'D$,$ ) A@ 

y 0 P 

¦ 

V(P) 

p i 

-$+&5'

(P, u) ¦ 

&@ D 7 ?=5 B 797859(+ " D@ +;5' P )&'OC*' # 0E Z1 4< # r p - P P '*);-=+ A0 )&':Y 7$ 4+&' # 

$(':+ 

u(v) := +1 @ )[-R

x 

¢*) ¥ 

x 

¢* ¥ 

3 

x 

¢ ¥ 

x 

¢ ¥ 

B ; 4 5D&'5>:D & 7E¥5850( &@ D 5D&'59( ;©785 £ 5 

¥ 

B 59( '587 


a 

y 

x 

y 

¢¨§©¥ 

¢¤£¦¥ 

y 

y 

y 

y 

Figure 40: 

785 

; 7E;5 ¢ ?=5 785 7950¥+; ©' DK( B 785E; 3=&

+;5 0E 3 )&'[+ $(5 /6X¦ 

0 

+&5':) 6 %$,'U+&5':)&' -/);' $ 3g'U1 0 $,-=+&1¥);-=+&' # Y!'*)&+ %C:'D$q-R< 0E E ' 

¤ 

x %$S+&5' 0 < F )&' ¦'*l Y!'*),+&':l 0 +&5' $(12 6 -=

¨ ¤ - $,'*+ 

S A@ # %$ ¢ 0 +S< 0 ' $(' E 3g' 0 +;$L- 0 # - 0 ' # E ' {y, z} @ C 0 Y (∂S) ¦ 

P ← C 0 Y (∂S) ¦ ¢¡¤£¦¥¨§© 

¨ 

w ← s¦ 25©;,©4(?5@68¨£;¨¡ p i 

: 

D 


 

0 @ ).-=3g' - @ +;'*)h+&5 7$ )&' rn ) %' 0 +;-/+ 0¦ Q )&' Y!'*) +;5'*)&' 7$ 0 

Y!':),+&':l +&5' 

x x 

# 0 - 

0 A0 0 +&5' ) 4' 0 +.-/+ 0 2d'DCD-=1$,'q2 +;5 -/)&'k$&-/+&1¥).-/+&' # -=

¢ 

) 

) 

¢ 

 

) ¦ 

D ¢ 

q 0 P 

¦ 

¢!1 $ ¨¡! &' - £#&'$ ¨ £ 

¤ 

¢!1 $%¨¡! &% £#&%$% ¨ £ $ 

¥ 

¢ ¢)! £#&%& 

p k 

$' £#& £ )¡./ 

¦ 

D ¥ 

(a 0 , . . . , a k ) k ¢ £ 

t £ r £ ¡ 

P 

+&5' 0 

t %$ 

∡ D p l 

$'- 

3.7. Algorithm Summary 

 

(p i , r, w) ← (a 0 , b, a 1 ) ¥6 5'*)&' b %$+&5' +;5'*) ¢= a 1 

J0 ' E 52 ) A@ 

a 0 

0 

P 

¦ 

¡>@ p -/);' @ 1< r £ -/+&1);-/+&'SC*)&'D-=+&' # -9);' ¦':lX+ 6 0 

0 

j ' # E 'O$ - 0 # '*l ?+ ¦ 

- ¡ ':Y!'*) ¨ 

2 ¡ @ 

p i = (p i , r, w) ← (b, a, c) 6 5'*)&' 7$U- )&' ¦'*l 

b 

D @ ) 3 


&'79 ?' ¡ ¡ 50703:;59( 

V(P) ¦ Q );' 0 ' 6 

0E 'D$SCD- 0H C:C:1¥) 0 - ¨ 0 4+&' 0 13q2d'*) A@ ?+;'*);-=+ 0 $ A0 < F!¦ 

$,1C.5HC;5©- 

+&5':) 6 %$,' ' ?+&5'*) £ +&' P ' ) £ +;' P @ %$g':lV'OC*1¥+;' # 0 ':Y!'*) F ?+&':);-/+ 0¦ 

¤ 

p i 

- 0 # +&5' 0 

p i 

7$ - $(+,) 7CN+&< F C A0 Y!'*l Y!'*),+&':l A@ +&5'U);'D$(1

(E? @D 5ED D@7 ¡ 

B ¡ 

¦ 

¦ 


D 

y z y z 

y z y z 

y z y z 

Figure 41: 

P 3¥6 

B '5 "#%$&@ ( 


D ! 

y z y z 

y z y z 

y z y z 

Figure 42: 

'507 65 

3 

5 3 

 

&'79 ?' ¡ ¡ 

P 

?=50 

P 

? ; 785 785 £ 52< (£ 5 

¦ 

D ¢ 

¨ 

D 7 ?=5 4507 B 50( ( 5 785¤£ 5 4 3 

p i 

p i 

3 

p i 

( "7 ( 5 785 £ 5 4 < 3 

(+ % < ?=5 =705¦ ( 507E; 

Proof. P 7$ -R< 6 - F $U- @ );-=3g'H- 0 # ¥ 

¢ 

 

 

¥ 

¥ 

¥ 

£ - 0 # 

¥ 

785 "507 5 B ?=50 50 ?=587 

P 

7 5D&'5 

P 

¥ 

¥ 5 < # @K )g+&5'H$,-=3g' 

¡ ; - 0 # 

¥ 

£ 5 < # - 0_F 

¡ 7$ - # 4)&'DC*+ C A0 $(' ¡ 1' 0 C*' @ i ) P $ 4+ A0 s¦ as¦ Z F i ) P $ r 

¥ 

D ¢ 

£ @ 0h@ -CN+ -RCDC ) # 0E 

P 

+&5©-/+ 7$ 0 + P '*) @ 'DC*+ ¤ +;5'' # E '92d'*+ 6 ':' 0 +&5'9+ 6 Y!':),+ 7C*'O$\+&5-=+ 

¦ 

¦ 


D 

Proposition 3.40 P 

P 705 3 

85 

3( 4'4 $;( 7E 5 :D ¨ 3(4'4 $;( 85E :D ( 

5 < 

 

&@70 ?' ¡ ¡ 785E '507 

?=5 # 4 

(£5 B 

3=&< #79 ( ?=# D $ 3 5 E507 

?=50 3( 

P 

( "7 © 

p i 

p i 

¨ £ ¡ ¨ ?=587855 (8( ( 5 ¥ B 3 

 

?=50785 

5 (8( ( 5705¤£ 5 4 3 3 

¨ ¡ ¡ 

r ( B '5 7 ( 53 P 

5 

¨ ¡ ¡ 7 4 '587 B 59( < 

B 5 7 

0K? 

3 B D 50¥ 5ED&'59( 785© 587 5 B 

p i 

4 K( ?' 9$ ?=5 7E$ 4F=& 4 ?' B ? ?=5D3(8( 5 B # $&@ 4 K( 

¨ £ ¡ K 8K? 5ED&'59( 3 B D 50¥ 

);'D-R$ A0 $-R$ 0 i ) P $ ?+ 0 s¦ R s¦\¡ +S);':3U- 0 $L+ $(5 =6 +&5-/+ ¨ £ I ¨ £ 5 < # 

- 0_F + 43g' - @ +&'*) £ +;' P 2 - 0 # +&5©-/+ 

¡ . 

+ 43g'f2d' @K );' £ +&' P - 0 " < E ) 4+&53 s¦¡ !_¦ 

':+o1$ -=) E 1' +&5©-/+f+&5' $ +;-=+&':3g' 0 +;$[5 < # @ )o+&5' ¨);$ + ?+&':);-/+ 0 A@ +&5' 

< _ P 0 " < E ) ?+&53 s¦ !¦ 

+ 0 V¦ ¢q0 # 12

¥ 

¥ 

%$g-/+U3 $ A0 'H$(1C.5 C 0 ¨ E 1¥);-=+ A0 - 0 # 

¡ +&5'*);' + 

P -/),+ A@ 4+ ¦ " $[+&5'q).- F 0 7$ +&5 7$ 4+&'*).-/+ 0 %$X$(5 + @ ) 3 

p i 

⇀ s 

¥ 

¡ 5 < # $ # 4)&'DC*+&< F - @ +&':) ¨ lV+&' 0 # r §S' ¦':l 0 £ +&' P 'S-R$\-[C 0 $(' ¡ 1' 0 C*' 

p i 

- 0 # 

r £ -/+;1¥);-/+;' # 'O$ 0 +[CN)&'O-/+&' - 0_F 1 0 $,- @ ' - 0 + r CD- P ) 

p i 

7$ $('*+ + 

t 0 £ +;' P 'S+&5' 0 

¥ 

D ¢ 


yz 2^'OC:-R1$(' -$ -/) E 1' # 0 i ) P $ 4+ 0 s¦¡ ¢ 

y - 0 # z 

0 ) ' ¡ 1-R< + 

0 C 0 Y!'*l $ 0E

¥ 

r -R$ 0 +&5' P )&':Y 1$ 4+&':);-/+ 0¦M£ %3 4

0 2 +&5 );- Er ¨ # E 'X- 0 # +&5' @

¢ 

¢ 

¥ 

¥ 

p i 

-/)&' E_ # @ ) 

¥ 

p i 

6 -R$ 5 4+ 2 F 


D ! 

E # -=2 Y!' +&5-=+ ?+ # # 0 +oC.5- 0E ' # 1¥) 0E +;5 %$ 4+&':);-/+ 0 - 0 # @ )[+&5' 

-=) 1' 

# E 'H-=< 0E +&5' E ' # 'O$ 7C + /6 -=) # $k+&5'H$,- @ ' CDC*1¥)&)&' 0 C:' A@ 

+&5 +&5'*)g' 7$ 

p i 

p l 

- 6 - F @ ) 3 

p l 

¦ " E - 0 +&5 7$ 

p l 

7$ C A0 Y!'*l )L$&- @ ' @ 4+ %$ 

p i 

0 +&5' @ );-=3g'f- 0 # ¢ E 1-/) # ' # ¤ 

p i 

¦\ 5 %$ P ) Y!'D$ ¨ ¡ ; ¨ £ ; - 0 # 

£ .¦ 

7$q' 0 $(1¥)&' # 

2 F ) 4' 0 + 0E +&5' 0 ' E 5_2 ) A@ 

Case 3: 

1 $,' +;5-/+g- # 7$,$('OCN+ 0 P < FsE0 

£ PP 

3g' )&' ¦'*l Y!'*),+&':l 

$ @ 

 

p i 

¢ D 

D 

¨ 7$ $ 43 P < F $ P 

p i 

+&5-=+ 6 -$J5 ?+ 2 F ⇀ e 

6 -R$C*':),+;- 0 < F E_ # 2d' @ )&' 4+ 

Either 2 +&5 ' # E 'O$ 0 C # ' 0 +L+ a -/)&' E # ¤ " $ +&5'¢);- F ⇀ s 

@ ) 3 

D 1 

7$ $ P 

¥ 

t - 0 # - 6 - F @ ) 3 

¡ 5 < # $ C:

@ 

p l 

0 < F!¦ Z FbpJ )

0 4+ 0 s¦ sR¦L 5'*)&' @ )&' -=

(a, c 1 , d 1 , . . . , c m , d m , b) ¦ ¤ 

B 5 B 5D 5D&'5 %3(8© op¨ 

¦ 

S )&'O$(1

'*Y!'*).-=< A@ +&5'h+;-R$ m¥$ # 7$,C*1©$,$(' # -/2 Y!' CD- 0 2d' %3 P

Figure 43: 

( 5 !( 5 & 50¥ ( 7 4 ?' B ? ?=5 (0 $ &7E ? (! K 

(£5 & 50¥( < ?=5 5 7 4 ?' B ? 1 7$ 

K 

¢ 


3.10 Remarks 

$[3g' 0 + 0 ' # 0 p 5- P +&':) R 4+ %$ 6 ':

Conjecture 3.45 ¡ 7 $ ¢ 5 ( 59A "37 4 ( 5D ( £ < 79 =&# 59( ¦ 

s 1 

 

s 3 

- 0 # 

¦ 

s 5 

+;5¥)&':' 3g' # %13 

s 2 

 

s 4 

- 0 # 

s 6 

- 0 # $('*Y!' 0 $(3U-=

6 ?=5 ( ¢ 7859( ( K 

0 

x 1 

A@ 


Da 

Theorem 3.46 

?=58785 ( ( 5 © ?'

1 i k/2¡ ¦ 

v 2i v 2i+1 ¢ S¦ 

¦ 

1 i (k − 1)/2¡ 

 

Figure 45: 

465879:; 3=& I; ? ?'78 &?!D ( £ 3¥ ( 5 & 58¥ ( 

(S) %$M-=

-/+&5 ¦ 5' @ 0 

43k13 0 13q2d'*) A@ Y!'*),+ %C:'D$+&5-=+C:- 0 2^'oY 7$ 4+&' # 2 F - 0 -=

FVE0 ¥ ¦ ¤ 2©$('*)&Y!' +&5-=+ ¥ 7$ 0 + 0 'OC*'O$,$,-/) 4< F -=

':l+ 6 ' ) 4' 0 + -=

j 

P ¦ 

v + P ¦ 

P ¦ 

P ← (v 0 , v 1 ) ¦ 

X # 

¢= u B0 ' E 5_2 ) A@ v 0 ¨ 

{u, v} 7$ -LY 7$ ?2 4

0 s < k $,1C.5 +&5-/+ (v 0 , . . . , v k−1 ) %$o- P -/+&5 ¦X¡>@ 

v 

s = 0 +;5' 0 

v k−1 v k 

C:- 00 + 2d' - 

Proposition 4.11 ¡ 7 $ '587 5 v 3 ?=5 8 D$ E4 B¦ 

Chapter 4. Alternating Paths 

 

v = ¥ ¡§¦ (B) 

u = ¢¡¤£ (B) 

Figure 48: 

843F 

B 

?=5 8 D$ (E? @D 5D¢¡ ( 4 4 ¡# £ 5 78 

Proposition 4.10 

?=5 < 

78 7 

¥( D 507 ?=5 ";? 

P B 5ED 9$ 

&'79 ?' £ 7 

(v 0 , v 1 ) :D X 4 ?=58705 f ¢ X ¤ ?=58 58 ?=507 P 

P 70 ( B $ B F5 ¦ 

7 

B ?=59(© '587 5 

705 

70 ( E4 F 

P 

X¦ 


4+\+&'*);3 0 -=+&'D$ 

X 

¡ @ +&5'-R< E ) 4+&53 # 'O$ 0 +);'D-RC.5 - 0_F Y!':),+&':l @ ) 3 

6 4+&5H- 6 -=

¥ 

£ ¡ 7 $ 

¡ ¡ 7 $ 

Proposition 4.13 ¡ 7 $ i¦ 2 i l¦ 

(S) 

?=55D&@5 

¦ 

(B i ) < 

|V( C ) (B i ))| ¥ 3 @ ) - 0F 

¦ 

4.1. Lower Bound 

 

B §78 5¥),+ (B) v 

( 45079:; 3=&> ";? 3 

4 ?' B ? (8+7 ( 4 ? 

( 9 $ 5D&'5 :D 50:DK(54 ? ( 5 & 58¥ 5D&@5 

 

£ 0 C*'M+&5'¢$(' E 3g' 0 +J' # E 'O$ -/);' P - ?) 6 %$(' # 7$3¢ 0 + +;5'*)&'¢C:- 0 0 < F 2d' 


'U$(' 3 0 ' E 0 # ' 0 + + - 0F Y!'*),+&':l ¦h 5' $(' E 3g' 0 +X' # E ' 0 C # ' 0 + 

0 E ' + # ' C 

A@ C ) 2 F 5¥),+ 

# '8¨ 0 4+ A0 £_5' 0 C:' +&5':)&'f-/)&' 0 < F Y 7$ 42 %< 4+ F 

+ P -=),+ 

0 C # ' 0 + + 5¥)&+ 

(B) (B) 

' # ¦ E 5'hC*

¡ ¡ 7 5¦'5879$ '587 5 

 

¢ u )¦ V( 

(B 1 ) u 6 

u ¢ 

4+&5 - Y 7$ 42 4

¢ 

P := (v 1 = κ, v 2 = µ, . . . , v k ) 2d'`+&5' 1¥+ P 1¥+ A@ " < Er 

(B 1 , . . . , B l ) ← ¦ 

v k ¢ V(B i ) V@ )9$ 3g' 1 i l +&5' 0 

← (B 1 , . . . , B l , P) ¦ 

κ ← -qY!':),+&'*l @ ) 3 2 # F (B |¤|) +&5©-/+3 0 43 %e:'D$ d(¥ ) ¦ 

µ ← -kY!'*),+;'*l 6 ?+&5 {κ, µ} ¢ E(¨ 

¥ 

) - 0 # d(µ) < d(κ) ¦ nC @¤¦§i ) P r 


?=5 85 &<

¢ 

P )&'D-C.5'D$ - Y!'*)&+&'*l v ¢ V( ) ¦ 

v k ¢ V( C ) (B i )) 8 E 1¥)&' ¢! 2 @ )S$ 3g' 1 i l ¦ ¢ 


a 

V( ) $ 0 C:' κ = v k ¢ V( 2 # F ( )) ¦ 

P 

{f} V( ) 

-C:C ) # 0E + i ) P $ ?+ 0 ¢¦ D V ' ?+&5'*)[)&'D-C.5'D$¢- Y!'*),+;'*l @ ) 3 

) @K );3g$ - 2©-R

4.1. Lower Bound 

 

e 0 

v k−1 

κ 

µ 

B 1 

B 2 

P 

v k 

¨ $ ¨ §¥ £ § £#&'&%¡( 

u 1 

v k = u s 

κ 

B 2 

e 0 B 1 

v k−1 

P 

µ 

¢¤£¦¥ 

P 

¨$% ¨)¡¡ £ § £#&%&' ¡ 

¢¨§©¥ 

P 

κ 

µ 

e 0 

B 1 

B 2 

P 

¢*) ¥ 

P 

¨$% $%& 

Figure 49: 

¨ 3 (0 7E; ( 7 

 

&@70 ?' £ £¦¥

¡ 

of Lemma 4.5 ¢ j 'q- PP < F " < E ) 4+&53 ¢¦7Da + 

¢ 

 

 

 

¢ 


 

':3g3U- ¢¥¦ ¦ 


(e 0 , e 1 ) - 0 # f 

" < EA ) 4+&53 ¢¦7Da %3g3 ' # 7-/+&'D< F P ) Y # 'D$+&5' P ) _A@ @ ) 

¦S¡ @6 ' 

= 

(B 1 , . . . , B l−1 ) 

0 $,5 /6 +&5-=+S+&5' -R< E ) 4+&53 -=< 6 - F $L+&'*);3 0 -=+&'D$ +&5'XC*

¢ 

¡ 

¦ 

 

 

4.2. Upper bound 

R 

? ; 3( ( ; F5K(8 

 

O(n) 3 5 

2 

2(n + 2)¡ − 3 

4507 B 50( B 05 B 5D 3 

< E 


H 

+ E ':+&5':) 6 4+&5 -=

λ k = 12< E 

¢*) ¥ 

¢¢¢ 

s 

¤¦¥¨§© 


 

s 1 

s 2 

s 3 

s 4 

s 5 

s 12 

s 1 

6 

s 3 

¢¢¢ 

s 2 

 

1 

¢¤£¦¥¡ 

 

2 

¢¨§©¥£ 

s 1 

s 2 s 3 

s 4 s 5 s 6 . . . s 12 

Figure 50: 

¤ ?=5 B (8 7 B 

 

k 

12 k − 6 ¦ k 

n k := 3 k+1 − 3 

Y!'*),+ %C:'D$ ¦ ' 0 C:' 

C A0 +;- 0 $L':ls-CN+&< F 

3 n k − 1 + < E 3 

3 k 

3 k − 

− 1 

6 

= 

< E 

12 

2 3 

< AE 

2 n k − 18 + 12 < E 3 

3 k 

3 k − 1 

0 # +&5'¢C:

5 %$ C.5- P +;'*) # %$&C*1$,$,'D$ +&5' P -=);-=3g':+,) 7CUC 3 P

Theorem 5.5 G /IHKJ 3( < W[1] 4 ? 7850( "5 B k 

k 0 G ¦ ¡ + %$ %3 P ),+.- 0 +¢+;5-/+ 

¢ 

¦ 

q r G ¨§ r 

W[1] r C 3 P 

 

Chapter 5. Chordless Paths 

! 

x A@ 

V(P) 

'*+&':)&3 ' # 2 F ?+.$M$(':+ A@ Y!'*),+ %C:'D$ ¦ 8 )['*l¥-=3 P

P %$S- C;5 ) #

(q u , □, r, □, −) ¢ ∆ V@ )-=

Figure 

( B ?=58 ; B D 50( B 79 ?=5 ¤ 79

¢*) ¥ 

¦¦¦ 

¦¦¦ 

¦¦¦ 

¦¦¦ 

G ¢¡ 

 


a 

# 7-=3 0 # ¦ " $¢+&5' 0 -R3 0EhA@ +&5' Y!'*),+ 7C*'O$o$(1 EE 'D$(+;$ 6 ' -$,$ C 7-/+&'g'D-C.5 

- 

+&5'D$,' P -/+&5$ 6 4+&5 - Y!':),+&'*l @ ) 3 0 - 2 ¢ 'DC*+ ?Y!' 3U- 00 '*) ¤ ) 1¥+ 0E - 

A@ 

-/+&5U+;5¥) 1 5 E 

v i ) $ 3g' 

P _@ 

 

j 

1 j n +&':) P )&':+&' # -$J$('D

{v p , v q } 0 G C 00 'DCN++&5'¢Y!':),+&'*l $,'*+;$ {v i p, ϕ i p} - 0 # 

 

7 3:D 5 "50:D 50 B 55D&@5 7 ( 5 5D&@5 

5ED&'5 

l = j ¦ 

 

 

¦ 

5.2. Hardness for W[1] 

! 

σ i 

ϕ i - 0 # 

j 

τ i ^@ ) 

0 # - E );- ¦o 5$ 

 

P 5' 

Γ t 'OCN+&' # 2 F # ' 0 + @ F 0E - 0 # ¦ 5' -/)&' C 0 $ +,);1CN+ C 0 00 A@ 

1 i k 1 j n k τ k 

C 0 # - %$ 

G 

+ -R$ 

3 P

¦ 

¦ 

¡ 

k = 2 


 

v 1 

v 2 

v 3 

¢¤£¦¥ 

G 

1 

1 

3 3 

s 1 σ 1 

2 2 2 2 

3 3 

1 1 

Figure 53: 

5 # F5 (8 7E;

5.2. Hardness for W[1] 

 

1 

1 

3 3 

s 1 σ 1 

2 2 2 2 

3 3 

1 1 

¢¤£¦¥ 

G 

¢*$% "¥ ) ¡& ¥ 

1 

1 

3 3 

s 1 σ 1 

2 2 2 2 

3 3 

1 1 

¢ $%¨¡! $% "¥ ) ¥ 

Figure 54: 

?=554 5 §70 ¡

Theorem 5.9 G /IHKJ 3( W[1] ? 7ED 4 ? 7859(+ "5 B k 

¢ 

¢ 

¢ 

¦ 

¦ 

 


¢! 

P 

1 j 

¡ +S)&':3U- 0 $+ C A0 $ # '*)S+&5'XC 0 $ 7$ +&' 0 C F ' # E 'O$ 0 C # ' 0 +S+ 

v 1 j 

ϕ 1 j 

ϕ 1 l 

- 0 # 

1 l n 6 4+&5 l ¢= j ¦ " 0 $(13 3U-=) F +&5' 0 ?+ %-R

S@ )h$ 3 ' 

Q ¦ 

6 4+&5 1¥+q1$ 0E - 0F C 0 $ 7$ +&' 0 C F! 

0 # ' P ' 0 # ' 0 C*' ) $('*+ ' # E ' - 0 # +;5' ¨ 0 -=

7$C 3 P 0 # ' P ' 0 # ' 0 + £ ':+ §6 'qC A0 $ +,)&1C*+9+&5' 

C +;- $ 0 0 

G 2k(n + 1) 

0 +&'U+;5-/+ +;5'*)&' -/)&' 

+ 1 

G 

0 C $ 7$ +&' 0 C F 

kn(n$('*+ − 

' 

1) 

# E 'D$ !6 5':)&' 

2nk(k − 1) 

G 0 

4kn 2k 'D$ +&5' # A0 $ P

W[1] r 5-/) # 0 'D$&$ # 'O$ 0 + n-/+

¦ 

H A@ G 7$X-!D ( £

(G, s, t, k) @ ¢ I/£¢J ¦ ¢ 

N G (s) = {x 1 , . . . , x d } 6 ?+&5 d = # ' E G(s) ¦ 

Figure 55: 

54 5 (8 78; 3=&?=5 B (8 7 B 

G 

 

5.3. Chordless Cycles 

¢ 

8 ) 

l > 2 l − 2 ## ?+ 0 -=¦ 

P 

@ -CN+ ?+ P ) Y # 'D$¢- 0 -=

l ¥ 2 

x i 

- 0 # 

C +;5¥) 1 E 5 s - 0 # t 0 G P -$,$('D$ +&5¥) 1 E 5 

5' 5-/) # 0 'D$&$ P -=),+ 6 -$ $(5 /60 0 5' );':3 ¦7 £ ?+ )&':3U- 0 $o+ 

 

) Y!'f3g':3q2d':);$(5 P V6 5 7C.5 %$ # 0 'f2 F )&' # 1CN+ 0 + £¢¥¤ ¦ 

P 

(G, s, t, k, l) A@ ¡¡ I/£¢J 6 5'*);' G = (V, E) 7$ 

U -$ -o$ 0E

t 0 G %$S-R

Problem 5.17 (Directed Chordless Cycle) ¡ 7A D3785 B 5D & 78 ? G = (V, E)¦ 

(s, t) r P -=+&5$ ¡ a ¢n¦ 

r C:1¥+,+ 0E -/).C:$ # 'O$ 0 +hC;5©- 0E '`+&5' P -=);-=3g'*+&) %C C 3 P

σ 1 + s 1 - 0 # 

l ¥ 2 

s = t k = τ k - 0 # t = s 1 ¦ ¡>0 

σ i @ ) -=

¦ 

¦ 

¡ 

k = 2 


¢! 

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Bibliography

Curriculum Vitae 

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2 ) 0bA0 Q - FHasD 0 Z\'*)&

On the Existence of Paths and Cycles

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