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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On the Existence
of Paths and Cycles
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Contents
Abstract
v
Zusammenfassung
vii
Acknowledgments
ix
1
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Introduction
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Basics and Notation
19
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Chapter 1
Introduction
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Chapter 2. Basics and Notation
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%$k- 0 0 # ' P ' 0 # ' 0 +k$(':+ 0 G \ E(P) " 0
B ?=7ED 59(E( @ - 0 # 0 < F @
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@ - 0 # 0 < F @ G[V(P)] = P ' 0 C*'$(1C;5 P -=+&5$\-/);'-=<7$ m 0/60 -R$ 3:D B 5
D
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2 F ω(G) G %$ B 5 B 5
D @ ω(G) = 1 - 0 # D ( B 5 B 65
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G
- 0_F C:1+ r Y!'*),+;'*l %$ B 5 B 5ED M 43 %<%-=)&< F - 0 ' # E ' e E @ - C 00 'DC*+&' #
E );- P 5 G = (V, E) 7$- B 5
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?+ 7$ )&' @ '*),);' # + -R$ +&5' B 5 5 &7E ? Kn " E ).- P 5 6 5 $(' Y!'*),+;'*lH$('*+fC:- 0 2d' P -=),+ 4+ A0 ' # 0 + -/+[3 $ +9+ 6 n' '
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- E );- P 5 %$ 2 P -/),+ 4+&' @ - 0 # 0 < F @ 4+ # 'D$ 0 +kC 0 +;- 0 - 0F ## C F C*<%' 5' 2 P -=),+ 4+&' E ).- P 5$ 6 4+&5 - 3U-/l %3 13 0 13q2d'*) A@ ' # E 'D$ >1 0 1' 1 P
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Special
of Graphs
+;5-/+ 7$ Classes
V 7$kCD-=<%<4'
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Chapter 2. Basics and Notation
7$- +&)&':' - $ C:-R<4<%' #
(+"3=& 785E5
2.2 Directed Graphs
" 0 # 4)&'DC*+&' # E ).- P 5 ) @ )M$(5 )&+ D<& 78 ? D = (V, E) %$ # 'E 0 ' # 0 - $('*+
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V
$,'*+ E C A0 $ %$(+;$ A@^ ) # '*);' # P - 4);$ A@ Y!':),+ 7C*'D$ +&5-=+ 7$ E V 2 5'L'D<4'D3 ' 0 +.$
A@ E -/)&'g)&' @ '*)&)&' # + -R$7 B $ " 0 -/).C (u, v) E 7$q$,- # + 2^' # ?);'DCN+;' #
@ ) 3 ?+.$ ( 7 B 5 u + ?+.$ +7+&@59 v 8 ) (u, v) E 6 'k-R<%$ $,- F (+&5'*)&' 7$
- 0 -/);C @ ) 3 u + v 0 D W$(1-=<%< F!^6 ' C 0 $ # '*) F 785E5 E );- P 5$ +&5-/+
7$ -/);CD$ A@ +&5'f+ F P ' (v, v) V@K )$ 3g' v V -/)&' 0 +-=<%< /6 ' # 5' 3 D 5 &785E5 # ' E − (v) := |{(u, v)|(u, v) E}| A@ -Y!'*),+&':l v V 7$ +&5'
' D 5 &785E5 # ' E + (v) :=
0 13q2d':) A@ 
A@ -LY!'*)&+&'*l v V %$c+&5' 0 13q2d'*) A@ &'3=& -=D);C:$B-/+ v E}|
|{(v,- u)|(v,
u)
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$&-=3g' Y!':),+&':l $(':+ 2 F $ P 'OC @ F 0E - # 4)&'DC*+ 0 @ ) 'D-C;5 ' # E ' @ G -C.5
A@ +&5'D$(' 2|E(G)| # T '*);' 0 + # E ).- P 5$ %$oCD-=<%<4' # - 0 70 50+; A@ G 43 r
%<7-/)&< F '*Y!':) F # E );- P 5 D = (V, E) 5-R$ - 0 :D 507 $K3=& 1 0 # 4)&'DC*+&' # E ).- P 5
) (v, u) E}) 9 ' 0 C:'q3 $ + A@ +&5' +&':)&3 0 < r
G = (V, { {u, v} | (u, v) E
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D 4 4 0 - # E );- P 5 D %$ - $(' 1' 0 C:' W = (v , . . . , v ) @K )
$ 3g' k @ Y!':),+ 7C*'O$S$(1C.5b+&5-=+S+&5':)&' 7$- 0 -/);C @ ) 3 v 1 + v k 0 D D3785 B 5
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D B $ B 59( - 0 # D3785 B 5
D 6 7 ( 2.3 Geometry
Q $(+ A@ +&5' @ <%< =6 0E +.-/m!'D$ P <7-RC:' 0 +&5' 1C*< # 'O- 0 i <%- 0 ' 2 j 'U-/)&'
P -/),+ 7C*1<%-=)&< F 0 +&':)&'D$(+&' # 0 +&5' @K <%< /6 0E + F P 'O$ A@ $(12$('*+.$ A@ 2 " 0 ':<%':3g' 0 + A@ 2 %$L);' @ ':),)&' # + -$- < " 3 5 ` 7$f- 0 ' r # %3g' 0 $ 0 -=<- 0 'k$(12$ P -C*' A@ 2 X + %$o1 0 1 'D< F
# '*+&':)&3 0 ' # 2 F + 6 # 7$ + 0 CN+ P 0 +;$ p - 0 # q -R$
` = { p + λ(q − p) | λ } .
2.3. Geometry
S':3
Y-R< A@ - 0F $ 0E <4' < 0 ' ` # 7$,C A00 'DC*+;$ 2 0 + + 6 C 3 P 0 ' 0 +;$ 5'D$('
-=)&' +&5' "50 ? 4 50( # '8 0 ' # 2 F ` 5' 4) )&'D$ P 'OCN+ 4Y!' C:< $(1);' ) ' 1 ?Y r
-R<4' 0 +;< F! 1 0 0 6 ?+;5 ` @K );3g$S- B F( 5
D? 4 5 " < 0 ' ` 7$ # %$,C 00 'DC*+&' # 0 + + 6 C 3 P A0 ' 0 +;$ 2 F )&'D3 Y-=< A@ - 0F
$ 0E <%' P 0 + p ` 5' ?))&'D$ P 'OCN+ 4Y!'fC*< $(1)&' ) ' 1 4Y-=<%' 0 +&< F 1 0 0 6 ?+&5
@K );3g$M- 7E$ ':3U- 0 -=+ 0EU@ ) 3 p -R< 0E ` 8 )- 0_F P 0 + q ` \ {p} 6 '
p
-R<%$ # ' 0 +&'[+&5'D$('f).- F $L2 F
−
:= { p + λ(q − p) | λ - 0 # λ 0} .
pq
" 0_FH0A0r 'D3 P + F! C 3 P -RC*+[- 0 # C 00 'OCN+&' # $(12$(':+ @ -g< 0 ' 7$f- 3 5 ( 5 & <%'*+ V(s) # ' 0 +&'k+;5' $,'*+
50 ) ( 5+& 50 @ ) $(5 ),+ 8 ) - $(' E 3g' 0 +
s
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0 'DCN+ s \L +&'L+&5-=++;5 %$ # E' 0 4+ 0 -R<4< /6 $ $,' E 3g' 0 +;$ +&5-/+JC 0 $ 7$ + A@ - $ 0E <%'
$ D 5 &@58 507E;65 P 0 + 0 < F 6 'XCD-=<%<c$(1C;5H$,' E 3g' 0 +;A
8 )- 0F );- F - 0 # - 0_Fb00sr # ' E ' 0 ':);-/+&' $(' E 3g' 0 ++&5'*)&' %$- 1 0 1'X< 0 '
C 0 +;- 0 0E ?+ \
5 %$L< 0 ' 7$)&' @ ':),)&' # + -$L+&5' :D 507 %$K<& %< 5 A@ +&5'[);- F
)J$(' E 3 ' 0 + j 'M$,- F +&5-/+ + 6 $(':+;$ A, B 2 -/);' B # 3 5
7 @ - 0 # 0 < F @
+;5'*)&' '*l %$(+;$B-M< 0 ' +&5-=+ C 0 +;- 0 $ 2 +&5 " 0_F + 6 # %$(+ 0 CN+ P 0 +;$ p, q 2
# E' 0 'X-q1 0 1'o< 0 ' $(' E 3g' 0 +
−
= { p + λ(q − p) | λ [0, 1]}
qp
+;$ p - 0 # q h
5' D "3 A@ -b$(' E 3g' 0 + pq 6 5'*)&' p =
%$S+&5' P 0 + ( 1 (p + q ), 1 (p + q )) q = (qx , qy )
x
x 2
y
y
2
pq := −
pq
6 4+&5 ' 0 # P 0
-0#
(px , py )
8' 0 '\- + +;-=< ) # '*) 0 +&5'':<%':3g' 0 +;$ A@ d -R$ @ <4< /6 $ 8 ) + 6 d
'D<4'D3g' 0 +;$ p, q d d 2 6 ?+;5 p = (p , . . . , p ) - 0 # q = (q , . . . , q )
# E' 0 ' p q @ - 0 # 0 < F @ ' ?+&5'*) p 1 < q d )h2 +&5 p =1 q - 0 d#
1
1+;-/m!'H+&5' 1$(1
1 -=<
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8
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d 5'X$ +,) 7CN+9<%'*l 7C E );- P 5 %C ) # '*) 0 d %$ # 8' 0 ' # - 0 -R< E 1$(< F
8 )B+ 6 0 +;$
F
p = (px , py )
2
2
P
Distances
and
Orientations
# ' 0 +&'[+&5' 4) 1C*< # 'D- 0 D3(0+ B 5k2
||p − q|| :=
%3 4<7-/);< F!s@ )9-k< 0 ' $(' E 3
8 )9- P 0 + p - 0 # - P $
-0#
q = (qx , qy )
(px − qx ) + (py − qy ) .
' 0 + s = pq # '8 0 ' 4+;$ F58=&K? -$ ||s|| := ||p − q|| ?+ 4Y!'f)&'O-=< 0 13q2d'*) ε <%'*+
ε (p)
:= {q 2
| ||q − p|| < ε}
Chapter 2. Basics and Notation
# ' 0 +&' +;5'2 P ' 0 E4 A@ );- # 41$ ε -=) 1 0 # p %3 4<7-/);< F! # ' 0 +&' 2 F
+&5' B 37 B 5 @ );- # 41$ ε -/) 1 0 # p 2
|
||q
−
p||
=
ε}
(p)
:=
{q
ε
8 )q+;5)&':' P 0 +;$ p = (p , p ) q = (q , q ) - 0 # r = (r , r ) 6 ' $,- F
x
y
x y
+;5-/+ p %$L+ +&5' 5 +&5x'[);y- F −
<
#
@
0
A
0
F
@
qr
# '*+ (p, q, r) := %3 4<7-/);< F! p %$S+ +;5' 79 &?
- 0 # p q - 0 # r =- )&' B # %<
5E7
px
qx
rx
py
qy
ry
1
1
1
>0.
+&5' ;) - F −
@ - 0 # A0 < F @ # ':+ (p, q, r) < 0 qr
@ - 0 # 0 < F @ # '*+ (p, q, r) = 0 8 )J+;5)&':' P 0 +;$ p q - 0 # r 6 4+&5 p = q = r <%'*+ s := (q) 5' 5
K( 785ED =&# 5 ](p, q, r) @ ) 3 p + i r -/) 11 0 # q qi7$
<%' 0E +&5 A@ +&5'gC 4);C:1<%-=) -/);C @ ) 3 s + s -R< 0E (q) 0
1
p
r+
C 1 0 +;'*);C:< C&m 6 7$(' # 4)&'DC*+ A0c 8 ) s = s ) q {p,
(
$
:
'
r}
](p,
q,
r)
:=
0
p
r
>@ ](p, q, r) < π 6 ' C:-R<4< ](p, q, r) (870 B $ B 45 L>@ ](p, q, r) > π 6 '
$&- F +&5-/+ ](p, q, r) 7$ 705 5 @ ](p, q, r) = π +&5'[- 0E <%' ](p, q, r) 7$ ; ) -=+ %$)&' @ '*)&)&' # + -R$ B '5 " 0 - 0E <%'q+&5-=+ 7$[' 4+&5':)o$ +,) 7CN+&< F C A0 Y!'*l 5' 50C=& 7 D 3 (p, q, r) @ p - 0 # r -=) 1 0 # q 7$+&5'f$(':+
A@ -=<%< P 0 +.$ s @ ) 6 5 %C.5 0 < (])
" 0 -=< AE 1$(< F! +&5'
](p,
q,
s)
<
](p,
q,
r)
B (5
D & 7 D 3
A@ p - 0 # r -=) 1 0 # q 7$9+&5'q$('*+ A@
(p,
q,
r)
[]]
-R<4< P 0 +;$ s @ ) 6 5 7C.5 ](p, q, s) ](p, q, r) 2Y 1$(< F ?+ 7$ ](p, q, r) + ](r, q, p) = 2π @ - 0 # 0 < F @ p, q, r -/);'
P - 4) 6 %$,' # 7$ + 0 C*+ _ 3g'*+ %3g'D$ 6 ' -=)&' 1$ + 0 +&':)&'D$(+&' # 0 +&5'o3 0 %3 13 A@
+;5'D$('o+ 6 - 0E <%'D$ ':+L1$ # E' 0 'f+&5' =&# 5 70 5
D 9$ p - 0 # r -/) 1 0 # q
-$
3 0 {](p, q, r), ](r, q, p)} .
(p, q, r) :=
p <%'D-=)&< F (p, q, r) %$9-=< 6 - F $9C 0 Y!':l Angles
@K ) i {p, r} # 'E 0 ' # -$o+&5'
" ($ '*+ S 2 7$ B '5 @ - 0 # A0 < F @@K )'*Y!':) F + 6 P 0 +.$
< 0 'L$(' E 3g' 0 + pq %$BC 0 +.- 0 ' # 0 S B
5' B '5 A? sC A0 Y (P)
2 7$f+&5' $(3U-R<4<%'D$ + 6 4+&5 )&'O$ P 'DC*+ + $(':+ 0 C*<%1$ 0= C A0 Y!'*l
$,1 P '*);$(':+ @ P H 0 C*'g+&5' 0 +;'*);$('OCN+ 0 A@ - 0 -/),2 ?+,).-/) F @ -=3 4< F A@ C A0 Y!'*l
$,'*+;$ 7$ - E - 0 C 0 Y!'*l C 0 Y (P) 7$ 6 'D<4< # '8 0 ' # - 0 # ?+ 7$f+;5' $,-R3g' -R$o+&5'
0 +&'*);$,'DCN+ 0 A@ -=<%< C 0 Y!'*l`$,1 P '*);$(':+;$ A@ P @ P %$ 0 ?+;' +&5' 0 C 0 Y (P) 7$
- 0 0 +&':);$('DC*+ A0bA@ C:< $(' # 5-=< @ P <7- 0 'O$S+&5-/+9-/);' # 'E 0 ' # 2 F P 0 +;$ @ ) 3 P Convex Hull
+&5'
p, q- $(':S+
A@
P
2.4. Topology
2.4 Topology
" # 8& B 4 (I B 5 7$9- P - 4) (X, O) 6 5'*)&' X %$- $('*+- 0 #
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+&5' 1 0 A0bA@ - 0 -/),2 4+,);-=) F @ -=3 4< F A@ P ' 0 ($ ':+;$ 7$ P ' 0
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X
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( A@ (X, O) @ - 0 # 0 < F @ '*Y!':) FX P ' 0 $('*+ 7$ -1 0 0qA@
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+ P < AEAF +&5-=+ %$ E ' 0 ':);-/+;' # 2 F +&5'2-R$ %$ { (p) | p 2 - 0 # 0 < ε } + 4<%< +&5'\+;'*)&3 P 0 + %$ 1$,' # 6 ?+;5 0 +;5 %$ $('OεCN+ 0 + )&' @ '*) + +&5' 'D<4'D3 ' 0 +.$
A@ X -R$[- 0 ':<%':3g' 0 + A@ - + P < E %CD-=< $ P -RC*' ).-/+&5':)[+&5- 0 -R$[- 0 'D<4'D3 ' 0 + A@
2
" $('*+ N X 7$o- 50<&? 87 ?= D A@ x X @ - 0 # 0 < F @ +&5':)&' 7$o- 0
P ' 0 $('*+ A $,1C.5 +&5-=+ x A N >0 +&5 %$oCD-R$(' x 7$ CD-=<%<4' # - 0 < 5070 7
P 0 + A@ N 5'[$('*+ A@ 0 +&':) ) P 0 +.$ A@ N 7$ # ' 0 +&' # 2 F N 43 %<%-=)&< F!
@K )q- 0 5 658797 P 0 + @ N +&5'UC 3 P <%':3g' 0 + X \ N 7$ - 0 ' E 5_2 )&5 # - 0 # +&5'f$('*+ A@ ':l+;'*) ) P 0 +;$ A@ N 7$ # ' 0 +&' # 2 F '*lV+ (N) 5' P 0 +;$ A@
+&5-/+ -=)&' 0 ' 4+&5':) 0 +&':) ) 0 ) '*lV+&':) ) + N C 0 $ + ?+&1+&'9+;5' 78507 )
X
6 "#FE& B 4 8 :D@79$ ∂N
" $,'*+ A X %$ B ( 5ED @ - 0 # 0 < F @ ?+;$C 3 P <%':3g' 0 + X \ A 7$ P ' 0
8 ) B X +&5' B ( 785 A@ B 7$ # ' 0 +&' # 2 F B := B ∂B p <4'O-/)&< F 2 +;5
- 0 # ∂B -/)&'fC*< $(' # " $ - 0 # X -/);'C 3 P <%':3g' 0 +;-=) F - 0 # +&5' F -=)&'2 +&5
B '
P 0c +&5' F -=)&'f-=<7$ 2 +&5 C:< $(' # \
5'[$ P -C*' X 7$$&- # + 2^' B 5 B 5
D @
- 0 # 0 < F @ - 0 # X -=)&'q+&5' 0 < F $(12$,'*+;$ A@ X +;5-/+ -=)&'q2 +&5 P ' 0 - 0 #
C:< $(' # +&5'*) 6 7$(' X 7$AD3( B 5 B 5
D " $(12$,'*+ U A@ - + P < E %CD-=<$ P -RC:' X 7$ B " B @ - 0 # 0 < F @ ':Y!'*) F
P ' 0 C Y!'*) A@ U C 0 +;- 0 $q- 0 4+&' $(12 C Y!'*) Q )&' P )&'OC 7$(':< F @K ) ':Y!'*) F
@ -=3 4< F (O ) A@ P ' 0 $(12$('*+.$ A@ X @K ) 6 5 %C.5 U +&5'*)&'M'*l %$ +.$
O
I
i
i
i
i
I
- 0 4+&'9$,12$(':+ J I $(1C.5 +;5-/+ U " $(12$('*+ @ n %$ C 3 P -RC*+
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i
@ - 0 # 0 < F @ 4+ 7$[C*< $(' # - 0 # 2 1 0 # ' i# Jf '*)&' 8 :D 5
D`3g'D- 0 $M+&5-/++&5'
$,'*+ 7$ C 0 +.- 0 ' # 0 - 2-=<%< @ );- # 41$ k -=) 1 0 # p c@K ) $ 3g' p n - 0 #
$ 3g' k g
-C.5 $(12$(':+ A X + E ':+&5':) 6 4+&5 O := {B A | B O} - E - 0
@K );3g$X- + P < E %CD-=< $ P -C*' h
5' $('*+ O %$XA)&' @ '*)&)&' # + -$ +;5' ( (+" B 5
A
a
Chapter 2. Basics and Notation
A @ A 0 (X, O) L +;' +&5-=+k+&5'h$,'*+;$ 0 O 0 P -/),+ 7C*1<%-=) A A
- 0 # (Y, B) +&5'
0 ':' # 0 + 2d' P ' 0 0 X 8 )J+ 6 + P < E 7C:-R<d$ P -RC:'D$ (X,
O)
P ) # 1CN+ X Y + E '*+;5'*) 6 ?+;5g+&5'$ r C:-R<4<%' # P ) # 1C*+\+ P < EF - E - 0g@ )&3U$
- + P < E %CD-=< $ P -RC*' 5' =78 D B A # 8& $ A@ (X, O) - 0 # (Y, B) %$ +&5'
+ P < AEAF E ' 0 '*);-=+&' # 2 F +&5'f2-R$ %$ {o b | o O - 0 # b B} " 3U- P f : X → Y 2d'*+ 6 'D' 0 + P < AE 7C:-R<$ P -RC*'O$ (X, O) - 0 # (Y, B) 7$
@ )X-R<4< b B " I; ? 2d'*+ 6 'D' 0
B 3 ( @ - 0 # 0 < F @ −1
f
(b)
O
- 0 # b 0 X 7$ - C A0 + 0 1 1$X3U- P γ : [0, 1] → X 6 4+&5 γ(0) = a - 0 #
a
@ a = b +&5' 0 γ 7$ - B (5
D ";? 5'h$ P -RC:' X %$ C:-R<4<%' #
γ(1)
=
b
";? B 5 B 5
D @ - 0 # 0 < F @ @ )o'*Y!':) F + 6 P 0 +.$
+&5':)&' 7$ a, b X
P -/+&5 2^':+ 6 ':' 0 a - 0 # b " P -/+&5 r C A00 'DC*+&' # $ P -C*' %$ -=< 6 - F $ C 00 'DC*+&' #
21+ 0 +J':Y!'*) F C 00 'OCN+&' # $ P -C*' 7$ P -=+&5 r C 00 'OCN+&' # Z1+J2 +&5 0 + 0 $J-/)&'
' 1 ?Y-R<4' 0 + @ ) P ' 0 $(12$('*+.$ A@ d 8 ) X = 2 6 ' 6 4<%< A@ +&' 0 )&' @ '*)9+ P -/+&5b-R$9- B
7 45 0b ) # ':)L+ C:<4'O-/)&< F # 7$ + 0E 1 7$(5 P -=+&5$ 0 +&5'[+ P < E %CD-=<
$,' 0 $(' @ ) 3 P -/+&5$ 0 +&5' E ).- P 5 r +&5' )&'*+ %Co$,' 0 $(' 6 C A0 + 0 1 1$B3U- P $ f, g : X → Y 2^':+ 6 ':' 0 + P < E 7C:-=<$ P -RC:'D$ X - 0 #
-/);' ?= K6 B @ - 0 # 0 < F @ +&5'*);' 7$-C 0 + 0 1 1$ 3U- P h : X [0, 1] →
Y $(1C.5 +;5-/+
- 0 # h(x, 1) = g(x) @ ) -=<%< x X 5'
h(x,
0)
=
f(x)
Y
3U- P h 7$ )&' @ '*),)&' # + -$U- ?= K6 =$ 2d':+ 6 ':' 0 f - 0 # g " 3U- P +&5-/+
7$q5 3 + P 7C + -`C 0 $(+;- 0 + 3U- P 7$kC:-R<4<%' # ?= K6 = B " $ P -RC*' %$
(E<
$ B 5 B 5ED @ - 0 # 0 < F @ 4+ 7$ P -/+&5 r C 00 'DC*+&' # - 0 # '*Y!':) F C:< $(' #
P -/+&5 0 ?+ 7$ 0 1<%<45 3 + P 7C 6 "#FE&$
2.5 Geometric Straight Line Graphs
n0 - &@5E 5 79 B & 7E ? +;5'Y!'*),+ 7C*'O$L-/)&' P 0 +;$ - 0 # ' # E 'O$-=)&' -$,$ C %-=+&' #
+ C*1)&Y!'D$UC 00 'DCN+ 0E +&5' 4) ' 0 # P 0 +;$ j ' -=)&' P -=),+ 7C*1<7-/)&< F 0 +&'*)&'O$ +&' #
0 +;5'`CD-R$(' 6 5':)&' -=<%< @ +&5'O$('`C:1),Y!'O$ -/);'`< 0 '`$,' E 3g' 0 +;$ Z1+ 2d' @ )&'
6 ' CD- 0 # E' 0 'k+&5 7$oC*<7-R$&$ A@JE );- P 5$ P )&'OC 7$(':< F!c6 ' 0 ':' # - 3 )&' 'D<%-=2 ).-/+&'
+;'*)&3 0 < EAF + # 'D$&CN) 42d'k+&5' 0 +&':);-RC*+ 0 2d':+ 6 ':' 0 + 6 < 0 'g$(' E 3g' 0 +;$ ) 3 )&' E ' 0 '*);-R<4< F + 6 $,12$(':+;$ A@ +&5' 1C:< # 'D- 0 P <7- 0 ' " $J- );$(+ $ +&' P <4':+J1$ # E' 0 '- 0 + 0 A@ 0 +;'*) ) P 0 +;$ +&5-/+ 7$ 3g'D- 0sr
0EA@ 1< @ ) 00sr>@ 1<%< # 43g' 0 $ 0 -R<$,12$(':+;$ A@ +&5' P <%- 0 ' $(1C.5 -R$ < 0 '`$(' Er
3g' 0 +;$ Definition
2.1 (
3(
p P
7 ( 5 B ?> ? ; P
2
?=5
relative interior P ( ?=5 ( 5 4 A
2.5. Geometric Straight Line Graphs
3 ( 35070 7 p
or
(E<
7
P
5'5879$
?=587855 3(0(
ε
>
0
δ
$ B 5 B 6 5
D (
ε
B ?
? ;
δ (p)
\P
( K
P =- ),+ 7C*1<7-/) @ )- 00sr # ' E ' 0 '*);-=+&' < 0 'S$(' E 3g' 0 + s ?+ %$ s
= s\V(s) @ s
# ' E ' 0 ':);-/+&' 6 ' 5-DY!' s
= s = V(s) 0 $ +&'D- # 5' @ %< < / 6 0E # '8 0 4+ 0
P ) Y # 'O$ + 6 )&'8 0 ' # 0 + 0 $ @ ) 0 +;'*);$('OCN+ 0
n 0
7$
4 >( 5 (
;5'5879$
2 705 ( D Definition 2.2 ?=5 ( 5 ( P, Q
overlap
:D
; 3
c P7>
Q
P
Q
cross
c
P Q
5'5879$
?=50785 5 (8 D 5 &@50 5878;
5 3 5 ( 5+& 50(
ε > 0:D
4 ?
( B ? ? ; V(s)
P
t
V(t)
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s
ε (c)
ε (c)
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P=
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" :C C ) # 0E + +&5 7$ # '8 0 4+ A0 - 0F + 6 $,'*+;$CN) $,$-/+'*Y!':) F C 3g3 0 0 +;'*) )
P 0 +21+ 0 + 0 'DC*'O$,$,-=) 4< F -/+f'*Y!':) F P 0 +M+&5-/+ %$ 0 +&5' )&':<7-/+ ?Y!' 0 +;'*) )
A@ 2 +&5 \S/6 '*Y!':) + 6 00sr # ' E ' 0 '*).-/+&'f< 0 ' $(' E 3g' 0 +;$ Y!':)&<%- P -/+- P 0 +
@ - 0 # 0 < F @ +;5' F CN) $,$S-=+ c c
/6 6 'b5-DY!'b-=<%<S+ <7$ + E '*+&5':)k+ # 'E 0 ' +;5'bC*<7-R$,$('O$ A@E ' 3g':+,) 7C
E );- P 5$+&5-/+ 6 ' 6 4<%<c2^'XC 0 C*':) 0 ' # 6 ?+&5 4 ?
& 7E ?
2 3( Definition 2.3
G
=
(V,
E)
V
geometric straight
. A D> $ ?=5( 5+& 50(
D E}
line
graph
S
:=
{uv
|
{u,
v}
K B < $ "3 78
K?=507 ? ?=583758:D 3( K ?=587
V
4 7EDK(
7 5 '5070$ ( 5 & 58 (
785E '507
uv S
V uv = {u, v}
4 (5 & 50( <
( B 4 F5
D
B 78(E( ?=58
S
G
planar straight line graph
& 7E ? (
:D $ ( 3( 7?' B . .
planar
2Y 1$,< F! - 0F $(12 E );- P 5 A@ -
%$[- . . -R$ 6 ':<%< 0 C:' 0 - . 2 +&5 - 0 ' # E 'X- 0 # - < 0 ' $(' E 3g' 0 +9-/)&' 1 0 1':< F # 'E 0 ' #
2 F + 6 P 0 +;$ 6 ' A@ +&' 0 # ' 0 + @ F ' # E 'O$o- 0 # < 0 ' $(' E 3 ' 0 .+ $ >0 +&5 7$ 6 - F
- 0_F . G = (V, E) 0 # 1C*'O$q-b$('*+ ∂G :=
A@ P 0 +.$ 0 +&5'
p
P <%- 0 ' L':3 YA-=< A@ ∂G # 4Y # 'O$ +;5' P <%- 0 ' 0 + - p0 V4+& ' E0 13 2^':) A@ 3U-=l %3U-=<
C 00 'DC*+&' # P ' 0 $(12$,'*+;$+&5-/+S-/)&' CD-=<%<4' # +&5' B 59( @ G \
5'f$(':+ @c@ -C*'D$
A@ G %$ # ' 0 +&' # 2 F (G) 507 ( 79 b$ +;-/+;'D$L+&5-/+
.
.
|V| − |E| + | (G) | − ω(G) = 1 .
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"& " # $%1$% $') #&'& ,&'$% 1 ! Chapter 2. Basics and Notation
" $ 6 'C 0 $ # ':) 0 4+&' . $ 0 < F 4+ 7$ C*<%'D-=)J+&5-/+ '*l-RC*+&< F 0 ' A@ +;5' @ -C*'D$
7$S1 0 2 1 0 # ' # + 7$$ 3g'*+ %3g'D$L);' @ ':),)&' # + -$L+&5'< 65 9 B 5 A@ G >0
- C 00 'DCN+;' # E );- P 5 ':Y!'*) F 0 4+&' @ -RC:' F (G) 7$S2 1 0 # ' # 2 F +;5'X' # E 'D$
A@ -oC:< $(' # +,);- %< 0 G \ 1C;5U-[+&);- %< 6 5 %C.5 7$+&5'L2 1 0 # -=) FkA@ - @ -RC:' @ ) 3
%$LCD-=<%<4' # - B 4 7E @
5' 0 0 4+&' @ -RC:' %$ 2 1 0 # ' # 2 F +&5'
(G)
G
' # E 'D$ @ -`C:< $(' # 6 -=<4m 0 G 6 5 %C.5 %$ -H+&);- %<L'*lC*' P + @ ) +&5' C:1+ r ' # E 'O$
6 5 %C.5 - PP 'D-=) + 6 7C*'h'D-C;5 5' ' # E 'O$ 2 1 0 # 0E - @ -C*'b-/);'H-=<7$ C:-R<4<%' #
3 B D 50+ +&5' @ -C*' " P -/+&5 P 0 - . G 7$(
< F5 @ - 0 # 0 < F @ (V(P), E(P)) %$gC*) $,$ 43 %<%-=)&< F 6 ' # 8' 0 ' $ %3 P <4'
. +&5-/+ %$ M0 + 6 ' # E 'O$ A@
P
C F C:<4'O$ 2VY 1$(< F ':Y!'*) F P -/+&5 )9C F C*<%' 0 - . 7$$ %3 P <4' j ' 6 %<%< 0/6 ':l+;' 0 # +&5' 0 + A0 A@ C*) $,$ 0E $ @ ) 3 $,12$(':+;$ A@ +&5' P <7- 0 '
+ 6 -R<?m$ 0 . $ " +,);- 4< T = (p , . . . , p ) 0 - . 0 # 1C:'D$- P 4'OC*' 6 %$(' < 0 'O-/)MC:1),Y!'
@ ) 3 p + p1 0 +&5n' P <%- 0 ' j 5 4<%'H+;5' ' # E 'D$ A@ T # 0 + CN) $&$
γT
'O-RC.5 +&5'*)o12 F # E' n0 4+ 0 +&5'gC*1),Y!' γ %$ 0 +X$ %3 P <4' +&5-=+ %$ 0 'DC*+ 4Y!'
0 E ' 0 '*);-R< -R$ +&5' $&-=3g'kY!':),+&':l`3U- F - PTP 'D-=) $(':Y!'*);-R<+ 43g'O$ -=< 0E T Z1+
5'*)&' 1) 0 +;'*)&'O$ + 7$ @ C*1$,' # -=+C*) $,$ 0E $ );-=+&5':)J+&5- 0 -=+$ %3 P < 7C 4+ F 8 )6 -=<4m W 0 - . <%'*+ ∂W := ∂(V(W), E(W)) 2 4 4 4 (
:D
3 .
B 78(E( ; "3
Definition 2.4
U
W
p :D> %
$ ?=85 705 5 3(8 ( 4 4 (
:D
∂U
∂W
U
U 78(E( ;W
W
7 4 ?' B ?
:D ( B ?>? ;
:D
B
|U |, |W | 3
∂U
∂W
p
j ' C 1< # )&' 1 4)&'\+&5' $(12 6 -=<4ms$ 0 'E 0 ?+ 0k_ + 2d' 00sr +,) 4Y %-R<2d'DC:-R1$('
@K )- +&) ?Y 7-=< 6 -=<4m U +&5' $(':+ ∂U C A0 $ %$(+;$ A@ - $ 0E <4' P 0 + " $+&5':)&' 0
6 - F + 21 4< # - 00sr # ' E ' 0 ':);-/+&'< 0 'L$(' E 3g' 0 + @ ) 3 -[$ 0E <%' P 0 + $,'*+B+;5'*)&'
7$ 0k6 - F +;5-/+9- 0_F $('*+ 0 +&5' P <%- 0 'XCD- 0 CN) $&$- $ 0E <4' P 0 + >0 - . + 6 6 -=<4ms$J+&5-=+ # q0 +L$(5-=)&'o- 0 ' # E '[C:- 0 C*) $,$ -=+ Y!':),+ 7C*'O$
0 < F! '*+ 1$ E ?Y!' - 0 -=<4+&':) 0 -/+ ?Y!' C.5-=);-RC*+&'*) %eD-=+ A0 @ )b$(1C.5 CN) $,$ 0E $ 6 ' $,- F +;5-/+ + 6 6 -=<4m$ U - 0 # W 36587 5
'5`-/+k-hY!':),+&'*l p A@ - .
@ - 0 # A0 < F @ U C 0 +;- 0 $S- $(12 6 -=<4m (q, p, r) - 0 # W C A0 +;- 0 $
G = (V, E)
-g$(12 6 -=<4m (s, p, t) $(1C;5 +&5-=+ p q r s - 0 # t -/);' P - 4) 6 7$(' # 7$ + 0 CN+f- 0 #
%$U$ +&) %C*+&< F 0 2d'*+ 6 'D' 0
- 0 # ](q, p, t) " 0 13q2d'*)
](q, p, r)7$ (8 79 %$ < 85 4 5E50 + ](q, p,
s)
B
6 0 13q2d'*);$ a, c @ - 0 # 0 < F @
b ) c < b < a a<b<c
4 4 4 (<
Proposition
2.5
5ED&'57 ?=58$<6587 5E45 .
B 70(E( D $ ?=50$ (E? 705
R
2.5. Geometric Straight Line Graphs
pJ0 $ # '*)M+ 6 6 -=<4ms$ U - 0 # W 0 - . G = (V, E) +&5-=+
P 0 + c ∂U ∂W q @ c / V +;5' 0 2 F # 'E 0 4+ 0 @ . c
< %'D$ 0 - 1 0 1'[' # E ' e E 5' 0 < FU6 - F c C:- 0 2d' 0 ∂U ∂W 7$ +;5-/+
- 'D-=);$ 0 2 +&5 U - 0 # W +&5'*) 6 %$(' 6 'f3U- F -$,$(13g'[+&5-=+ U - 0 # W
e PP
C*) $,$J-=+ Y!':),+ 7C*'O$ 0 < F 0 P -=),+ 7C*1<7-/) c V pJ0 $ # '*) +&5'M2-=<%< (c) 6 4+&5
δ ' $(1)&'O$
).- # %1$ δ := 1 3 0
2
(
$
*
'
,
)
!
Y
M
'
&
+
5
=
J
+
&
+
5
'
.
C
5
7C
*
'
A
@
0
||p
−
c||
δ
p
∂(G\c)
2
+;5-/+ V
- 0 # +&5-/+[-=<%< ' # E 'O$ @ G 6 5 7C.5 0 +;'*);$('OCN+ (c) -=)&'
δ (c) = {c}
δ
0 C # ' 0 +S+ c Z F # E' 0 4+ 0 +;5'*)&' -/)&' $(12 6 -=<4m$ U A@ U - 0 # W A@ W 6 4+&5 2 $(1C.5 +&5-/+ ∂U - 0 # ∂W CN) $,$ -=+ c " $ U - 0 # W CN) $,$
|U |, |W | 3
-=+fY!'*),+ %C:'D$ 0 < F U - 0 # W 3 1$ +f2 +;5 2d' A@ $ %e:'k+&5)&'D' k 0 C*' U - 0 #
C*) $,$ -/+ c +&5'*)&' %$ - 00sr # ' E ' 0 '*);-=+&' $,' E 3g' 0 + s = uv 6 ?+;5 V(s) W
$(1C.5 +&5-/+ s
∂W = s @ +&5'*);' 7$ - 0 ' # E ' e E $(1C;5
∂U
(c)
δ
+;5-/+ s
e
+;5' 0 2 F # E' 0 4+ 0 @ δ +&5' P -/+;5 U Y 7$ 4+;$ e Z\1+ 0 +&5'
+&5'*)95- 0 # 6 'X5-DY!' e
∂W = s C 0 +,).-/) F + 1)M-R$,$,13 P + A0 +&5-/+ U
- 0 # W CN) $&$f-/+fY!'*)&+ %C:'D$ 0 < F ' 0 C*' +&5':)&' 7$ 0 ' # E ' A@ G C 0 +;- 0 0E
Z F +&5'fC.5 7C*' A@ δ +&5 %$ 43 P < %'D$J+&5-/+ u v - 0 # c -=)&' P - ?) 6 7$(' # 7$ + 0 C*+ s81),+&5':)&3 )&'q+&5':)&'k3k
1$ +o2d' + 6 # 7$ + 0 C*+ ' # E 'O$ 0 U +;5-/+X-=)&' 0 C # ' 0 +
+ c 0 +&5':) 6 ) # $ U = (p, c, q) - 0 # - 0 -R< E 1$(< F W = (x, c, y) @
- 0 # -=)&'MC <%< 0 'O-/) +;5' 0 ∂U pq %3 P < %'D$+;5-/+ x - 0 # y 3 1$ + < 4' A0
c p'*)&' +fq$ 'D$
# T 0 # A@ +;5' < 0 ' +&5) 1 E 5 p - 0 # q +&5':) 6 %$,' 6 ' 3U- F -$,$(13g'
6 4+&5 1+ < $,$ A@Ê ' 0 '*);-R< 4+ F +&5-=+ q 7$+ +&5'S<4' @ + A@ −
u cp - 0 # v cq cp
5' 0 s
(p, c, q) 43 P < 4'O$L+&5-=+ (p, c, q) ∂W
'b- 0 ' # E ' f E(W δ) (c)
s ' 0 C:' 2 F +&5'h(])C.5 %C:' A@ δ +&5'*);' 3 1$ + 2d(])
6 ?+&= 5
j 4+&5 1+g< $,$ A@E ' 0 '*);-R< 4+ F <%'*+ f = cx pJ0 $ # ':)
f
(]) (p, c, q)
+ 6 CD-R$('O$ q @ y 7$ <%' @ + A@ −
8 E 1)&' n- +&5' 0 2 F +&5' $,-R3g'g-=) E 13g' 0 +
cx
-$q-/2 Y!' 4+ %$ cq
(x, c, y) +&5'*) 6 7$(' 8 E 1);' >2 . 6 ' 5-DY!'
>0 (])' ?+;5'*)gC:-R$,' U - 0 # W - 0 # +&51$ U - 0 # W
(y,
c,
x)
cp+&
'*)&<%'D-DY!' (])
-=+ c 0
Proof.
C*) $,$o-=+X- ⇒
@
c
u
p
x
y
c
y
u
cx v
p
q
Figure 8:
y
v
$% &% q
x
y
$% $% cx 87 (E(E<=& 4 4
( 36587 5 '5 D &@59( 5 4 4 785 E( ?= 4 9$( #% D %< 59( 5
D&@59( ?=5 K ?=85 7 4 4 9$ D K 5
D 3 59( Chapter 2. Basics and Notation
SpJ0 $ # ':)9+ 6 6 -R<?m$ U - 0 # W 0 - . G = (V, E) 9p <4'O-/)&< F!
@ U 0 # W $(5-=)&'o- 0 ' # E ' e E +;5' F C*) $,$'D-RC.5 +&5':)L-=+L':Y!'*) F P 0 + @
\ ' 0 C*'[-R$,$,13g'M+;5-/+ U - 0 # W 0 +&':)&<%'D-DY!' - 0 # <%'*+ U := (q, p, r) 2d'fe$,1
2 -=<4m
6
A@ U - 0 # W := (s, p, t) 2d'M-X$(12 6 -=<4m @ W $(1C.5U+&5-=+ p q r s - 0 # t -=)&' P - 4) 6 7$(' # 7$ + 0 CN+\- 0 # ](q, p, r) %$$ +,) 7CN+;< F 0 2d'*+ 6 'D' 0 ](q, p, s)
- 0 # ](q, p, t) BpJ0 $ # ':)B+&5' 2-=<%< (p) @ )\$ 3g'L-/),2 ?+,).-/) F ε > 0 - 0 # +;5'
ε
C ?).C*<%' (p)
5
:
'
&
)
'
6
0 {ε, 3 0
':+ f := q r
1 3
δ := 2
δ
ε (p)
p V ||p − c||} - 0 # g := s t 6 5':)&' i := (p) @ ) i {q, r, s, t} p <4'O-/)&< F V(f) pi
δ
- 0 # V(g) ∂W
\j ' $(5 /6 +;5-/+ f
∂W = V
∂U
(p)
(p)
ε
ε
>@ p q - 0 # r -/)&'qC <%< 0 'O-/)[+&5' 0 p f
∂W @ ](p, q, r) 7$M$ +&) %C*+&< F
C 0 Y!':l +&5' 0 = f
−
+&' +&5-/+ −
ps
ps
∂W
∂W
δ (p)
8 0 -R<4< F! @ ](p, q, r) %$\);' ':l +&5' 0 = f
−
" 0 -=< EA 1$,< Fk0 '
∂W
pt
CD- 0 C A0 C*<%1 # '+;5-/+ g
∂U = sB
5V1$ U - 0 # W - 0 # +&5'*)&' @ )&' U - 0 #
CN) $,$ W
⇐
-
@
" 6 -=<4m W 0 - . 7$(5 B 78(E(
3=& @ - 0 # 0 < F @ ?+oC A0 +;- 0 $M+ 6 C*) ,$ $ 0E $(12 r>6 -=<4ms$ +&5-/+ 7$ W = (. . . , p , . . . , p , . . . , p , . . . , p , . . .) 6 4+&5 1 i < j k < ` n - 0 # +;5' 6 -R<?m$ i(p , . . . j, p ) - 0 k# (p , . .` . , p )
C*) $,$ Jj 5' 0 '*Y!'*) 6 ' 6 - 0 ++ 'D3 P 5-R$ %e:' +&5-/+Mi- 6 -=<4m j%$ 0 +M$('D< k@ r C*) $,$ 0È
6 'M':l P )&'D$,$J+&5 7$ 2 F )&' @ ':),) 0E + 4+-R$ B 78(E(E<=& 2$,'*),Y!'+&5-=+ - 6 -R<?m
+;5-/+ 7$ 0 +9- +,);- 4< 7$S-=< 6 - F $9$(':< @ r CN) $,$ 0E
2.6 Visibility Graphs
8 ) - 0F + 6 P 0 +;$ p - 0 # q +&5'\$(5 ),+;'D$ + C*1),Y!'2d'*+ 6 'D' 0 p - 0 # q -C:C ) # 0E
+ +&5' 1C*< # 'O- 0 3 ':+,) 7C 7$ # 'D$,C*) 42^' # 2 F +&5'k< 0 ' $,' E 3g' 0 + pq Z\1+f+&5'
$ ?+&1-/+ 0 C.5- 0E 'O$X$,< E 5_+&< F @6 ' 0 +,) # 1C:' - 0 2$(+;-RC:<4' )&' E 0 O 2 $&- F! -g< 0 ' $(' E 3g' 0 +[- 0 # -R$ m @ )[-U$(5 )&+&'D$ +fC*1),Y!' 2^':+ 6 ':' 0 p - 0 # q +&5-/+
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2.6
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−
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ε (γ(x))
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2.9
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α
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k
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2.10
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(
( =? 5 5EG
D&'59(
G
T
05 7 4 =&
? ; 3( 5
B ? 5 D&@5 3( 79505ED
C
C
3( 7E
'587 ( 5
D 9$
?=50 ?=5 07 50( 3( D 3785 B 5ED
T
59
Proposition 2.11
:D 5 (V,
E)
C
3 ?=50<7 7ED
C
3 ?=5 D<705 B B 70(E(
3=& 7E
T
!
2.7. Polygons
' 0 +&'
- 0 # <%'*+
2^'9+&5' ) %' 0 +;-=+ 0UA@
0 # 1C*' #
Proof.
2 F T -R$ # 'O$,CN) ?2d' H# :=
-=2 G[V(C)]
Y!':),+&':l v A@ H - 0 # +&5'fC F HC*<7C ) # '*)
Y!' pJ0 $ # '*)- D
A@ +;5'JY!':),+ 7C*'O$ 0 N (v) -/) 1 0 # v ':+ u - 0 # x 2^' + 6 Y!'*),+ %C:'D$ A@ H $(1C.5
+;5-/+ u v - 0 # x -=)&'fG- # (-RC:' 0 + -=< 0E C - 0 # 5' 0 C*' 0 +&5 7$ ) # '*) -=) 1 0 # v 1 PP $,'X+;5-/+M2 +&5 ' # E 'D$ (u, v) - 0 # (x, v) 0 D -/)&' ) %' 0 +&' # + /6 -/) # $ v '*+ w N (v) 2d'q+&5' Y!':),+&':lbY 7$ 4+&' # 2 F T # 4)&'DC*+&< F - @ +&':) (u, v) - 0 # <%'*+
G
2d'9+&5'Y!'*)&+&'*l Y 7$ 4+&' # 2 F T # 4)&'OCN+&< F - @ +&':) (x, v) 2VY 1$(< F u y NG (v)
- 0 # y -/)&' P - 4) 6 7$(' # %$ + 0 C*+ v w x
" $ T 7$ 00sr C*) $,$ 0E 2 F i ) P $ ?+ 0b u - 0 # w -$ 6 ':<%< -R$ x - 0 # y
=- )&' 0 ' E 5_2 );$ 0 +&5'qC F C:< 7C ) # '*)[-/) 1 0 # v 51$+&5'qC 1 0 +&':);C*< C;m 6 %$('
) # '*) -=) 1 0 # v 7$ 6 4+&5 1+k< $,$ A@9E ' 0 '*);-R< 4+ F u, w, y, x -$ # ' P %C*+&' # 0
8 E 1)&' sXL/6 );'DC:-R<4<B+&5-=+ T = (. . . , u, v, w, . . . , x, v, y, . . .) f>0 P -=),+ 7C r
u
Q
x
v
w
y
P
Figure 13:
?=50785 B 85 '587 5 v
4 ?
#'E
−
D (v)
=2
1<%-=) T C 0 +.- 0 $q-HC*< $(' # $,12+,);- 4< P @ ) 3 v Y 7- w + x - 0 # 2-C;m + v %3 4<7-/);< F T C A0 +;- 0 $k- C*< $(' # $(12+&);- %< Q @ ) 3 v Y 7- y + u - 0 # 2-RC;m
+ v Q );' Y!':) P - 0 # Q @ )&3 - P -=),+ 4+ 0 @ T 0 P -/),+ 7C*1<%-/) +&5' F -=)&'
' # E ' r # 7$3 0 + " <%$ 0 +&' +&5-=+ P - 0 # Q CN) $,$ -/+ v - 0 # +&5-/+ '*lC*' P + @ )
A0 P - 0 # (u, v, y) 0 Q -=<%< +&5'*)f$ %e:' +&5)&':'q$(12+,);- %<7$ A@ P - 0 #
(x, -=v,<7$ w)'O-/) 0 ' 0 C*' - 0_F CN) $,$ 0EÀ@ P - 0 # Q +&5':)q+&5- 0 -=+ v
Q 1< -=<7$ PP 2d'q-gCN) T $,$
6 # 0E @ T 0 C A0 +,);- # 7CN+ 0 + U 1)[-R$,$,13 P + A0 +&5-=+
%$ 00sr CN) $,$ 0E j ' 6 %<4< $(5 =6 $(1C.5 - CN) $,$ 0E ':l 7$ +;$o- 0 # +;51$+;5'*)&'
TCD- 2d'
0
0 Y!':),+&'*l v 6 4+&5 # ' E − (v) = 2 8 ?).$ + 6 'q3U- F -R$,$(13 ' +;5-/D+ # ' E (v) = 2 -R$ T 7$ 0A0r C*) $,$ 0E +&5':)&'
7$ 0 $,12+,);- 4< A@ $ 4eD'b+&5)&'D' 0 P +&P5-=+ 0 +&'*);<4'O-:Y!'O$ 6 ?+&5 0 ' A@ (x, v, y)
) (u, v, w) Q )&' Y!'*) u v - 0 # x -/)&' - # (-RC:' 0 + -=< A0E - @ -RC %-R<oC F C*<%'
- 0 # 5' 0 C*'U+&5'*)&' 7$ 0 ' # E ' 0 C # ' 0 +q+ v 0 (x, v, u) pJ 3q2 0 0E
+;5'D$('X$ +;-/+;':3g' 0 +;$ 6 'XC 0 C*<%1 # 'o+&5-/+ 0 $(12+,);- 4< (])
@ P +&5'*)L+&5- 0 (x, v, w)
0 +&'*)&<%'D-DY!'O$ 6 ?+;5 (u, v, y) pJ0 $ # '*) +&5' $(12+,).- %< P A@ P +&5-=+k' 0 # $k-=+
+;5' );$ +JY 7$ ?+ A@ v 2 F P 6 4+&5 0 (y, v, u) 1C.5 -X1Y %$ ?+L'*l %$(+;$ 2d'OC:-=1$('
(])
#
0
L/6 <%'*+ P 2d'b+&5'H$(12+,).- %< A@
P = (. . . , x, v)
x (y, v, u)
(])
2
Chapter 2. Basics and Notation
+&5-/+[$ +;-=),+;$ 6 ?+&5 +&5' <7-R$ +Y 7$ 4+ A@ v 2 F P 6 4+&5 0 (u, v, y) " E - 0
-kY 7$ 4+M'*l 7$ +;$S2d'OC:-=1$(' P = (v, w, . . .) - 0 # w (]) (u, v, y) L/6
1'
=
&
)
'
+
*
C
<
(
$
'
'
%$
3 0 +o+,);- 4<7$X+&5-=+ CN) $&(])
$X-=+X'*l-RC*+&< F 0 '
#
#
#
#
0
6
E
r
P
Q
2
Y!':),+&':l V0 -R3g':< F v - 0 #H# ' E (v) = 2 '*lV+ 6 '`$(5 /6 +&5-/+ 4+ P7$ %3 P $&$ ?2<4' +&5-=+ + 6 C*< $(' # ' # E ' r # %$3 0 +
+&);- %<%$ CN) $,$b-=+H'*l-RCN+;< F A0 ' Y!'*),+;'*l pJ0 $(+,)&1C*+b- C F C*<%' P @ ) 3 P 2 F
+&);-DY!'*);$ 0E P - 0 # <%'D-DY 0E 'O-RC.5 Y!'*),+&':l -RC:C ) # 0E + 4+;$ <%-$ + C:C:1),)&' 0 C*'
-R< 0E P p <4'O-/)&< F P = (v, w, . . . , x, v) " $ P %$X- C F C:<4' 0 - . +&5'
C:1),Y!' γ +&5-/+9+,);-DY!'*);$,'D$ P 1 0 @ )&3g< F %$M-kC*< $(' # = ) # - 0 C:1),Y!' Z F +&5'
= ) # - 0sr Pp 1),Y!' 5' )&':3 γP # 4Y # 'D$ +;5' P <7- 0 ' 0 + -b2 1 0 # ' # - 0 # - 0
1 0 2 1 0 # ' # C 3 P 0 ' 0 + " $ P CN) $&$('D$ (u, v, y) ':ls-CN+&< F A0 C*' -/+ v $ # 'D$ γ - 0 # +&5V1$ u - 0 # y -=)&'U$(' P -/);-=+&' # 2 F γ >0 P -/),+ 7C*1<%-=) +;5'*)&'
7$q-H$('OC P0 # CN) $&$ 0E 2d'*+ 6 'D' 0 γ - 0 # Q - P -=),+ P@ ) 3 +&5' CN) $,$ 0E -/+ v
6 5 %C.5 C ),)&'O$ P A0 # $+ -g$('DC A0 # PC*) $,$ 0E A@ P - 0 # Q " $ P - 0 # Q -/)&'
' # E ' r # 7$3 0 + +&5 7$SCN) $,$ 0E CDC*1).$-/+9- Y!'*),+&':l z = v 4+&5':) +&5 7$XC*) $,$ 0E A@ P - 0 # Q -/+ z C ),)&'D$ P 0 # $ + -bCN) $,$ 0E @
- 0 # Q - 0 # 6 ' -/)&' # 0 ' +&5':) 6 %$,' ?+ 7$ P = (v, . . . , a, z, b, . . .) - 0 #
P
q>0 +&5 %$oCD-R$(' 6 ' 5-DY!' @ 1 0 # - C*< $(' #
P
=
(v,
.
.
.
,
a,
z,
c,
.
.
.
,
d,
z,
b,
.
.
.)
$,12+,);- 4< P := (z, c, . . . , d, z) A@ P - 0 # +&51$ A@ T $(1C.5 +&5-/+ P - 0 # Q CN) $&$
-=+c':ls-CN+&< Fo0 'Y!'*),+;'*l z p <%'D-=)&< F |P | < |P| - 0 # 6 ' C 0 C:<41 # '2 F 0 # 1C*+ A0
0 |P| +&5-=+ +&5':)&'J':l 7$ +;$ -M$('DC 0 # CN) $&$ 0E 2^':+ 6 ':' 0 P - 0 # Q 6 5 7C.5 43 P < %'D$
-M$(':< @ r CN) $&$ 0Ef@ T L' E -/) # 0E +&5'J2-R$('C:-$(' @K ) +;5' 0 # 1C*+ A0q0 +;' +&5-=+
- C F C*<%'[C:- 00 + 2d'C*) $,$(' # 2 F - 0 ' # E ' r # %$3 0 + C*< $(' # +,);- 4< 0 ':ls-CN+&< F 0 '
Y!':),+&':l -R$-kC 0 $(' 1' 0 C*' @ +;5' = ) # - 0sr p 1),Y!' 5' );':3 P
$,11C.5
D- Y 0 E 'O$ +;-/2< 7$(5' # +&5 7$ 0 %C:' P ) P '*)&+ F A@S00sr CN) ,$ $ 0E 1 <4':)q+ 1);$ 6 '
6 - 0 +U+ ' 0 $(1);'b+&5-=+ $(1C.5 - + 1) -=< 6 - F $U'*l 7$ +;$ - $g< 0E R- $U+&5' E );- P 5
1 0 # '*)9C 0 $ # '*);-=+ A0 7$ 1<%'*) 7- 0 -/+9-=<%< 59
85 F5879 .
Theorem
2.12
G
=
(V,
E)
7
3 ( B ?> ? ;
( B 78(E(E<=& 507
T
G
T
Proof.
# 'E 0 4+
?=50
?=507853( '*+ T = (p , . . . , p ) 2d'`- 0 -/)&2 ?+&);-/) F
0 A@M 1<%'*) + 1) CD- 00 +kC*) $,$ 4+;$('D< @ -=+
1
0 +&'*) ) A@ ?+;$' # E 'D$ " $&$(T13g'
+;5-/+ 7$ n
T
1 <4':) + 1) A@ G Z F
P 0 +;$ 0 +;5'U)&'D<%-=+ ?Y!'
5-$M-U$,':< @ r NC ) $,$ 0E -/+f- Y!'*),+&':l p = p T = (p1 , . . . , pi , pi+1 , . . . , pj−1 , pi , pj+1 . . . , pn ) ,
i
j
2.7. Polygons
K@ )S$ 3g' 2 < i + 1 < j < n JpJ0 $ +,);1CN+S- 0 +&5'*) 1<4':)+ 1) T @ ) 3 T 2 F
+&);-DY!'*);$ 0E +&5' $(12 r P -/+&5 (p , . . . , p ) 0 );'*Y!'*).$(' # ) # ':) \
5 -=+ 7$ i+1
j−1
T = (p1 , . . . , pi , pj−1 , . . . , pi+1 , pi , pj+1 , . . . , pn ) .
pJ0 $ # ':)f+&5' $ ),+&' # 0 C*)&'D-$ 0E < F" Y!'DC*+ ) a(T ) A@ - 0E <%'D$ @K );3 ' # 2 F
+&) P <4'O$ A@ P 0 +;$ +&5-=+ -/)&'fC A0 $('DC:1+ 4Y!'-R< 0E T 0 C*<%1 # 0E p p p
n−1 n 1
*
'
)
*
'
l
R
*
C
&
+
<
+
4
<
O
'
$
#
#
0
T
0
F
6
0
E
5'f+ 1);$
pn p1 p2
T
T
-0#
-0#
)
α := (pi−1 , pi , pi+1 )
β := (pj−1 , pi , pj+1 )
γ := (pi−1 , pi , pj−1
δ := (pi+1 , pi , pj+1 )
0
0
T
T
-=<%<
-0#
Y!':);$(1$
j ' C*<7- %3 +&5-/+ a(T ) < a(T ) 0 +&5'U<%'*l %C AE );- P 5 7C ) # ':) 0E 5' 0
6 5' 0 ':Y!'*)+&5'*);' 7$\-f$(':< @ r CN) $&$ 0E 0 T 6 'C:- 0 1$('L+&5'S-/2 Y!' P '*);-=+ A0 + C 0 $ +,)&1CN+- 0 +&5':) 1<%'*)+ 1) T A@ G 6 5 $(' Y!'DC*+ ) A@ - 0E <4'O$ %$$ +,) 7CN+;< F
$,3g-R<4<%'*)X+&5- 0 +&5' 0 ' @ T 0 C*'k+&5':)&' -=)&' 0 < F 0 ?+;':< F 3U- 0F # 7$ + 0 C*+
1<%'*)S+ 1);$ @ ) G +&5':)&'o3 1$ +S2d' 0 'f+&5-=+ %$ 0A0r C*) $,$ 0E
+k)&'D3U- 0 $q+ P ) Y!' +&5' C*<7- %3 +&5-/+ 7$ 3 0 {γ, δ} < 3 0 {α, β} " $
5-R$b- $,':< @ r CN) $,$ 0E -/+ p 6 ' 3g- F -$,$(13g' 2 F i ) P $ 4+ 0 +&5-=+
T
i$(+,) 7CN+&<
7$
F 0 2d'*+ 6 'D' 0 γ := ](p , p , p )
α
:=
](p
,
p
,
p
)
i−1
i
i+1
- 0 # ](p , p , p ) 5-=+ 7$ +&5'LC F C:< 7C ) # ':) A@ p p i−1 p i j−1
-0#
i−1
i
j+1
i−1
i+1
j−1
-/) 1 0 # p %$ ' 4+&5':) p , p , p , p ) p , p , p , p pj+1
<4' µ @ j−1
)&3g' # i+1
2 F - 0_j+1
+&5'O$(' j+1
+;$B-/) j−1
10#
p <%'D-=)&< F +;5'J3 i0 43k13 - 0E i−1
F + 6 [A@i−1
P 0 i+1
%$ @ )&3g' # 2 F + 690 ' E 52 );$ +;5-/+ 7$ P 0 +;$+&5-/+B-=)&'JC A0 $('DC:1+ 4Y!' 0q0 '
pi
A@ +&5'D$,' C F C:< 7C ) # '*).$ n0H@ -CN+ +&5' 0 ' E 52 ).$9-/);' +&5' $,-=3g' 0 2 +;5 ) r
# '*);$ " $ T 7$- 0 1<4':)+ 1) +&5'LY!':),+ 7C*'O$ p p p - 0 # p -/)&'
-Rj−1
$ 6 ':<%<i+1
-$ p - 0 j+1
# p
P - 4) 6 %$,' # 7$ + 0 CN+ n0 P -/),+ %C:1<%-=) pi−1 - 0 # pi−1
i+1
j−1
-=)&' 0 + 0 ' E 52 ).$ 0 +&5' C F C*< %C ) # '*)-=) 1 0 # p \
51$ µ < 3 0 {α, β}j+1
i
@ µ {γ, δ} +;5' 0 +&5' C:<%- 43 %$ %3g3g' # 7-/+&' +&5'*) 6 7$(' -$,$(13g' 6 4+&5 1+
< $,$ A@ E ' 0 ':);-=< 4+ F +&5-=+ µ = (p , p , p ) p <%'D-=)&< F 4+ %$ µ < π j '
i−1
i
j+1
# %$ + 0E 1 7$(5h+ 6 C:-$('D$ UpJA0 $ # '*)o+&5'k< C:-=+ A0 @ p o
>@
Case 1: µ :=+&](p
,
p
,
p
)
<
π
i−1
i
j+1
i+1
5' 0 δ < α 8 E 1);' n- .B <7$(' 6 'S5-:Y!' α > π - 0 # γ < α
α
π
8 E 1)&' >2 . 43 %<%-=)&< F! @
+&5' 0
β := ](pj−1 , pi , pj+1 )
π
8 E 1);' n C . - 0 # @ )
?+ 7$ δ < β 8 E 1)&' # 6 .
γ<β
β >π
UpJA0 $ # '*)o+&5'k< C:-=+ A0 @ p o
>@
Case 2: µ := +&](p
i−1 , pi , pj+1 ) > π
i+1
5' 0
<7$(' 6 ' 5-:Y!' α > π - 0 # δ < α L %3 4<7-/);< F! @
α
π +&5' γ < α 0 δ < β 0 # @ ) β > π ?+ 7$ γ < β β
π
!
Chapter 2. Basics and Notation
5 %$SC 3 P <%'*+;'D$+&5' P ) _A@ A@ 2 +;5h+&5'oC*<7- %3 - 0 # +;5'f+&5' )&':3 pj−1
γ
pi = pj
pi−1
pi+1
pi−1
α
α
pi+1
δ
pj+1
α
π
pi = pj
pj+1
α >π
pj−1
γ
pi+1
β
pi−1
pi = pj
*) =&# 59(
78
β
D
π
pj−1
β
δ
pi+1
β >π
B 78(E(E<=& ?=5 6 7 3 K( 5
?=52=&;F50( 785 (8?= 4 )
<
π
T
=&# 59( 9$ &7850$ 7 B (
T
,p ,p 5
i−1
](p
B 7 B ( i ?=j+1
pi−1
pj+1
pj+1
Figure 14:
pi = pj
( 5 0$
5 ' )&':3 -R$,$(':),+;$ +&5''*l 7$ +&' 0 C*' A@ - 00sr CN) $&$ 0EX 1<4':)\+ 1) 0 ':Y!'*) F
1<%'*) %- 0 . 9@ C 1).$(' - $ P 'DC C . 3 E 5+fC 0 +;- 0 $(':Y!'*);-R<$,1C.5
+ 1);$ " )&' +;5'*)&' E );- P 5$ 6 5 7C;5 5-DY!' '*l-RC*+&< F 0 ' 00sr CN) $,$ 0E 1<4':)
+ 1
) @ - + 1) 7$kC A0 $ # '*)&' # -R$ - $,' 1' 0 C:' A@ Y!':),+ 7C*'O$ +&5' 0 +&5' 0 < F
$,1C.5 E ).- P 5$f-=)&'q+&5 $,' 6 ?+;5 A0 < F 0 ' Y!'*),+&':l X
5 7$ 7$2^'OC:-R1$(' @K )o':Y!'*) F
1<%'*) + 1) T @ - E );- P 5 G +,);-DY!':);$ 0E +&5'MY!'*),+ 7C*'O$ 0 )&':Y!'*);$(' ) # ':) F 4'D< # $
- 0 +&5':) 1<%'*)\+ 1) T @ G 6 5 %C.5 %$ # 7$ + 0 C*+ @ ) 3 T @ 4+ C 0 $ 7$ +;$ A@ 3 )&'
+;5- 0U0 'SY!':),+&':l +&5':) + 1);$ CD- 0 2d' E ' 0 '*);-=+&' # 2 F $,5 @ + 0E +&5'M$(' 1' 0 C*'
A@ Y!':),+ 7C*'D$SC F C:< 7C:-R<4< F!
':+ 1$f':lsC:<41 # ' +&5'O$(' E ' 0 '*).-=<%< F - PP < 7C:-=2<4' P + 0 $[2 F # 'OC*<7-/) 0E + 6 C:< $(' # 6 -R<?m$ U = (u , . . . , u ) - 0 # W = (w , . . . , w ) s@ )M$ 3g' k 0 - E );- P 5 G 5 #40F58 @ k- 0 # A0 < F @ +&5'*)&0' 7$o$ 3gk' j 6 ?+&5 0 j k
$,1C.5 +;5-/+ w = u @ )q-R<4< 0 i k B6 5':)&' ' 4+&5':) ϕ(i) = (i + j)
3 # (k + 1) i ) ϕ(i)ϕ(i)= k − ((i + j) 3 # (k + 1)) j ' 6 %<%< # 8' 0 ' P < FsEA0 $ + 2d' . $+&5-=+c5-:Y!' -SC:'*),+;- 0 + F P ' A@ 1<4':)
+ 1) Z1+ 6 ' 6 - 0 +[+ -=<%< /6 P < FVE0 $9+ # ' E ' 0 ':);-/+&' 0 + < 0 'k$,' E 3g' 0 +;$
< CD-=<%< F! +;5-/+ %$ 0 +;5'+&'*)&3 0 < EAFqA@ . $ 6 ' 6 - 0 ++ -R<4< /6 C*1+ r ' # E 'D$ p <%'D-=)&< F - E );- P 5 +&5-=+ 5-R$q-HC:1+ r ' # E ' %$ 0 + 1<%'*) 7- 0c ' 0 C*' 6 ' -=<%< =6
2.7. Polygons
+;5' + 1)k+ +,).-:Y!':);$(' 'D-C;5 C*1+ r ' # E ' + 6 7C*' W9$ 0E +&5' @ <4< /6 0E $ %3 P <4'
C 0 $ +,)&1CN+ 0 6 ' C:- 0 C 0 $ # '*) P < FsEA0 $+ 2^' 1<%'*) %- 0 E ).- P 5$ @ )f3 $ +
P 1) P $('D$S-R<?+&5 1 E 5H$(+,) 7CN+&< F $ P 'O-/m 0E +&5' F -=)&' 0 + 0 'OC:C*'O$,-/) 4< Fb 1<%'*) %- 0
7 .
(
Definition ( 2.13
G
=
(V,
E)
2-edge-connected
closure G =
D 5 5
D>K( # F 4 ( 50 K5 9$
?=5 ( 5 B 5
D2&@50(
, E )
C E
(V
G
?=5 '507 65 ( 5 (
4 ?=50785
( B G
V
:=
V
{v
|
c
C}
v
c
c
(0 7 B 5
D ( B ? ? ;: ( D 2(8+ B 5 ?=5 D "3
3(
:D
3(
m
c
δ
v
c
c
6 ?=579 &? −
4 ?=50785 x 3( ?=5 F5 B E&7E ?' B 4 $(
43F587 50:D <
xy
D
3 0
c
δ := 12
p ∂(G\c) ||p − mc ||
?=5 5
D&@5 ( 59 (
- 0 # x V(c)} G2
E := E {{x, vc } | c C
2$('*)&Y!'b+&5-=+
7$ -=< 6 - F $ - .
+ 1) A@ G 6
F $(12 P -=+&5 A@
A@ G 7$q)&' P <%-C*' # 2 F +;5' P
cC =
xy
),)&'D$ P 0 # 0E + 1) A@ G 7$ -f+,).- %<^+&);-DY!'*);$(' # + 6 7C*' 5':)&G' 2'*Y!':)
" 0 1<4':) + 1) 0 G 0 # 1C:'D$ +&5'X+ F P ' (x, v , y) @K )[$ 3g2 ' C:1+ r ' # E '
-/+&5 (x, y) Z cF # 'E 0 4+ 0 A@ G +&5'
P -/)&+ @ ) 3 +;5' @ -CN+ +&5-/+ C:1+ r ' # E 'D2$ -=)&'
3(
:D %$ 4 B
.
Definition
2.14
P
=
(V,
E)
polygon
5ED&'59( 7853 B D 50 6 ( 3 5 B 5 D
B +3 ( 5
P
P2
6 5 #4F58 B 5 B 78(E(
3=& 507 7 ?=5 B 707050(+ :D3=&
( D 58 K5
D 9$
?=5 ( 5 6 7 2 3( B 4 5ED ?=5
P
P
∂P
boundary
P
?=5 "#%$&'
(
:D $ ( B $ B F5 (
3
5
P
simple
P
# $K&' ?'785E5 '587 B 59( 3(2
(
<
F5 "#%$&' 7
triangle
'507 B 59( ( 705 5879785
D K( quadrilateral
+&' +&5-/+ 6 ' -=<%< /6 D 5+&'50 507E;5 P < FsE0 $d+&5-/+ C 0 $ 7$ + @ -L$ 0E <%' P 0 +
) < 0 ' $,' E 3g' 0 + A0 < F! " $[+&5' $ F 3q2 < 0 # 7C:-/+;'D$ 6 ' -R< 6 - F $X-R$,$(13 'k+&5-=+
Y 7$ 4+;$J+&5'Y!':),+ 7C*'O$ A@ +&5' 0 0 4+&' @ -RC*' A@ P 0 C 1 0 +&'*);C:< C;m 6 7$(' ) # '*) P
8 ) P = (p , . . . , p ) 6 ?+&5 p = p - 0 # - Y!'*),+&':l p @ ) 0 i k 0 +&5' k (
# ' 0 +&' 2 F p 0 := pk
BEB 59(E( 7 A@ i - 0 # 2 F
pi +;5-/+
pi ) 1 :=
i 1 +&5' =7(i+1)
85
D 5 B 50(8(7 k @
g
'
,
$
0
j
F
@ 0 p(i+k−1)
p
P
p
i
i
k
7$J- B '5 '507 65 0 P @ - 0 # 0 < F @ ] p := ](p , p , p )
i7$S<C k Y!'*l
0 " 0 -=< E 1$(< F # E' 0 ' (0 79 B %$ B '5 P705 i 5 - 0 # i 1; 4i507 i B 150(
A@ P 2$(':),Y!' +&5-/+X2d' 0E - $ +,) 7CN+&< "F C 0 Y!'*l );' ':l c ) -=+ Y!':),+&'*l %$
0 +- P ) P ':),+ FhA@ +&5' P 0 + p -=< 0 ' 6 5 %C.5 3 E 5_+M- PP 'D-=)[$(':Y!'*);-R<c+ 43g'O$
0 P >0 $(+&'D- # +&5 7$ P ) P '*)&+ F i %$ + 4' # + - $ P 'OC C C:C:1),)&' 0 C*' @ p -=+
i
P $ 4+ A0 i 0 P Chapter 2. Basics and Notation
) "& "& )2! $' $%)$ 2 $ $% #)#
"& $' $%) )2
$ Figure 15:
4
59( !&'
!
3 # $&@ (
Z FHi ) P $ 4+ A0` '*Y!':) F @ -RC*' A@ P %$L2 1 0 # ' # 2 F - # ?)&'OCN+&' # 00sr
C*) ,$ $ 0E +,).- %< 0 +&5' ) 4' 0 +.-/+ 0 A@ P 0 2# 1C*' # 2 F +&5' 1<%'*)+ 1) @ +&5 7$
# ?)&'OCN+&' # +,);- %< 7$ ) 4' 0 +;' # C 1 0 +&'*).C*< 2C;m 6 7$(' 6 'HC:-R<4<M+&5'bC ),);'D$ P 0 # 0E
@ -RC*' "(E '5 %$ 795065
D " $q'O-RC.5 @ -C*' A@ P # 4)&'OCN+&< F C ),)&'D$ P 0 # $ + @ -RC*' A@ P _6 '[-=<7$ E ':+ - 0 ) 4' 0 +;-=+ A0 @ )J+&5' @ -RC:'D$ A@ P 0 +&5 %$ 6 - F
5'
+ 1) P A2@ P +,);-DY!'*);$,'D$B+&5'2 1 0 # -/) F @ 'D-C;5 @ -C*'SC 0 $ %$ +;' 0 +&< F @6 'S-=<%< =6
C:1+ r ' # E 'O$+ 2d' +,);-DY!'*);$,' # 0 C*' 0 2 +&5 # 4)&'DC*+ 0 $ ' 0 +&'b+&5' $('*+ @ P $ 4+ 4Y!':< F ) %' 0 +&' # 0 ?+&' @ -RC:'D$ A@ P 2 F + (P) ' "#%$&' 4 D < (P) +&5-/+ 7$
':Y!'*) F P < FVE0 P 6 ' -$,$ C %-=+&' +&5
# E' 0 ' # -R$
(P) := ∂P
F F.
+ (P)
5'350797 A@ - P < FVE0 P 7$ # ' 0 +&' # 2 F P := (P) \ ∂P 43 %<%-=)&< F
+;5' 55070 7 A@ P 7$ # 'E 0 ' # -R$g'*lV+ (P) := 2 \ (P) 5' @K <%< /6 0E
P ) P $ 4+ 0 1$ + 'D$L+;5' 0 -=3 0EU@ +&5'O$(' )&' E A0 $ " 7< $ U 2$(':),Y!'X+&5' 43 r
P < 7C:-=+ 0 +&5-/+ (P) %$9C 3 P -RC*+ Proposition 2.15 ∂P
( ?=5
78507
# 8& B 4 8 :D@70$ (P)
2.7. Polygons
L'DCD-=<%<
+&5-/+[+&5' @ ) A0 + %'*) A@ - $('*+ M 2 C 0 $ %$ +.$ @ -R<4< P 0 +.$
0 2 +&5-/+ -=)&' 0 ' 4+&5'*) 0 +&'*) ) + M 0 )q+ 4+;$ C 3 P <4'D3 ' 0 + 2 \ M ' 0 +&' +&5' @ ) 0 + %'*) A@ P 2 F B(P) p <%'D-=)&< F 4+ 7$ B(P) ∂P 2d'DCD-=1$('g-=<%<
@ -RC*'O$ -=)&' P ' 0 - 0 # 5' 0 C*'S-=<%<s+&5' ?) P 0 +;$ -/);' 0 +&'*) ) 0 +&5' +&5':)B5- 0 #
C 0 $ # '*)9- P 0 + p ∂P @ p < 4'O$ 0 +&5'[)&'D<%-=+ ?Y!' 0 +&'*) ) A@ $ 3 ' ' # E ' e E(P) +;5' 0 C A0 $ # '*)
+;5' ) 4' 0 +;-=+ A0 A@ e 0 # 1C*' # 2 F P @ e %$g- C:1+ r ' # E ' A@ P +&5' 0 2 F
# E' 0 4+ 0 A@ P < FsEA0 e 7$ 0 C # ' 0 + + +&5' 0 0 4+&' @ -RC:' A@ P 6 5 %C.5 %$
':l+;'*) ) + P +&5':) 6 7$(' 2 F # 8' 0 4+ A0 A@ (P) +&5' @ -C*'f+ +&5'f<%' @ + A@ e
2d'D< 0E $M+ P 6 5 4<%'q+&5' @ -RC:' + +&5'X) E 5+ A@ e 2d':< 0E $M+ '*lV+ (P) [
5V1$
':Y!'*) F 0 ' E 5_2 )&5 # A@ p 0 +&':);$('OCN+;$L2 +&5 (P) - 0 # '*lV+ (P) @ 0 +&5' +&5'*)o5- 0 # v V(P) +&5' 0 +&5':)&' 7$o-/+X<4'O-R$ + 0 'k' # E ' A@ P
0 C # ' 0 + + v 2^'OC:-R1$(' P %$ 1<%'*) 7- 0cH>0 P -=),+ 7C*1<7-/)q'*Y!':) F P ' 0 0 ' E 5 r
2 )&5 # N A@ p C 0 +.- 0 $ 2 - P 0 + q 0 +&5'9)&':<7-/+ 4Y!' 0 +&'*) ) A@ $ 3g'M' # E ' " $ N 7$S-=<7$ - 0 ' E 52 );5 _ # A@ q 6 'XC:- 0 -/) E 1' -$-/2 Y!' _ 3g':+ 43g'O$ ?+ 7$JC 0 Y!' 0 4' 0 +\+ $(12 # ?Y # 'M-X<%-=) E ' P < FsEA0 0 + $,'*Y!'*).-=<
$,3g-R<4<%'*) P < FsEA0 $L'O-RC.5 A@ 6 5 7C.5 7$ P +&' 0 + %':<%< F 'D-R$ 4':)M+ 5- 0 # <%' +&5- 0 +&5'
) E 0 -=<c<%-=) E ' P < FsEA0c\ # 'D-R<4< F $(1C.5 -k$,12 # ?Y 7$ 0 7$ 0 +S+ );' # 1 0 # - 0 +
0 C Y!'*) 0E $ 3g' P -/),+;$ A@ +&5' P < FsEA0 3U- 0F + %3g'D$ Proof.
7
$&@
#
( 5 ( 59
Definition
2.16
P
dissection
{PD 1, .50.( . P
}
"#%$&'( ( B ? ? ;
D nK
(Pi ) 4 =?=50785 (P) (Pi )
'507 75 405 70$ 1 i n
(Pj )
1 i, j n
i=
j
" P -=),+ @ ) 3 $ 43 P <%'
C:<%-$,$('O$ A@ P < FsEA0 $ j
C 00 'DC*+&' # P < FVE0 $3
P < FsE0 $ +&5'*);' -/)&' - 0 13q2d'*) A @ &+ 5':) %3 P )&+;- 0 +
4+&5 0 +&5 7$ 6 ),m 6 ' 6 4<%<2^'C 0 C:*' ) 0 ' # 6 ?+;5 $ %3 P < F
$(+ A@ +&5'f+ %3 ' "#%$&@
Definition
2.17
( (
<
%$ B 5 B 5
D B 45 (
D $ P# $&@simply
connected
(P)
(
:D $ (
P
convex
(P)
5 ' C 0 Y!'*l 5V1<%< A@ - 0 ?+&' $('*+ 6 5 ($ ' P 0 +;$q-/)&' 0 + -=<%<C <%< 0 'O-/) %$ C 0 Y!':l P < FsEA0
Chapter 2. Basics and Notation
Chapter 3
Hamiltonian Polygons
5 %$cC.5- P +;'*) 7$ # '*Y +&' # + +&5'$ +;1 # FfA@s -R3 4<4+ 0 7- 0 P < FVE0 $ 0 $(' E 3g' 0 +
' 0 # P 0 +LY 7$ 42 %< 4+ F E );- P 5 $ 7 ( 59
"<74 3( 5!D ( < 3 5 ( 5 & 05 ( Definition 3.1
S
Ha3( E( <
5A # $K&'C? ; 3 ( (+I < & ( +& 78 ?2
miltonian
polygon
1 %$
(S)
6 2"#%$&@
7 (
Figure 16: ( 5+& 50( 78
H
785 (E?= 4 9$ ?'
S
# $&@
0$ ?'< ( #% D B 3 50(
H
9$ D K 5
D 3 59(
5 (5 & 50( S
B ( # D&7E$ 3 59(
:D ?=55
D&@50( 1
?=5
?=5
7$ (S)
5 '`$(' E 3 ' 0 +.$ @ ) 3 S DC - 0 2^' E ) 1 P ' # 0 + +&5);':' # T '*);' 0 +hCD-/+&' EA ) %'D$
-C:C ) # 0E + &+ 5' 4) P $ ?+ 0 &) ':<7-/+ ?Y!'\+ - -R3 4<4+ 0 7- 0 P < FsEA0 H $ 3g' @
!a
Chapter 3. Hamiltonian Polygons
+;5':3 3g- F - PP 'D-=)-R$\' # E 'D$ A@ H _ +&5'*).$ -/)&'D &@:4%( +&5-=+ %$ $(' E 3 ' 0 +.$
6 5 $,' )&':<7-/+ ?Y!' 0 +;'*) ) %$bC A0 +;- 0 ' # 0 (H) - 0 # +&5' +&5 ?) # m 0 # 7$
CD-=<%<4' # 5 = &@:4%( +&5-/+k-=)&' $,' E 3g' 0 +;$ 6 5 $(' );':<7-/+ 4Y!' 0 +;'*) ) %$ # %$3 0 +
@ ) 3 (H) 2Y 1$(< F +;5' '*l %$ +;' 0 C:' @ - -R3 4<4+ 0 7- 0 P < FsEA0 43 P < 4'O$M+&5-/++&5'
C ),)&'D$ P 0 # 0E Y %$ 42 %< 4+ F E );- P 5 %$ -R3 %<4+ 0 %- 0 5':)&' @K );' +&5' 3g- 0 +&5' r
);':3 @ +&5 7$ C.5- P +;'*) $ +;-=+&' # 2d'D< /6 %3 P < %'D$ +&5-/+ @ ) n-=<%3 $(+ - 0F 0 ?+;'
$,'*+ A@ P - 4) 6 7$(' # 7$3 0 + < 0 'b$(' E 3g' 0 +;$ +;5' ' 0 # P 0 +kY 7$ ?2 4< ?+ F E );- P 5 %$
-=3 %<?+ 0 7- 0c
7 $
5 ( 5 "<74 3( 5 D ( < %< 5 ( 5+& 50(
Theorem
3.2
K 4 B # 3 5
7 ?=50785 5 3(8 ( "#%$&@ 5' P ) A@ @
5' )&'D3 s 7$9-R< E ) 4+&53 7C +&5-/+ 7$ @ )+&5' E ?Y!' 0 $('*+ S @
$,' E 3g' 0 +;$ 6 ' C 0 $ +,);1CN+M- -R3 %<4+ 0 %- 0 P < FsEA0c $ 43 P < @ F +&5' # 7$,C*1$ r
$ 0 <%'*+k1$q-$,$(13g' +&5-/+k-R<4<< 0 ' $(' E 3g' 0 +;$q-=)&' 00sr # ' E ' 0 ':);-/+&' +&5-/+ 7$ -R<4< $(' E 3 ' 0 +M' 0 # P 0 +;$9-/)&' P - 4) 6 7$(' # 7$ + 0 CN+ 5'X'*lV+&' 0 $ 0 + # ' E ' 0 ':) r
-=+&'f$(' E 3g' 0 +;$ %$L$ +&);- E 5_+ @K ) 6 -/) # - 0 # 6 %<%<d2d' # %$&C*1$,$,' # -/++&5'[' 0 # A@ +&5 7$
C.5- P +&':) 5' 0 '*lV+M$('DC*+ A0 P ) Y # 'D$- 0 Y!':),Y 4' 6 - 0 # # 'D$,C*) 42^'O$+;5' -R< E ) 4+&53
- 0 # 4+;$ # T ':)&' 0 + P 5-$('D$ @ ) 3 -q5 E 5b<%'*Y!':< P 0 + @ Y %' 6X
3.1 Algorithmic Overview
5' E ' 0 '*);-R< # 'D- 2d':5 0 # 1)-R< E ) 4+&53 7$J+ 3U- 0 +.- 0 - P < FsE0 6 4+&5 0
+;5'\Y 7$ 42 %< 4+ F E );- P 5X+;5-/+BC A0 +;- 0 $ -R<4<s$(' E 3g' 0 + ' 0 # P 0 +;$ 0 4+;$ P < FsE0 -=<
# 3U- 0 0 + 0 'DC:'D$,$&-/) %< F -R$ Y!':),+ 7C*'D$ - 0 # 6 5 $('hY!':),+ 7C*'D$g-=)&' n$ 3g' A@
+;5' $(' E 3 ' 0 +' 0 # P 0 +;$ 5'XC A0 Y!'*l 51<%< C 0 Y (∂S) $(':),Y!'D$9-R$- $ +;-/)&+ 0E
P 0 + ?+ %$ C*'*)&+;- 0 < F -X$(12 E );- P 5 A=@ 1 %$ (S) 4+ C 0 +;- 0 $ -=<%<^$,' E 3g' 0 +;$ - 0 #
4+;$9Y!'*),+ %C:'D$-/)&'q$(' E 3g' 0 +' 0 # P 0 +;$ 9 1) 0E +&5' C A0 $ +,)&1C*+ A0 6 'q- %3 -=+
0 C*)&'D-$ 0E +&5' 0 13q2d':) @ $,' E 3g' 0 +o' 0 # P 0 +;$f+&5-/+ +&5' C:1),)&' 0 + P < FVE0
Y 7$ ?+.$2 F 3 # @ F 0E +&5' P < FsE0 < CD-=<%< F -RCDC ) # 0E + C:'*),+.- 0 )&1<%'D$ " +J+&5'[$,-=3g'+ %3 ' 6 '[3U- 0 +;- 0 - # 7$,$('OCN+ 0 A@ +&5' P < FsE0 +&5-=+J+,) 4'O$
+ f0 %C:':< F E ) 1 P +&5'$(' E 3 ' 0 +.$ +&5-=+\-=)&'S$ + %<4< 0 +&5' 0 +&'*) ) @ +&5' P < FsEA0
'*)&' 0 7C*'D< F 3g'D- 0 $X+&5-/+ '*Y!':) F $(' E 3g' 0 + +;5-/+ %$ 0 +&5' 0 +&':) ) @ +&5'
P < FVE0 %$ C 0 +;- 0 ' # 0 '*l-RCN+;< FU0 ' P < FVE0 A@ +&5' # 7$,$('DC*+ A0c 1) E -=<
7$S+ 6 r>@ < # 0H0 ' 5- 0 # + 0 C*<%1 # ' 3 )&'X- 0 # 3 )&'X$,' E 3g' 0 +9' 0 # P 0 +.$
3.1. Algorithmic Overview
-$ !Y '*),+ %C:'D$ 0 + +;5' P < FsEA0 - 0 # 0 +&5' +&5':)b5- 0 # + $ 43 P < @ F +&5'
# %$,$,'DCN+ 0 $(1C;5 +&5-/+ ':Y!' 0 +&1-R<4< F -R<4< P < FsEA0 $ 0 +&5' # 7$,$('DC*+ A0 -/)&'C 0 Y!'*l 5' 0 %C:' P ) P '*),+ FU@ C 0 Y!'*l # 7$,$('DC*+ A0 P < FsE0 $ 7$ +&5-/+ +&5' F P 1+1$
0 + ':ls-CN+&< F +&5'X$ ?+;1-/+ 0 6 5'*)&' 6 ' $ +.-/),+&' # - C A0 Y!'*l $,'*+ 0 4+ 7-=<%< F +&5'
C 0 Y!':l 51<4< 0/6 +&5' # %$&$('DC*+ 0 P < FVE0 +&5-/+ P $,$ 42< F C 0 +;- 0 $f$ 3 '
$,' E 3g' 0 +;$ @ ) 3 S 0 4+;$ 0 +&'*) ) " +f+&5 %$ P 0 + 6 ' CD- 0 - PP < F 0 # 1CN+ 0
+ $ <?Y!' +;5' P ) 2<%':3 @ ) 'O-RC.5 A@ +&5' # %$&$('DC*+ 0 P < FsE0 $ $,' P -=);-/+&'D< F!
_ 3g'fC:-=)&'f5-R$J+ 2d'[+;-=m!' 0 +&5-/++&5'O$('[)&'OC*1);$ ?Y!'D< F 2+;- 0 ' # -=3 %<?+ A0 %- 0
P < FVE0 $ @ ) +&5'$(12 P ) 2<4'D3g$C:- 0 2^'MC 00 'OCN+&' # + +&5' E < 2-R< P < FVE0
+ 0 -R<4< F @K );3 - -=3 %<?+ 0 7- 0 P < FVE0 @ )S+&5' 6 5 <%' $(':+ @ $,' E 3g' 0 +;$ ':+ 1$ E );- # 1-=<%< F - ## 3 )&' # '*+;- 4<+ +;5' # 'O$,CN) P + 0 A@ +&5'9-R< E ) 4+&53
1 0 + %< 6 ' 0 -=<%< F E ':+ + - @ 1<%< F ' # E ' # < %$(+ A@ 0 Y-=) %- 0 +.$ B
5'C 0 $ +,);1CN+ 0
P ) C:':' # $ 0 $ ?l P 5-R$,'D$L+&5-/+9-/);' 4<%<%1$ +,);-=+&' # 0 8 E 1)&' O
Phase 8 1 Initialisation:
E 1)&' n - +.-/),+ 6 4+&5 +&5' P < FVE0
P←
C A0 Y
(∂S)
-$S$(5 /60 0
8 )J+&5 $('[$(' E 3g' 0 +;$ @ ) 6 5 7C.5 0 < FU0 'M' 0 # P 0 +- P r
A@ P 0 C:<41 # 'g+&5' +&5'*) ' 0 # P 0 + -R$ 6 ':<%< 1$ 0E
P
-h< C:-R0 +&5'C 1);$(' A@ +&5'O$('3 # CD-/+ 0 $ P 6 4<%<
0HE ' 0 ':);-=< 2^'OC 3g' - 0A0r $ 43 P <%' P < FsEA0c
5' 0 +&':) ) A@ +&5' P < FsE0 P 7$ # 7$,$('DC*+&' # 0 + C A0 Y!'*l
Phase 3 <Dissection:
P FsEA0 $ 2 F C*1+&+ 0E ?+-=< 0E );- F $L$ +;-/)&+ 0Eq@ ) 3 4+;$J)&' ':lUY!'*),+ %C:'D$ j 5' 0 '*Y!'*)g$(1C.5 - );- F 5 ?+;$k- $(' E 3 ' 0 +k+&5-/+ < %'D$ 0 +;5' 0 +;'*) ) A@
+;5 %$U$,' E 3g' 0 + 7$ 0 +;' E );-/+;' # 0 + P - E - 0 2 F 3g'D- 0 $ A@ - < CD-=<
P
3 # CD-/+ 0 " + +;5' ' 0 # A@ +&5 7$ P 5-R$,' -=<%<S$(' E 3g' 0 +;$ @ ) 3 S -=)&'
' 4+&5':)' # E 'D$ A@ P # %- EA0 -R<%$ A@ P ' P EA0 -=<7$ A@ P V ) +&5' F < %' 0 +&5'
0 +&':) ) A@ 0 ' A@ +&5' # %$&$('DC*+ 0 P < FVE0 $ 8 E 1)&' nC 6.
W$ 0E 0 C:'- E - 0 < CD-=<d3 # CD-/+ 0 $ +&5' P < FsE0
Phase 4 %$Simplification:
C.5- 0E ' # $(1C.5X+&5-/+ 4+2d'OC 3g'D$ -S$ %3 P <4' P < FsE0 8 E 1)&' # P
8 )q+&5 7$ + 6 ),m 6 ' 5-DY!'U+ C:-=)&' @ 1<4< F C 0 +&) < +&5'U+ F P ' A@ 00sr
$ %3 P < 7C 4+ %'D$ +&5-/+9-/) %$(' 0ì 5-R$(' - 0 # s
8 )['*Y!'*) F C 0 Y!':l P < FVE0 C 0 +&5' # 7$,$('OCN+ 0À@ 1)
Phase 5 <Induction:C 3 1+&'
0 # 1C*+ ?Y!'D< F - -=3 4<4+ 0 7- 0 P < FsE0 +&5-/+ Y 7$ ?+.$
P FsEA0 P P
+&5' ' 0 # P 0 +.$ @ -R<4<L$(' E 3g' 0 +;$ +;5-/+ < 4' 0 +&5' 0 +&'*) ) A@ C 8 Er
1)&' >' +93 E 5+95- PP ' 0 +&5-/++&5'*)&' %$ 0 $(' E 3g' 0 + ) 1$ + 0 '
Phase 2'D-=Saturation:
);$ -R$ - Y!'*),+;'*l
!
Chapter 3. Hamiltonian Polygons
$(' E 3g' 0 + 0 +&5' 0 +&':) ) A@ $(1C.5 - P < FsE0k6 5 7C.51$(+ $ 43 P < 'D$+&5'
C 3 P 1+.-/+ 0 0 +&5 7$ P 5-R$(' 5' -=3 %<?+ 0 7- 0 P < FsEA0 $XC 3 P 1+&' # 0 i 5-$(' @ )
Phase +&65' Bridging:
$(' E 3 ' 0 +.$ 0 +;'*) ) + +;5' # %$&$('DC*+ 0 P < FsEA0 $c-/);' C 3q2 0 ' # 6 ?+;5
+&5' P < FsE0 P + h@ )&3 - -=3 %<?+ A0 %- 0 P < FsE0 @ ) +;5' 6 5 <4'U$(':+
8 E 1)&' @ .[
' 0 $(1)&' +&5-/+[+&5 %$M2) # E 0E 7$f-=< 6 - F $ P $,$ ?2<%'
S '3U-/m!'1$('
6
A@ $ 3g' @ )&':' # 3 );' E -/) # 0E +&5'MC 0 $(+,)&1C*+ 0UA@ P 6 '
C:- 0 l - 0 -/),2 4+,);-=) F ' # E ' A@ +&5'[C 0 Y!':lU51<%< $(1C.5 +&5-=+ 4+ 7$ P -/),+ A@
+&5' -R3 4<4+ 0 7- 0 P < FsE0 + 2d' C A0 $ +,)&1C*+&' # 0 +&5' +&5':) 5- 0 #
6 '5-DY!'+ 3U-=m!'M$(1);'9+&5-/+J'O-RC.5 # 7$,$('OCN+ 0 P < FsE0 5-R$ -XC 3 3 0
' # E ' 6 ?+&5 P 5 'f)&'O$ + A@ +&5 7$SC;5- P +&':) %$ ) E - 0 %e:' # -$ @ <%< /6 $ >0 'OCN+ 0 s 6 '`< 7$ + $ 3g' P ) P '*),+ %'D$ A@X 1) 6 ),m 0E P < FsEA0 P
3g' 0 + 0 ' # -/2 Y!'L2 F 0 +&) # 1C 0E -oC:<%-$,$ A@ P < FsEA0 $CD-=<%<4' # 78 59( 'DC r
+ 0 s s 'DC*+ 0 V - 0 # 'OCN+ 0 s # 'D$,C*) ?2d'\+&5' -=+&1);-=+ 0 %$,$,'DCN+ 0
- 0 # %3 P < C:-=+ A0 i 5-$(' )&'O$ P 'DCN+ ?Y!'D< F! 'DCN+ 0 s a )&'*Y 7$ ?+.$f- 0 # )&'8 0 'O$
+;5' -/+&1).-/+ 0 -R< E ) 4+&53 -RCDC ) # 0E + +&5' 0 ':' # $ @ +&5' 0 -=<Z ) # E 0E
+;' P 5' 0 'DCN+ 0 s P ) Y # 'O$ - $,13g3g-=) F @ +&5' # %$,C:1$,$ 0 0 +&5'
@K 1) P )&'DC:' # 0E $('DC*+ 0 $ - 0 # # 'O$,CN) ?2d'D$U+&5' 0 -=<M-R< E ) 4+&53 @ ) -=+&1);- r
+ 0\ %$&$('DC*+ 0 - 0 # %3 P < CD-/+ 0 8 0 -R<4< F! 0 'OCN+ 0 s H6 ' - PP < F
+;5 %$-=< E ) ?+;53 + P ) Y!' +;5' '*l %$ +;' 0 C:' @ - -R3 %<4+ 0 %- 0 P < FVE0 0 # 1C r
+ ?Y!'D< F!j 'JC 0 C:<41 # '\2 F P ) Y # 0E -S$(5 )&+)&1 0 + %3g' - 0 -R< F $ 7$ 0 'OCN+ 0 s V
- 0 # E 4Y!' $ 3 ' 0 -R< )&':3U-=),ms$9)&' E -=) # 0E -=3 4<4+ 0 7- 0 P < FVE0 $ @ ) +&5':)
+ F P 'O$ A@ 0 P 1+ I2 'OCN+;$ 0` 'DC*+ 0 s :s
3.2 Frame Polygons
5' C:' 0 +,);-=< 2I'DC*+ 0 +&5' -=< E ) ?+&53 %$M- P < FVE0b6 5 7C.5 0 ?+ %-R<4< F %$ 1$ +
+;5' C 0 Y!':l 5V1<4< A@ +;5' 0 P 1+M$(' E 3g' 0 +;$21+ 0 +&5'X' 0 # 2d'DC 3 'O$S+&5' - r
3 %<?+ A0 %- 0 P < FsEA0 +&5-/+ 7$f+&5' 0 -R< E -=< A@ +&5' 6 5 <4' C 0 $ +,);1CN+ 0 n0
+;5 %$ $('DC*+ A0 6 ' @ )&3U-=<%< F # 8' 0 '+;5' P ) P ':),+ %'D$ A@ +&5'D$,' P < FsEA0 $\+&5-=+ 6 '
CD-=<%< 78 59( j ' 6 %<4< 3U-/m!' $(1);' +&5-/+9+;5' 0 +;'*)&3g' # %-=+&' P < FVE0 $L2d':< 0E
+ +&5 7$C:<%-$,$L+&5) 1 E 5 1+ +&5' -=< E ) ?+&53 (
3
$ B 5 B 5
D
$K&'
#
3( B 43F5
D
Definition
3.3
P
=
(V,
E)
frame
7 A( 5 D 3( 3 %< 5 ( 5 & 85 ( :D $ - ? K( ?=5 # 4 3=&
S
=70 "507 59(
3.2. Frame Polygons
z
y
$%$ #&%$' $'(
z
y
* ) $%)2$'(
Figure 17:
y
z
!)$'(
? K( 59( 3
?=5 B (87
B z
y
!
$'(
z
y
$%"&'$ ) $' (
y
z
$ $% "#%$&@
Chapter 3. Hamiltonian Polygons
V(S)
P
3( (
(P)
+&7E
?
1
7$ (S)
'507 65
@"5
7 ( 4 B 5 < ?=50 =
? 5 =& 7
v
P
V
D < 87 :D
<6587 ( 5 B (
< 4 (870 B $ B 45 >=&# 59(
v
P
? ; ( 7
'
507 65
v V
@ 5
7 ( 785 ? 4 B 5<
P
P = (. . . , pi = v, . . . , pj = v, . . .)
8K ?
7
:D
5
B '
](pj
( 5
:D
uv S
?=85 5 B 4 - '587 5
4 ?' 5 4507 5
P
single vertex
P
1 , v, pi 1 )
](pi
1 , v, pj 1 )
785 (870 B $ B '
5
3(
:D
? 50 B 5
=
9$
V
/ V
P
u
v
v
@ 5
7 ( $ B 5 3
:D
3(
u
P
]P u
? ;
@ 5 7 ( 4 B 5 3
v
P
double vertex
V
( 785 5070785ED K(
? ; @ 5
7 ( $ B 5 3 P
8 E 1)&' : 4<%<%1$ +,);-=+&'D$S+&5' # T '*)&' 0 + P ) P ':),+ %'D$L< %$ +;' # 0 'E 0 4+ 0 s s n0
;+ 5' E 1);'D$ # 12<%'Y!':),+ 7C*'O$J-/);'M1$,1-=<%< F 0 # 7C:-=+&' # 2 F + 6 $(< E 5_+;< F $(5 @ +&' #
C ?).C*<%'D$\+ 0 CN)&'O-R$('9+&5' 4) Y %$ 42 %< 4+ F @ )&3U-=<%< F 2 +&5 C:C*1),)&' 0 C*'D$ @ - # 12<4'
Y!':),+&':l C ),)&'O$ P 0 # + +;5' $,-R3 ' P 0 + 0 +&5' P <%- 0 ' 2$(':),Y!'+&5-/+ C A0 Y (∂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 -=<%< /6 1$+ C 0 +,) <
0 13q2d':) - 0 # + F P ' A@ )&' ':l Y!'*)&+ %C:'D$+&5-/+ - PP 'O-/) 0 +&5' @ ).-=3g' >0 +&1 4+ ?Y!'D< F
$ P 'O-/m 0E C 0 Y!'*l - 0E <%'D$ -/)&' # 'D$ 4);-/2<4' E 4Y!' 0 +;5-/+ 6 'U-/);'U5'D- # 0Eb@ ) C 0 Y!':l # 7$,$('DC*+ A0 0 +&5'f' 0 # " $ i ) P ':),+ F $(1 EAE 'O$ +;$ +&5' + 1) P A@ - @ );-R3g' P 3U- F Y 7$ 4+
$,' E 3g' 0 +9' 0 # P 0 +.$L+ 6 7C*' Z1+ 0 +&5'X' 0 # 6 ' -/)&' 0 +&':)&'D$ +;' # 0 - $ %3 P <4'
P < FVE0 +;5-/+ 7$ _6 '5-DY!'+ 3U-/m!'$(1)&'9+&5-=+ @ )J'*Y!':) F # 12<4'9Y!'*),+;'*l 0 '
A@ +;5' C:C*1),)&' 0 C*'D$ 0 P C:- 0 2d'':< 43 0 -/+&' # 0 i 5-R$,' A@ +&5'-=< E ) ?+&53 ' 0 C:'o<%'*+1$S< _ m -=+S+&5'O$('oY!':),+ 7C*'D$L3 )&'oC:< $('D< F!
E( D 507 D F5 '507 65
7E 5 ?=50 05 =
? 587
Proposition
3.4
b
P
@ 5
7 ( ; F5
K(8 B 5 K( 87 5 5 '507 65 <
7
@ 5 79( 4 B 5
bK( ; '507 653
P
b
P
V
3.2. Frame Polygons
pj+1
pj−1
α
v
pi−1
s
1
pi+1
#1 .$'
& ) !
V(s) * (P)
s
*) #
.$'& v
v
#1 . $'& "" 0$%1 $ #
P
./
pj+1
pj−1
v
α
u
v
pi−1
* pi+1
#1 . $'& ) ! α > π Figure 18:
#
59(
7
# .$'& $' u
v P
7E 50(
#./
Chapter 3. Hamiltonian Polygons
" C:C ) # 0E + hi ) P '*)&+ F 0 Y!'*)&+&'*l @ P 3U- F - PP 'O-/)X3 )&'
0 6 %C:' 0 P JpJ0 $ # '*9) - # 12%< 'fY!'*)&+&'*l b - 0 # %< '*+
Proof.
+;5- +
P = (. . . , a, b, c . . . , d, b, e, . . .)
$,1C.5h+&5-/+ ](c, b, a) 7$ 0 +L)&' '*l @ # ' E (b) = 2 +&5' 0 b 7$ 0 C # ' 0 + + + 6 C*1+ r ' # E 'D$ @ P +&5-=+ %$ c = d
- 0 # a = Pe kp <%'D-=)&< F ' ?+&5'*) ](c, b, a) ) ](e, b, d) = ](a, b, c) 7$[)&' '*l
)S2 +&5b- 0E <%'D$-=)&' -/+ @ # ' E (b) = 3 +&5' 0 b %$ 0 C # ' 0 +[+ '*l-RC*+&< F`0 ' C*1+ r ' # E ' 0 P - 0 #
6 4+&5 1+ ](c, b, a)
+;5' 0 cb %$h- C:1+ r ' # E ' A@ P +;5-/+ %$ 0 + 0 C # ' 0 +h+ +&5' 0 0 ?+;' @ -RC:' 0 C 0 +,);- # %C*+ 0 + +&5' @ -RCN+g+&5-/+ P 7$U- P < FsEA0c ' 0 C*' ](c, b, e) <
- 0 # ](e, b, d) = ](e, b, c) %$);' ':l ](c, b, a) π
+&5':) 6 %$,' 6 ' 5-:Y!' # ' E (b) = 4 - 0 # 2 Fì ) P $ ?+ 0 4+ %$[' ?+;5'*)
P )
d, e d, e (]) (c, b, a)
(]) (a, b, c)
pJA0 $ # '*)\+&5'SC:-$(' d, e (a, b, c) 5' 0 2 F i ) P $ 4+ A0 A +;5'
C ?).C*1<7-/) ) # ':) @ a, c, d, e -=) 1 (])
0 # b 7$ a, e, d, c -R$ $(5 /60 0 8 E 1)&' D n- .
':l+ (P) - 0 # (a, b, e)
n0 P -=),+ 7C*1<7-/) 2 +;5 (d, b, c)
(b)
ε
(])
(])
*
'
V
l
+
H
)
(
$
1
gC
4'
&
+
<
,
$
g
3
R
4
<
<
V
5
1
$
&
+
5
' C:1),Y!' +&5-=+
M
K
@
0
F
(P)
ε
>
0
(b)
ε
+&);-DY!'*);$('O$+&5' C*< $(' # 00sr C*) $,$ 0E $(12+,);- %< (b, c, . . . , d, b) @ P 1 0 @ )&3g< F
7$ 0 + 0 1<4<%5 3 + P %C 0 C 0 +,).- # 7CN+ 0 + +;5' @ -RC*+f+&5-=+ P -$o- @ ).-=3g' %$
$ 43 P < F_r C A00 'DC*+&' # M0 +&5' +&5'*)[5- 0 # @ d, e (])(c, b, a) +&5' 0 2 F i ) P $ ?+ 0 +;5' C 4);C:1<%-=) ) # ':) A@ a, c, d, e -=) 1 0 # b %$ a, c, d, e -$ $(5 /60 0 8 Er
1)&' D 2 . >0 P -=),+ 7C*1<7-/) b - PP 'O-/);$9-R$- );' ':l Y!'*)&+&'*l 6 4+&5 0 (d, b, e) >0 $(13g3U-/) F 6 ' 5-:Y!'X$(5 /60 +&5-/+ b - PP 'D-=);$9-=+<4'O-R$ + 0 C:' -R$9-q)&' '*l
Y!':),+&':l 0 P 1 0 <4'O$,$ b 7$ 0 C # ' 0 +S+ + 6 C <%< 0 'O-/)[C*1+ r ' # E 'D$ A@ P +&' +;5-/+X- Y!':),+&'*l`3U- F - PP 'O-/) + 6 7C*'g-R$ - )&' ':l !Y ':),+&':l 0 - @ );-R3 '
-$+&5'o':ls-R3 P <4' 0 8 E 1)&' D n- $(5 /6 $ Z1+ i ) P ':),+ F P 1;+ $SC:'*),+.- 0
);'D$ +,) %C*+ 0 $ 0 +&5' E ' 3g'*+,) 7Cf$ 4+&1-=+ 0 -=) 1 0 # ($ 1C.5H- Y!'*,) +&':l 3.3 Saturation
5 %$$('DC*+ A0 @ C*1$('O$ A0Ui 5 -R$,' MA@ +&5'L-R< E ) 4+&53 5' E -=< A @ +&5 7$ P 5 -R$('
7$f+ 3 # @ F +&5' C*1),)&' 0 + @ ).-=3g' P < C:-=<%< F 0 ) # '*)o+ 0 C:<41 # k
' +&5' &+ 5':)
3.3. Saturation
d
P
e
P
c
P
d, e
Figure 19:
d
P
a
a
b
e
b
(]) (a, b, c)
5 '507 65
D d, e
c
(]) (c, b, a)
@ 5
79( B 5 K( 785 5
'507 65 ' 0 # P 0 + A@ +&5 $('g$(' E 3g' 0 +;$ @K ) 6 5 7C;5 0 < F 0 'g' 0 # P 0 +X- PP 'O-/);$ -$ Y!':),+&':l A@ P 5'k$(' E 3g' 0 +;$+&5-/+ < 4' C 3 P <%'*+&'D< F 0 +&5' 0 +;'*) ) @ P -/)&'
0 +5- 0 # <%' # 0 +;5' -=+&1);-=+ 0 i 5-$(' -R<?+&5 1 E 5H$ 3g' A@ +;5':3 3 E 5+S2^'
&CD-=1 E 5+ 2 F
-RCDC # ' 0 +;-=<%< F >0 E ' 0 '*);-R< +&5' @ );-R3 ' 6 4<%<J2d'OC 3g' - 00sr
P
$ 43 P <%' P < FVE0 # 1) 0E +&5 7$ P 5-R$(' +&5-=+ 7$ $ 3g' # 12<%'MY!':),+ 7C*'O$J3U- F 2^'
C*)&'D-=+&' # pJA0 $ # '*)- $(' E 3g' 0 + p q S @K ) 6 5 7C;5 p V(P) - 0 # q / V(P) >0
+;5' @ <%< /6 0E 6 ' CD-=<%< $(1C.5Hi - $(' E 3g' 0 + ( ; i7E;65
Dh2 F P " 0 -=< AE 1$(< F $,' E 3g' 0 +S+&5-/+ %$ 0 +91 0 $,-=+&1);-=+&' # 2 F P 7$S)&' @ '*),);' # + -$ ( ; 7E;65
D S 43 r
%<7-/)&< F - Y!'*),+;'*l 0 V(P) 7$[C:-R<4<%' # ( ; 78;5
D @ 4+;$ 0 C # ' 0 +[$(' E 3 ' 0 +
@ ) 3 S 7$ >1 0s+r $,-=+&1);-=+&' # 2 F P 1'+ i ) P ':),+ F B6 '[m 0/6 +&5-=+ @K )
':Y!'*) F 1 0 $,-/+&1);-/+&' # $(' E 3g' 0 + p q S 4+;$ ' 0 # P 0 + p 7$L- $ 0E <%'[Y!'*),+;'*l A@
i
i
,
$
1
.
C
h
5
&
+
5
/
+
7$
C
!
Y
*
'
l
0
P
]P p i
5' P <7- 0 %$+ 0 +&' E );-=+&' q -R$9- Y!'*),+;'*l 0 + P 2 F 1$ 0E +&5'X' # E ' p q
- 0 # @ ) 3 q E 0E 2-C&m + P Y %-- E ' # 'O$ 7C n0X@ -RCN+ +;5'*)&' -=)&' + 6 P $,$ ?2i<%'
6 - F $L+ C 0 + 0 1' @ ) 3 q + /6 -=) # $ p )9+ =6 -/) # $ p 5':)&' @K );' 6 '
i 1
-$,$ E0 - 0b ) %' 0 +;-/+ 0 + 'D-RC.5bY!':),+&':l i 0 1P 7 78 5 "#%$&'
(
Definition 3.5
P
orientation
u(P)
7 '587 5 D 58 K65 9$
u : P → {−1, +1}
pi
P
Y (p ) := pi 1 , u(pi ) = −1,
u i
pi 1 , u(pi ) = +1.
?=5 '
587 5
towards
4 ?' B ?
p
(
oriented
B 5 ' ) 4' 0 +;-=+ A0 A@ -gY!':),+&'*l # *' +&':)&3 0 D' $ 6 5 7C.5 0 C # ' 0 +o' # E ' A@ P %$[);' r
P <%-C*' # 6 5' 0g6 '9$,-/+;1);-/+;'M- P $,$ 42< F 0 C # ' 0 + 1 0 $,-/+&1;) -/+&' # $(' E 3g' 0 + 1$ 0E
Chapter 3. Hamiltonian Polygons
+;5' @ <4< /6 0E < C:-R0 +&5' E 1)&'D$ ;+ 5' ) 4' 0 +;-=+ A 0 A @ - Y!':),+&'*l
6 %<%< A@ +&' 0 2^' 0 # %CD-/+&' # 2 F -o($ 3U-=<%< P <%1$ ) 3 0 1$ r $ F 3q2 < 0 $ # 9' +&5' # 7$,C
+;5-/+S)&' P )&'O$(' 0 +;$ +&5'oY!'*),+&':l <& 785 Operation 1 (Build-Cap(P, u, pi ))
- @ );-=3g' - 0 ) 4' 0 +;-=+ A0
- 0 # - 0 1 0 $&-/+&1).-/+&' # $(' E
P
u(P)
@ ) 6 5 7C;5 p V(P) pi q S
i
"507E; 2+;- 0
@ ) 3 P 2 F )&' P <%-C 0E +&5'M' # E ' (p , r) 2 F +&5'
P
i
'
Y
5
:
'
&
)
'
:
'
+
E
6
@
(p+&i'*,) q) )S Y!'*)&+ (q,
p
r
:=
u(p)
:=
u(p
i , r)
i)
u (pi )
%C:'D$ A@ E ' (q, p , r) 0 C:<41 # 0E q 0 i
(P , u)
q
3 '0+
P /- +&5
)q-=<%<
q
pi−1
pi
pi−1
pi+1
P
pi
pi+1 = r
u(pi ) = +1
q
pi−1 = r
pi+1
pi
) Figure 20:
) -/%'+;'*0 ) +;-=6 + ' 0 6
7E
u(pi ) = −1
5
:D ?=5 7850( P
(E(
F570 58 +; ( pi
D
(P, u, pi )
7
?=5 4 %<4<X1$(' +&5' - # # 4+ 0 R- < # ' E );':' A@X@ );':' # 3 P ) Y # ' # 2 F &+ 5'
+ 3U-=m!'X$(1);' &+ 5-/+M- C*':),;+ - 0 lV' # ' # E ' A@ +&5 ' 0 ?+ 7-=< C A0 Y!*' l
3.3. Saturation
5V1<%< @ );-=3g' 6 %<%< 0 '*Y!'*)S2d'X)&' P <%-C*' # 0 +&5' C 1);$(' A@B 1)9C 0 $ +,)&1CN+ 0 " $
':l P <%- 0 ' # 0 'OCN+ 0 s +;5 %$ %$ %3 P ),+;- 0 + @ ) +&5' 0 # 1C*+ 0 $(+&' P 8 )
0/6 6 ' 1$ + 6 ),m 6 ?+&5 -q1 0 @ )&3 ) %' 0 +;-=+ 0 u +1 +&' +&5-=+ P %$ 0 + 0 'OC*'D$&$,-/) %< F $ %3 P <%'q2d'DC:-R1$('k$ 3g' A@ +&5' 0 +;'*) )
Y!':),+ 7C*'O$ @ ) 3 E ' (q, p , r) 3 E 5+-=<4)&'D- # F 5-DY!'f2d'D' 0 0 V(P) i
?=5 Proposition 3.6
P
D
7E
( 5
P < FVE0 + $ +;-/)&+ 6 4+&5 - 0 # +&5V1$B2 +;5 - 0E <%'D$
F i
p
P
i
- 0 # ](p , p , q) -=)&' $ +&) %C*+&< F C 0 Y!'*l X ' 0 C:' p C:- 00 +
](q, pi , pi 1 )
i
i 1
C:C:1)c-$- Y!'*)&+&'*l A@E ' (q,
0 P -=),+ 7C*1<7-/) p - 0 # q -/)&' $ 0E <4'i Y!':),+ 7C*'D$
p
,
r)
A@ P - 0 # {p , q} %$ 0 + - C:1i + r ' # E ' A@ P " $o-=i<%<\' # E 'O$ 0 $('*),+;' # 0 + P
-R< 0E E ' (q,ip , r) 2 1 0 # +&5' 0 0 4+&' @ -RC*' P 7$o- P < FsE0 p <4'O-/)&< F P
7$$ %3 P < F_r C 00 i'DCN+;' # - 0 # @K <%< /6 # ?)&'OCN+&< FH@ ) 3 +&5' # 8' 0 4+ 0 A@ E ' # 'O$ 7C:$f- 0 #
@ ) 3 +&5' @ -CN+9+;5-/++;5' 0 P 1+ P < FVE0 P 7$9- @ );-R3g' +)&'D3U- 0 $S+ C.5'OC&m
P ) P ':),+ %'D$ I &
':+ E ' (q, p , r) = (q = q , . . . , q = r) @ )X$ 3g' k j ' 6 %<%<
-=) E 1'U$(' P -/);-=+&':< F i @ )X'O-RC.5 @ +&05' Y!'*)&+ %Ck:'D$ 6 5 $(' 0 ' E 52 )&5 _ # C.5- 0E 'O$
@) 3 P+ P 7$- $ 0E <%' Y!':),+&'*l 0 P L
51$ - 0 # " $9-=) E 1' # -/2 Y!'
pi
pi
5 < # " $J+&5'f$(' E 3 ' 0 + @ ) 3 S 0 C # ' 0 ++ p 7$L- 0 ' # E ' A@ P i ) P r
i
':),+ F 7$ @ 1< <%<4' # -R$ 6 ':<%< Proof.
Z )
P ':':),++ F 1$ ); $( +C 0 %$Y -f0 C C:' 0 Y!1'*).l $(':$ <4Y!0'DE $B<4'+&5Y!-='*+)&+&P'*l 7$A\@ q
r
qj
-/) E 1'X-R$ @K )
pi
p %< 'D-=)&< F r - P P 'D-=);$ - $ A@ +&' 0 0
%3 P < 4'O$ I ,
@ )
P
-$ 0
P
-0#
]P r < ] P r
5 7$
'*),+&':l q 7$ 0 $('*),+;' # -R$q-bC 0 Y!':l Y!'*),+&':l +&5-=+ 7$ - 0 # 4+;$L);' j ':l # 3U- 0 (q , q , q ) 7$ 0 +&' r
1 -=<%j<
%' 0 +&< F j $(3U
P - 0 # ':l+;'*) ) + P 0 $ 3g'$(1 UC (])
0 ' j E 152 )&5 _ #
A@ q 5' ,'*lV+&':) ) + P Rr P -/),+ )&'D- # 4< F 43 P < %'D$ &6 5 %<4' +&5'
0 +;'*j) )J+ +&5-=+
=r P -=),++&':<%<%$1$ + E '*+;5'*) 6 ?+&5 i ) P '*),+ F A@
P
P
@ q - PP 'D-=);$ 0 P +&5' 0 4+ 7$\-[$ 0E <%'L)&' '*lqY!':),+&'*l i ) P '*),+ 4'O$ - 0 # j @ <%< =6 %3g3g' # 7-/+&'D< F!
1
]
) P)\q+ j
%$MjC <0 Y!k'*l
1
a
Chapter 3. Hamiltonian Polygons
5' -=+&1);-=+ A0Ui 5-R$,' %$$(13g3U-/) %e:' # 0 +&5' P ':);-/+ 0 2^'D< =6XB +\C 0 $ 7$ +;$
A@ 4+&'*).-/+&' # - PP < 7C:-/+ 0 $ @ Z\1 %< # r p - P + +&5' C:1),)&' 0 + @ ).-=3g' P 1 0 + %< -=<%<
$,' E 3g' 0 +;$ -/)&'$,-/+;1);-/+;' # 2 F P " +\+&5 7$ P 0 + '*Y!'*) F $(' E 3g' 0 + @ ) 3 S ' 4+&5'*)
5-R$L2 +&5h' 0 # P 0 +.$ 0 V(P) ) 4+< %'D$C 3 P <4':+&':< F 0 P Operation 2 (Saturate(P, u))
9- @ );-=3g' - 0 # - 0b ) %'
P
"507E; " $o< 0E -R$ +&5':)&'
0 ;+ -=+ 0 u(P) ':l 7$ +;$X- 0 1 0 $,-=+&1);-=+&' # !Y '*),+;'*l
4< ':+ (P, u) ← Z\1 4< # r p - P (P, u, p ) i
(P, u)
pi V(P)
5' @ <4< /6 0E %$S- 0 43g3g' # %-=+&' C 0 $,' 1' 0 C:' A@ i ) P $ 4+ 0 V as
Corollary 3.7
?=5 3
; E7 ;65 ( 7E
5
5 ' ':ls-R3 P <4' # ' P %C*+&' # 0 8 E 1);' s $(5 =6 $U+&5-=+ -=+&1);-=+&' 3U- F CN)&'O-/+&'
# 12<4'LY!':),+ 7C*'O$-$ 6 ':<%<-$ C*1+ r ' # E 'D$ 0 +;5' @ );-R3 ' B
5 %$ 7$ 0 + - P ) 2<4'D3 -$L+&5'O$(' @ 'O-/+&1)&'D$ 6 %<4<c2d' # 'O-=<4+ 6 4+&5 0 +;5' %3 P < C:-=+ A0 i 5-$(' " +k+&5 7$ P 0 + 6 'hC 1< # ' 0 # +&5 7$ $('OCN+ 0 -$q+&5' # 'O$,CN) P + 0 A@ +&5'
-R< E ) 4+&53 @K )[+&5' -/+&1).-/+ 0 i 5-$(' 7$fC 3 P <%'*+&' oS/6 '*Y!':) 0 - 0 + %C P - r
+ 0hA@ +&5' C 3 0E P 5-R$('O$S<%'*+1$S- 0 -=< F eD'o+&5' 0 13q2d'*)- 0 # + F P ' A@ )&' '*l
Y!':),+ 7C*'O$ +&5-/+ -/+&1).-/+&'L3U- F CN)&'O-/+&' L' '*l Y!':),+ 7C*'D$-/);' 43 P ),+;- 0 +-$ +&5' F
3U- F 2d' );'*Y %$ 4+&' # 2 F - E ' # 'O$ 7CJ);'D$(1<4+ 0E 0 - # 12<4'Y!'*)&+&'*l A@ +&5' @ );-=3g' " $ 6 ' 5-DY!' + # 'D-R< 6 4+&5 # 12<4' Y!'*),+ 7C*'O$ # 1) 0E +&5' 43 P < C:-/+ 0ki 5-R$,' 6 ' 0 'D' # + 5-DY!' $ 3g'oC 0 +,) < 0 +&5'D3 7 # $&@
( I; ?
( Definition
3.8
P
(p
,
.
.
.
,
p
)
P
reflex
i
k
:D %$ :D
705 0K? 785 5 3 :D
( ; 7
twin
pi
pk
P
pj
4 i<j<k
p %< 'D-=)&< F +&5' 0 ?+ %-R< @ );-R3 'oC 0 Y (∂S) # 'O$ 0 +SC 0 +;- 0 - 0_F );' ':lU+ 6 0 +
# 'D$ 0 + 5-:Y!' $ )&' '*l Y!':),+ 7C*'O$ Z1 4< # r p - P P ) # 1C*'D$g':ls-CN+&< F 0 '
0 ' 6 )&' '*l`Y!'*),+;'*l -/+ q 2$(':),Y!' +;5-/+o+;5'qY!'*)&+ %C:'D$ 0 E ' (q, p , r) -=)&'
) %' 0 +&' # &- 6 - F g@ ) 3 +&5 %$ 0 ' 6 )&' '*lbY!':),+&'*l [
E '*+&5':) 6 4+&5 -U$(i1 4+;-/2<4'
) %' 0 +;-=+ 0 +&5 %$X-:Y # $[)&' '*l`+ 6 0 $ gQ )&' P )&'OC 7$(':< F 6 'gC:- 0 2 1 0 # +&5'
0 13q2d':) @ );' ':l + 6 0 $ 0 +;'*)&3U$ A@ +&5' 0 13q2d'*) A@ -=<4+&'*) 0 -/+ 0 $ 0 +&5'
) %' 0 +;-=+ 0
3.3. Saturation
pi
!
pi
*) Figure 21: 3
pj
! $%& pi
!$%& "
pj
(P, u, pi )
"
pj
(P, u, pj )
; E7 ;65 $ B 785
;65 B
5
D&@50( <2 ?=5
7E
5
Chapter 3. Hamiltonian Polygons
< 7950+;
7E 5
( I37
Definition
3.9
alternation
u
P
4 B (5 B '5 '587 B 59(
<
( B ? ? ;
(p
,
p
P
u(pi ) = +1
)
i
i
1
D
785 ( ; 7E;5
D :D 8K ?
u(pi 1 ) = −1
pi
pi 1
(
D 507 78 5
4 ? 70 50+;
;
Proposition
3.10
P
u
3
?=50 < ?=5 7859( <=& 7E 5
?=50785 3 (;
7E;65 ( @ 5
D>
(P, u) 7 5
? '85 7 5 P
(8 5 5 4 785 5 4 3 3
( B ?>? ;
B
v
P
( ( ; E7 ;5
D :D 79 5865
D 4 7EDK( 785 5 4507 65
3
v
r
P
32 ?'3( B K( 5
3 ( I7 ?=5785 5 4 < A ( B 785
;65
D
r
7
( "7 45070:; 3
:D
v
u
u(v) = +1
5 4
4 <( 3
P
705 ?=( 5 4 ?' B ?!785 K 4%( =7859( 50 3
P
pJ0 $ # ':)[- $ 0E <%' Z1 %< # r p - P - 0 #`# ' 0 +&' +&5' P -/),+ 7C P -=+ 0E Y!'*) r
-=2 Y!'H2 F p q - 0 # r '*+ Q # ' 0 +&'H+&5' 0 P 1+ @ );-=3g' - 0 #
<%'*+ Q # ' 0 +&'[+&5' 1i+ P 1+ @ );-=3g' A@ +&5 %$ P -=),+ 7C*1<7-/)oZ1 %< # rp - P P '*);-=+ 0
" $,$(13g' 6 4+&5 1+S< $,$ A@ E ' 0 ':);-=< ?+ F u(p ) = +1 i
5' P '*).-/+ 0 C*)&'D-=+&'D$X'*l-RC*+&< F 0 ' 0 ' 6 )&' '*l Y!'*)&+&'*l c0 -=3g'D< F -=+ q 5-$\+ 6 0 ' E 5_2 );$ 0 Q 0 ' @d6 5 %C.5 7$ p Z Fgi ) P '*),+ F 1 '*),+;'*l
q
i
%$
q
C
!
Y
:
'
l
!
Y
*
'
&
)
&
+
*
'
l
d
2
O
'
:
C
=
1
(
$
'
?+ 7$q$ +,) 7CN+;< F
#
0
0
0
p
Q
]
p
<
]
p
i
P
i
P
i
C 0 Y!':l 0 Q [>@E ' (q, p , r) * qr +;5' 0 +&5':)&' %$[- 0 0 +&'*) )MY!':),+&':l A@
+;5' E ' # 'D$ 7C\+&5-=+ %$ $(+,) 7CN+&< F iC 0 Y!'*l 0 Q + 3 E 5+2d'J- # 12<%'\Y!'*),+&':l 21+L+&5' P -=),+ 7C*1<7-/) C:C:1),)&' 0 C*' -=< 0E +;5' E ' # 'D$ 7C 7$-=< 6 - F $C 0 Y!'*l +\)&':3U- 0 $ + C 0 $ # '*)\+&5'C:-R$,'S+&5-=+ E ' (q, p , r) qr \ 1 PP $('L+;5-/+
7$)&' '*l 0 Q +&5-=+ %$ p 7$ ) %' 0 +&' # + =6 -/) # $Si -q)&' '*l Y!'*),+;'*l r
i
@ r 6 -R$S-R<?)&'O- # F P )&'O$(' 0 + 0 P +&5' 0 +&5':)&'f3 1$ +L5-DY!'[2d':' 0 - 0 1 0 $,-=+ r
1);-/+&' # Y!':),+&'*l ) %' 0 +&' # + /6 -=) # $ ?+ pJ0 $ # '*) +&5' P 0 + 6 5':)&' p 2^'OC:-R3 '
- 0 ' E 52 ) @ r 5 7$ 3 E 5+ -=<4)&'O- # F 2d'+&5'[C:-$(' 0 P @0 + +&i5' 0 p %$
Y 7$ ?+;' # 2 F - E ' # 'O$ %C 0 $ 3g'kZ\1 %< # r p - P P ':);-/+ 0 -PP < 4' # + - Y!':),i+&'*l
8 ) p + 2d'OC 3g' - 0 ' E 5_2 ) A@ r -R$[- )&'O$(1<4+ @ +&5 7$ P ':);-/+ 0 p
p
k
3k1$ +f2d' i) %' 0 +&' # + /6 -=) # $ r k ' 0 C*' 6 ' 3U- F C 0 C:<41 # ' 0 # 1C*+ ?Y!'D< F +&5-=k+
+;5'*)&' 7$ - Y!'*),+;'*l v 0 P +&5-/+ 7$ ) %' 0 +&' # + /6 -/) # $ r - 0 # +&5' P +&' 0 + 7-=<
);' ':l + 6 0 (p , r) 7$C Y!'*)&' # 2 FHpJ0 # ?+ 0 R
+&5':) 6 %$,' i r 5-R$k2d':' 0 CN);'D-/+;' # 0 +&5 7$ -=+&1);-=+&' $(+&' P 2 F - P )&'*Y 1$
Z\1 %< # r p - P P ':);-/+ 0 " +[+&5-=+ P 0 + r 6 -$[' 0 # P 0 + A@ - 0 1 0 $&-/+&1).-/+&' #
$,' E 3g' 0 + p r @ ) 6 5 %C.5 p 6 -R$ - Y!':),+&'*l A@ +&5' @ );-R3g' - 0 # u(p ) = −1 pJ0 $ # ':) +;j5' +&5'*) = p j 0 ' E 52 ) p A@ r # 4)&'DC*+&< F - @ +&':) +;5 %$JZ\j1 4< # r p - P
j
k
Proof.
+ %C:'D$ -$
3.4. Dissection
P *' ).-/+ 0 >@ p %$ $,-=+&1);-=+&' # -/+ +&5 7$ P 0 + ) @ u(p ) = −1 +&5' 0 +&5'
' # E ' p r )&'D3U- 0 k$\- 0 ' # E ' @ +&5' @ );-R3 '+&5) 1 E 5 1+ +&5 7$k -=+&1);-=+&'S$ +&' P 0
C 0 +,).- #k 7CN+ 0 + p 2d' 0E - 0 1 0 $,-=+&1);-=+&' # 0 ' E 52 ) A@ r -=+o$ 3 ' P 0 + 5'*)&' @ )&' ?+ 7$ u(pi ) = +1 - 0 # 6 ' @ 1 0 # - 0 -=<4+&':) 0 -/+ 0 0 +&5' @ ).-=3g' " $ Z1 %< # r p - P P );'D$(':k),Y!'D$ ) %' 0 +;-/+ 0 $ +&5 %$ -R<?+;'*) 0 -=+ A0 3k1$ +q5-DY!'U2d':' 0
P )&'D$(' 0 + 0 P -R<?)&'O- # F - 0 # 4+ 7$ # 'D$ +,) /F ' # -=+J+&5' P 0 + 6 5':)&' r %$JY %$ 4+&' #
- 0 # 5' 0 C:' $&-/+&1).-/+&' # >0 $ +&'D- # @ ) 3 +;5 %$ P 0 + 0 +;5'*)&' 7$M- Y!'*)&+&'*l +&5-=+
7$ ) 4' 0 +&' # + =6 -/) # $ - )&' '*l Y!'*)&+&'*l 0 +&5' @ ).-=3g' 6 5 %C.5 3g- F <%'D- # + -=+
3 $ + 0 ' 0 ' 6 )&' '*l + 6 0 -$S'*l P <7- 0 ' # -/2 Y!' n0 P -=),+ 7C*1<7-/) i ) P $ 4+ A0 s7D + E '*+&5':) 6 4+&5 pJ ) <%<%-=) F s %3 P < %'D$J+&5-=+
+;5' );'D$(1<4+ A@o -/+;1);-/+;'h- PP < 4' # + +&5' @ );-R3g'hC 0 Y (∂S) 6 4+&5 -`1 0 @ )&3
) %' 0 +;-=+ 0 u +1 %$S- @ );-=3g'f+;5-/+ # 'D$ 0 +C 0 +;- 0 - 0F );' ':l + 6 0
3.4 Dissection
5 %$L$('OCN+ 0 # %$&C*1$,$,'D$L5 /6 + # %$&$('DC*+S+&5' @ );-R3g' P
+;5-/+ '*Y!' 0 +&1-R<4< F C:- 0 2d' P ) C*'O$,$(' # 0 # 1C*+ ?Y!'D< F! C:1),)&' 0 + @ );-R3 ' P 6 4<%< 2d' # ' 0 +&' # 2 F " $ @ )M+;5'
-R<%$ $,'*Y!'*).-=< P ) P ':),+ %'D$L+;5-/+ 6 ' # ':3U- 0 # @ ) 3 Definition
3.11
D3(8( 5 B 5 4
5070$ "#%$&'
5 4
5070$ "#%$&'
(
D 05 7 7E 5
7 ( 5 P
S
3(
D $ P
nice
( 50 ?=507 (
3
? K( B D
D
0 + C 0 Y!'*l P < FsE0 $
5' # %$&$('DC*+ 0 A@ +&5'
P < FVE0 P +&5':)&'q-/)&'
D3( 3 ( 5 & 58 (
57 %< 5 (5 & 50
5ED&'5 4 ?
P
?=507853( 785 5 4 <
D
7A5 '5879$705 5 4507 6 5 r ( 5 D ?=50785 3 ( 3 B D 50
5
D&'5
? ; ( B 8K?
D
:D ( B ? ? ;
(
rq
D
P
q
B '5 4507 5 D
7 5'5879$ s S s P ?=50C?=50785 3( D ( B ? ? ; ?=587 4 3( 5
743
s D
s D =
D
3
S'DC:-R<4<M+&5-=+
# $&@
6 ' 2^' E 0 6 4+&5 - +,) 4Y 7-=< # 7$,$('DC*+ A0 A@ +&5' 0 4+ 7-=< @ );-R3 '
C 0 Y (∂S) +&5-=+ %$ = {P} Sp <%'D-=)&< F +;5 %$ 0 ?+ 7-=< # %$,$,'DCN+ 0 @ 1< <4<7$ , 1) E -R< 7$\+ -C;5 4':Y!'M- 0 # 3U- 0 +;- 0 - 0 7C*' # 7$,$('DC*+ A0 @
# 1) 0E
P
a
Chapter 3. Hamiltonian Polygons
+;5' 7$,$('OCN+ 0 i 5-R$(' 8 )M+;5 %$ 6 'q3U- F 5-DY!' + 1 P # -=+&' -=<4)&'O- # F # 1) r
0E +&5' -=+&1);-=+ 0 i 5-R$(' >@ ' # E 'D$ A@ +&5' @ );-R3 ' P /- )&'h3 # ' # +&5-=+
-=)&'g-=<7$ ' # E 'O$ A@ - # 7$,$('DC*+ A0 P < FsEA0 D +&5' 0 6 ' 1$ X
+ - PP < F +&5'g$,-R3 '
3 # C:-/+ 0 $ + D -R$ 6 ' # + P 3.4.1 Canonical Dissections
$ +&5' @ ).-=3g'U3 E 5+X2d'DC 3 ' 0A0r $ 43 P <%' # 1) 0E +&5' -=+&1);-=+ 0 i 5-R$,' " X
6 'k5-DY!'k+ # $ 3g':+&5 0E 0 ) # '*)f+ ' 0 $(1)&' ;gpJ0 $ # ':)X- # 12<4'
Y!':),+&':l q A@ P " C:C ) # 0E + gi ) P ':),+ F +;5'o- 0E 1<7-/) # 3U- 0 -/) 1 0 #
0 +;'*);$('OCN+;$ P 0 + 6 $(+,) 7CN+&< F C 0 Y!':l - 0E <%'D$ " $ 6 ' -/)&'k5'D- # 0E @K ) q
C 0 Y!':l # %$,$,'DCN+ 0 +&5 7$[$(':'D3U$f- EA # P 0 +M+ $ P < ?+f+;5' # 7$,$('DC*+ A0 P < Fr
E0 +&5-/+oC A0 +;- 0 $ q 0 + + 6 P < FsE0 $M-=+ q M 0E $ @ )f'D-C;5 # 12<4'
Y!':),+&':l CN)&'O-/+&' # 2Y 1$,< F F %':< # $ - # 7$,$('OCN+ 0 0 + $ %3 P <4' P < FVE0 $ - 0 #
< 0 ' $,' E 3g' 0 +;$ B6 5'*);' +&5' < 0 'h$(' E 3g' 0 +;$qC ),)&'D$ P 0 # + C*1+ r ' # E 'D$ A@ P p <%'D-=)&< Fbi ) P '*),+ F %$M$,-/+ 7$ ' # -R$ 6 'D<4< 6 5 %<4' - 0 # -/);' - 0
'O-R$ F C 0 $(' 1' 0 C:' A@Bi ) P $ 4+ A0 s DV
Z1+ 0 ) # ':)q+ ' 0 $(1)&' 6 ' 5-DY!'U+ C 0 $ # '*) - 0 +&5':)q+ F P ' A@
':Y!' 0 +;$ +&5-/+ 3 E 5+ C:C:1) # 1) 0E +&5' -=+&1);-=+ A0 i 5-$(' ) ':Y!' 0 -=<4)&'D- # F
@ ) 3 +&5'fY!'*) F 2d' E 00 0E +&5' @ );-=3g' CD- 0 5-DY!' $(' E 3g' 0 +;$ @ ) 3 S -R$ # 7- Er
0 -R<%$ ':+X1$o)&' @ ':) + $(1C;5 $(' E 3g' 0 +;$X-R$ (5 & 50 D&':4( ' E 3g' 0 +
# %- EA0 -R<%$93U- F - PP 'D-/) 0 +&5' 2d' E 00 0E -R$ # %- EA0 -=<7$ A@ +&5'XC 0 Y!'*lh51<4<
8 E 1);' A n- . - 0 #b# 1) 0E Z1 %< # rp - P @ - 0 ' 0 # P 0 + A@ - 0 1 0 $&-/+&1).-/+&' #
$,' E 3g' 0 + 7$Y %$ 4+&' # 2 F - E ' # 'D$ 7C 8 E 1)&' 2 6 ) - E ' # 'O$ %CSY 7$ 4+;$\2 +&5
' 0 # P 0 +;$ A@ - $(' E 3g' 0 + +;5-/+ 6 -R$ 0 +&5' 0 +&'*) ) A@ +&5' @ );-R3g'H2d' @ )&'
8 E 1);' C .
s
s
s
Figure 22:
( 5 & 50 D&':4
( s
D?=5 7859(
) 3=& B B 4 D (E( 5 B
" $9- $(' E 3g' 0 + # %- EA0 -R< %$ 0 +C A0 +;- 0 ' # 0 +&5' 0 +&':) ) A@ +&5' @ );-R3 '
a
3.4. Dissection
4+M$(5 1< # 0 + 0 +&'*);$,'DCN+9+&5' 0 +&'*) ) A@ - 0F # %$,$,'DCN+ 0 P < FVE0 -RCDC ) # 0E
+ ; 5 7$ C:- 0 2d' -C;5 4':Y!' # 'D-$ %< F @6 'h$ P < ?+k+&5' # %$,$,'DCN+ 0 P < Fr
E0 $ -R< 0E -R<4<o$,' E 3g' 0 + # 7- E0 -=<7$ 5 %$ - ## 4+ 0 -=<o$(' P -/).-/+ 0 3U-/m!'O$
-R<%$ $,' 0 $(' @J6 ' m!'D' P 0 3 0 # 1) 0 -=< E -R< A@ C 0 Y!'*l # %$,$,'DCN+ 0 P < Fr
E0 $ Z F i ) P '*),+ F - 0 # +&5' 6 - F E ' # 'D$ %CD$f-/)&'q1$,' # 0 Z\1 4< # r p - P
+;5'[' 0 # P 0 +;$ A@ -q$(' E 3g' 0 + # %- EA0 -R<-=)&'oC 0 Y!'*lgY!'*),+ %C:'D$ A@ 2 +&5 # 7$,$('DC r
+ 0 P < FVE0 $9+&5' F - PP 'D-=) 0c ':+1$$,13g3g-=) %e:' +&5'O$(' 2$(':),Y-/+ 0 $ 0
+;5' @ <%< /6 0E P ) P $ ?+ 0
5 B ! B ?' 5'5 B 5 D (E( 5 B ?=5 7E 5 ; ?=5
Proposition
3.12
50:D> ?=5
; 87 ; ? K( 5 0$ (+ <& ?=5 D3(E(5 B $&@ ( ;
#
3
4 D 5 '507 B 95 ( D 4 =& 43 ( 5 & 50 D &@:4%( i ) '*)&+ 4'O$A I -=)&'kC*<%'D-=) -R$ # 7$,C*1$,$(' # -/2 Y!' 8 ) +&5'g' 0 # A@ +&5' -/+;1);-/+ 0 i 5-R$('U+;5'*)&' 7$ 0 1 0 $&-/+&1).-/+&' #
0
$,' E 3g' 0 + 5-/+ 7$ 'D-C.5 $(' E 3g' 0 + @ ) 3 S ' ?+;5'*) 7$ 0 +&5' 0 +&'*) ) A@ +&5'
@ );-=3g' P )k2 +&5 A@ ?+;$k' 0 # P 0 +.$k-/);' 0 V(P) >0 +&5' <%-=+,+&'*)gC:-$(' +&5'
$,' E 3g' 0 + %$ ' 4+&5':)- 0 ' # E ' A@ P )- $,' E 3g' 0 + # 7- E0 -=< - 0 # 0 2 +;5 C:-$('D$
4+ 7$ # %$ 0 + @ ) 3 +;5' 0 +&'*) ) A@ 'O-RC.5 # %$,$,'DCN+ 0 P < FVE0 >0 +;5' @K );3 ':)
CD-R$(' )&'OC:-=<%<M+&5-/+g+&5'H$(' E 3g' 0 +;$ @ ) 3 S -/);' # 7$3 0 + >0 P -=),+ 7C*1<7-/) 0
$,' E 3g' 0 + C:- 0 C 0 +;- 0 - # 12<%' Y!':),+&'*l 0 ?+.$f)&'D<%-=+ ?Y!' 0 +&':) ) 0 )XC:- 0 4+
C*) $,$S-k$,' E 3g' 0 + # %- EA0 -R< \
5 %$ P ) Y!'D$ &
P
Proof.
+;' +;5-/+q-/+
(
D 507 7E 5
D D E( ( 5 B ?=5
Definition 3.13
P
P
4 ? 7859(+"5 B 6
3( + 3 5 D 87 K
( #
canonical
dissection
P
F 4 ( A?=5 D (E( 5 B C"#%$&@ (; 4 D 5 4507 B 50( :
D
3
P
4F=& 4 (5 & 50 D &@:4%( L +&'L+&5-=++;5'$(' E 3g' 0 + ' 0 # P - 0 # -$Y!':),+ 7C*'O$ A@ +;5' # %$,$,'DCN+
- &C 0 Y!'*lkY!':),+&':l X6 '5-DY!'M+ Y!':),+&':l A@ +&5' @ );-=3g' ) A@B0 '
P ) P $ 4+ 0 C.5-/).-RCN+;'*) %e:'O$+&5'
P < FVE0 $ 0 .+ $- PP 'D-=) 2 +&5U-$\Y!'*),+ %C:'D$ A@ +;5' @ );-R3g' P
0 P < FVE0 $ ' 0 C:' 6 5' 0H6 'X+.-=<4mb-=2 1+
$ P 'DC @ FU6 5':+&5':) 6 'M)&' @ '*) + ?+-R$-XC A0 Y!'*l
A@ +&5' # %$&$('DC*+ 0 P < FVE0 $ 5' @ <4< /6 0E
)&' '*lbY!'*)&+ %C:'D$ A@ +&5'qCD- 00 7C:-R< # 7$,$('DC*+ A0
(
D 507 E7 5
7
:D Proposition
3.14
P
S
D (E( 5 B ? ; % (
:D
P
'507 65 3((
3=&;F5 <
D
P
# $&@
87 D
?=58 5 45079$ 87 5 5
a!
Chapter 3. Hamiltonian Polygons
Proof.
%$ -=<7$
p
i
0
pJA0 $ # '*)-k)&' '*l Y!'*),+;'*l p A@ $ 3g' D Z FHi ) P *' )&+ F -hY!'*),+;'*l A@ P M0 +&5' i +&5':) 5- 0 # 2 F i ) P '*),+ F - 0 #
# 12<%'fY!':),+&'*l @ P CD- 0 2d'o);' ':l 0 - 0F D 3.4.2 Extension to Interior Segments
p <%'D-=)&< F +&5' P < FsEA0 $ 0 +;5'SC:- 0A0 %CD-=< # 7$,$('OCN+ 0 -/)&' 0 + 0 'DC:'D$,$,-=) %< F C 0sr
Y!':l " );$ + # 'D- + 2+;- 0 - # 7$,$('OCN+ 0 0 + C A0 Y!'*l P < FVE0 $ @ ) 3 7$[+&5' @ <4< /6 0E C.5 _ $(' -U)&' '*l Y!'*),+;'*l p A@ $ 3g' D - 0 # # ).- 6 ).- F @ ) 3 p +&5-=+9$ P < ?+.$ ] p 0 + + 6 $ +&) i%C*+&< F C 0 Y!'*l - 0E <4'O$ J] -=< 0E
+;5'h);- F 1 0 i+ %< ?+U5 4+;$ ∂D D ) i - P )&':Y 1$(< F # );- 60 );- F -=+ $ 3 ' P 0 + x >@ +&5' $(' E 3 ' 0 + p x # 'D$ 0 + C*) $,$ - 0F $(' E 3g' 0 + @ ) 3 S +&5' 0 6 ' )&'8 0 '
+;5' # %$&$('DC*+ 0 2 F i $ P < 4+,+ 0E D -R< 0E p x [L +;'X+;5-/+ x 7$ 0 + 0 'DC*'O$,$,-=) 4< F
' 0 # P 0 + A@ -q$(' E 3g' 0 + @ ) 3 S \
5'M);'Di$(1<4+ 0E # 7$,$('OCN+ 0 # ' P ' 0 # $ A0 +&5'
) # '*) 0 6 5 %C.5 +;5'J).- F $B-=)&' # );- 60 21+- 0F ) # '*) 6 1< #k# -=+B+&5 7$ P 0 + Z1+ @ - 0FUA@ +&5'$(' E 3 ' 0 +.$ p x CN) $,$('O$ -q$,' E 3g' 0 + s @ ) 3 S $ P < 4+,+ 0E
=
<
1
<
Y
7
<
/
&
+
'
i& ' 0 C:' 6 ' 5- 0 # <%' +&5 7$ CD-R$(' 0 - # @ r
#
0
E
6
D
@ '*)&' 0 + 6 - pFi x2 F '*lV+&' 0 # 0E P + 0 C ) P );-/+&' s 5 7$<%'D- # $S+ - 0 ' 6 2-R$ 7C
P '*).-/+ 0 l+;' 0 # r L' '*l <& 785 Operation 3 (Extend-Reflex(P, u, , pi, r, s ))
- @ ).-=3g' - 0 ) %' 0 +;-=+ 0
- # 7$,$('OCN+ 0 A@ P -`)&' '*l
P
u(P)
Y!'*),+&':l p A@ $ 3g' D - C 0 Y!':l Y!'*),+&':l r A@ D - 0 # -`);- F s
':3U- 0 -=+ 0iEk@ ) 3 p i
705 B :D ( pir %$f- C 3g3 0 ' # E ' A@ D - 0 # P s C*1+.$ ]Dpi 0 + + 6 $ +,) 7CN+;< F C A0 Y!'*l - 0E <4'O$ - 0 # s 5 4+;$ +&5'f$(' E 3g' 0 + qt @ ) 3 S +&5-/+
< %'D$ 0 D -/+M- P 0 + x qt
-=)&' A0
"507E; " $,$,13g' 6 4+&5 1+ < $,$ A@ E ' 0 '*);-R< 4+ F +;5-/+ - 0 #
r
t
+&5'U$,-=3g' $ # ' A@ +&5'U$(1 PP ),+ 0E < 0 ' A@ s 2+.- 0 P @ ) 3 P 2 F
)&' P <7-RC 0E +&5'' # E ' p r 2 F +&5' P -/+&5 E ' (p , x, q) (q, t) E ' (t, x, p , r) 7$,$('OCN+ CD- 00 7C:-R<4i< F! ':+ u( ) := −1i @K ) -R<4< 0 +;'*) ) Y!'*),+ %C:'Di$ A@
E ' (p , x, q) 0 C*<%1 # 0E q - 0 # u( ) := +1 @ )X-=<%< 0 +&':) )oY!':),+ 7C*'O$ A@
E ' (t,ix, p , r) 0 C:<41 # 0E t i
(P , u, )
0"
)$'& pi x (∂S ∂D) =
a
3.4. Dissection
t
t
x
q
s
q
pi
pi
r
Figure 23:
r
658:D
5
5
(P, u, , pi, r, s)
5'*)&' -/)&'U+ 6 Y-=) %- 0 +.$ @M lV+&' 0 # r L' ':l # ' P ' 0 # 0EÀ0 6 5':+&5':) r
@K <%< /6 $ ) P )&'OC*' # 'D$ p 0 P j 'U5-DY!' # 'D$&CN) 42d' # 0 < F +&5' );$ + -=2 Y!'
- 0 # )&' @ ':)+ +&5 7$ Y-=) 7- 0 i+ 0 +;5' @ <4< /6 0E 5' +;5'*)Y-/) 7- 0 + %$ C 3 P <%'*+&'D< F
$ F 3g3g'*+,) %C m 0E -/+[+&5' P )&'DC A0 # ?+ 0 $ A@ lV+&' 0 # r L' ':l 6 'q5-DY!' + ' 0 $(1);'U+&5-=+ @K )q'O-RC.5 )&' '*l Y!':),+&'*l A@ - P < FsE0 D @ ) 3 +;5'*)&' %$q- 0
0 C # ' 0 + ' # E '+&5-/+ 7$ C 3g3 A0 + 2 +;5 D - 0 # P - 0 # 6 5 $(' +&5':)c' 0 # P 0 +
7$S- C A0 Y!'*lUY!':),+&':l @ D \
5 %$ 7$L'*l-RC*+&< F 6 5-/+ i ) P '*),+ F # ':3U- 0 # $ +,) 7CN+&< F $ P 'D-/m 0E +&5' E ' # 'D$ 7C:$ -/2 Y!'g-/);' 0 + 6 ':<%< # '8 0 ' # 2d'DC:-R1$('
+;5' P 0 + x %$ 0 + - Y!':),+&':l @ 1 %$ (S) Z\1+ $ 0 C*' {q, t} 7$ - 0 ' # E ' A@
- 0 # +&5' P %'DC:' 6 7$('k< 0 'D-/) C:1),Y!'O$ (p , x, q) - 0 # (t, x, p , r) # 0 +
1 %$
(S)
P ' 0 '*+,).-/+&' - 0_F $(' E 3g' 0 + @ ) 3 S +&5' # E' 0 ?+ i0 C:-=),) %'D$ Y!':) i
" $ # 7$,C:1$,$(' # 0 'DC*+ 0 as +&5' E ' # 'D$ 7C A@ - $ %e:' @ 1) P -=+&5 %$ 0 +
0 'DC*'O$,$,-=) 4< F $ %3 P <4' " <7$ 0` lV+&' 0 # r L' '*l 0 'X' 0 # P 0 + A@ +&5'X$(' E 3g' 0 +
5 ?+M2 F +&5'X);- F s CD- 0È 1 P @ ) 3 # ' E )&':' e:':) + # ' E )&':' @ 1) 0 P 8 Er
1)&' 4.H '*Y!'*)&+&5':<%'D$&$ 6 ' 6 4<%< $(5 /6 2d':< /6 +&5-=+X+&5'g)&'D$(1<?+ 0E P < FVE0
7$f-=< 6 - F $f- @ ).-=3g' " $f8 E 1)&' # ':3 0 $ +,);-=+&'D$[- 0 # -R$ 6 ' 5-DY!' -=<4)&'D- # F
$,':' 0h@K )[Z\1 %< # r p - P lV+&' 0 # r L' ':l 3 E 5+CN)&'O-/+&' $ 3 ' 0 ' 6 1 0 $&-/+&1).-/+&' #
$,' E 3g' 0 +;$ 6 5 %C.5 P )&':Y!' 0 +B+&5' # %$,$,'DCN+ 0 @ ) 3 2d' 0Eo0 %C:' 5 7$ P ) 2<4'D3 %$
'O-R$ %< F )&'O$ <?Y!' # 2 F - PP < F 0EM0 C:' - E - 0U -/+&1).-/+&'J+ +;5' @ );-R3g' +&5-=+ )&'D$(1<?+;$
@ ) 3 lV+&' 0 # r L' '*l 5' ) %' 0 +;-/+ 0 -R< 0E +;5' E ' # 'O$ 7C:$ %$ C.5 $(' 0 0
$,1C.5H- 6 - F +;5-/+ 0 Y!'*)&+&'*l A@ P +;5-/+ 7$ 0 C # ' 0 +S+ - 0 1 0 $&-/+&1).-/+&' # $(' Er
3g' 0 + 7$ ) %' 0 +&' # + /6 -/) # $9- )&' '*lhY!':),+&'*l L'DCD-=<%< +&5-/+-R$- P );'DC 0 # ?+ 0
A@ lV+&' 0 # r L' ':l r 7$- C 0 Y!'*l Y!':),+&':l A@ D ' 0 C*'X2 Fbi ) P $ 4+ 0 s7D
+;5'*)&' 7$ -=+3 $ + 0 ' 0 ' 6 )&' '*l + 6 0 0 -=<%< # 7$,$('DC*+ A0 P < FsEA0 $ @ )B+;5'J);' r
$,1<?+ 0E @ );-R3 ' (q, t) p <%'D-/);< F 2 +&5g' 0 # P 0 +;$ A@ +&5'S$(' E 3 ' 0 + 6 '9'*lV+&' 0 #
a
Chapter 3. Hamiltonian Polygons
x
t
r
pi
Figure 24:
587 5
t
$? '5 D 5 &785E5
7 E 507
50:D
5
5
+ 3U- F 2d'f)&' '*l 0 +&5'f)&'D$,1<?+ @ );-=3g' P @B lV+&' 0 # r S' ':l " 0 +&5':) %3 P < 7C:-=+ A0 7$ +&5-/+ @[6 'b1$(' l+&' 0 # r L' '*l +&5' 0 6 ' 6 4<%<
0 +[2d' -=2<%'q+ 3g- 0 +;- 0 - 0 %C:' # 7$,$('OCN+ 0 A@ +;5' @ );-R3g'q+&5) 1 E 5 1++&5'
7$,$('OCN+ 0gi 5-$(' \S=6 '*Y!':) +&5' )&' '*lq+ 6 0 (q, t) %$ +&5' 0 < F A0 ' P );'D$(' 0 +
0 +;5' 6 5 <%' @ );-R3 ' " +9+&5 %$ P 0 + 6 ' 3U-/m!'X1$(' A@ +&5' @ -CN+9+;5-/+ 6 ' # #
0 +9$ P 'OC @ F $ k@ -/) @ ) 3 6 5 7C.5b)&' '*l Y!':),+&':l 6 ' $(5 +S+;5'f);- F! @ 6 ' 1$ +
C.5 $(' 0 'MY!'*)&+&'*l A@ +&5'M)&' '*lg+ 6 0 CN)&'O-/+&' # 2 FU lV+&' 0 # r L' ':l +;5' 0 +&5 7$
+ 6 0k6 %<4< 0 < 0E '*) C ),)&'O$ P A0 # + + 6 C 0 $('OC*1+ ?Y!')&' '*lqY!':),+ 7C*'D$-=< 0E P < FVE0 0 2d'DCD-=1$,'+&5'LY!':),+&':l @ ) 3 6 5 7C.5g+&5');- F 7$ $(5 + 6 4<%<2d'DC 3 '
C 0 Y!':l 0 +&5' P ) C*'O$,$ A@ 5- 0 # < 0E +&5'[);- F
5'[)&'O$(1<4+ 0E # 7$,$('OCN+ 0h6 4<%<
0 +2d' 0 7C*' 0 +&5'$ +&) %C*+ $(' 0 $,' 21+ ?+ C 3g'O$ P )&'*+,+ F C*< $(' 6 5 7C.5 E 4Y!'D$ ) 7$('
+ +&5' @ <4< /6 0E # E' 0 4+ 0
Definition
3.15
D (E( 5 B D
(
D 05 7 7E 5
7 ( 5 P
S
(
:D $ P
almost
nice
K5
( 4 5 3 K( ?=5 B :D 85 4
D3( 3 ( 5 & 58 (
% (
?=5 0587 785 5 4 3 ( B 5
D '5074 "#%$&'( 78
( ; (0 5
L +&' +&5-/+B+;5'*)&'3U- F 2d'L3U- 0F )&' '*l + 6 0 $ 0 P 21+ @ ) +&5'J3 3g' 0 + 6 '
-=)&' C A0 C*'*) 0 ' # 6 4+&5h+&5 $(' 0 < F +&5-=+9-/)&'X-=<7$ C 0 $('DC:1+ 4Y!'[Y!'*),+ %C:'D$ 6 4+&5 0
- # %$&$('DC*+ 0 P < FsE0 '*+f1$ # 7$,C*1$,$f- @ ' 6 $ P 'OC 7-=<\$ 4+&1-/+ 0 $+&5-/+f3U- F
-=) 7$(' 0h l+&' 0 # r L' '*l 8 ?);$(+ A@ -R<4< @ p r 7$- $,' E 3g' 0 + @ ) 3 S +;5' 0 ?+ %$
- 0 ' P E0 -=< 0 P 'OC 0 # $('*Y!':);-=<$ 4+&1i-/+ 0 $[3U- F <%'D- # + +&5' CN)&'O-/+ 0
A@ C:1+ r ' # E 'O$ 6 ?+&5 0 +&5' @ );-R3 ' p <4'O-/)&< F -XC*1+ r ' # E '93U- F 2d'MC*)&'D-=+&' # 6 5' 0 Y!'*),+ 7C*'O$ A@ -f)&' '*l + 6 0 - P r
P 'O-/)SC 0 $('OC*1+ 4Y!':< F -=< A0E - E ' # 'O$ %C " CN+;1-=<%< F! 0 +&5'M' 0 # 6 '[5-DY!'+ 2^'
a!
3.4. Dissection
Y!':) F C:-=)&' @ 1< 6 4+&5H$(1C.5h'*Y!' 0 +.$J2d'OC:-=1$('f+&5' F -/);' P +;' 0 + %-R<4< F +,) 12<4'O$ 3g'
0 +&5' %3 P < CD-/+ 0 +&' P <%-=+&'*) 21+ @K ) 0/6 +&5 7$ 7$ 0 + 1) C 0 C*':) 0
/6 ':Y!'*) +&5':)&'X-=)&' -R<%$ - @ ' 6 +&5'*) 6 - F $+ 0 +,) # 1C*' C*1+ r ' # E 'D$ 0 +&5'
@ );-=3g' 6 5 7C.5 6 4<%<c2d' # 7$,C*1$&$(' # 2d'D< /6X
@ p 5-R$U- 0 ' E 52 ) w 0 D +&5-=+ 7$g-`)&' ':l Y!'*),+;'*l A@ D +&5' 0 w
3U- F - PiP 'O-/)q-R$ - $,'DC 0 # Y!'*)&+&'*l 0 E ' (p , x, q) g>0 +;5 %$XC:-$(' p w 7$XC:1+ r ' # E ' A@ P - 0 # # ' E (p ) = 1 43 %<%i-=)&< F! @ r 5-$q- 0 ' E 5_2 i ) v 0
i
;
+
5
/
+
7$
U
&
)
'
':lbY!'*),+;'*P
l
+&5' 0 v 3 E 5_+f- PP 'O-/) -R$f-U$,'DC 0 # Y!':),+&'*l
A
@
D '
D
0hE (r, p , x, q) +&5'*)&':2 F );'*Y!'*)&+ 0E rv + - C*1+ r ' # E ' A@ P " 0 ':ls-R3 P <4'
%<%<41$(+,);-/+ 0E i 2 +&5HC:-$('D$ 7$$(5 /60 0 8 E 1)&' A
t
t
x
s
Figure 25:
r
pi
w
q
50:D
5
r
pi
w
q
5 $ B 785E;5 B
5
D&@59( ;
pi
7
r
5 '*)&' 7$M- 0 +&5':) P $,$ 42 %< 4+ F + 0 +,) # 1C:' - 0 ' 6 C:1+ r ' # E ' 0 P @ t
- PP 'D-=);$L+ 6 %C:' -R$S- Y!':),+&':l A0bE ' (r, p , x, t) - 0 # +&5'o$(' E 3 ' 0 +.$ @ ) 3 S 0
+;5' 0 +&'*) ) @ +&5' 1- # ) 4<7-/+&':);-=< (r,ip , x, t) -/)&'qC <%< 0 'O-/) o>0 +;5 %$C:-$('
-R<4< +&5'D$,'[$,' E 3g' 0 +;$ -$ 6 ':<%< -$J+;5'9Y 7$ ?2 4< i?+ F ' # E 'D$C 00 'OCN+ 0E +&5':3 -=< 0E
+;5' 4)1 0 # ':)&< F 0E < 0 'X-=)&'XC:1+ r ' # E 'O$ 0 P -R$9$(5 /60 0 8 E 1)&' as
5' # 7$,C*1$&$ 0UA@ +&5'O$(' # ' E ' 0 '*);-=+&'9CD-R$('O$ %$ 1$ + @ ) 4<%<41$ +,);-=+ 0 @ ) 3
- 0 -R< E ) 4+&53 %C P 0 + A@ Y %' 6 Y!':),+ 7C*'O$ 0 C # ' 0 +B+ C*1+ r ' # E 'D$B-=)&' 0 0[6 - F
3 )&' P ) 2<%':3U-/+ %CM+&5- 0 - 0_F +&5':) # 12<4'Y!'*)&+ %C:'D$ L/6 6 'f-=)&'[)&'D- # F + P ) Y!'[+&5' P ) 3 %$(' # 0 Y-/) %- 0 + A@B l+;' 0 # r L' '*l ?=5 3 D (E( 5 B Proposition
3.16
50:D
P
5
5
3( 7E
5
( 4% (8: B 5 ?=50 ?=5
" $ @K )[Z\1 %< # r p - P 6 ' 6 4<%< );$ +9-=) E 1'o+&5-/+ P %$- P < FsEA0 8 ?);$(+
F s 7$ 0 +SC 4< < 0 'D-/)S+ qt -R$S- P );'DC 0 # ?+ 0h@B lV+&' 0 # r
Proof.
+;'[+&5-/+L+&5'[);-
0
aa
Chapter 3. Hamiltonian Polygons
q
q
x
s
pi
pi
t
t
r
5 5
658 D Figure 26: ( 5+& 50( $ B 785E;5A $ B
E'
(r, pi , x, t)
r
5
D&@50(4 & B ; %< 5E7
S' ':l ' 0 C*'+;5'S+ 6 E ' # 'O$ 7C:$ 0 l+;' 0 # r L' '*l # X0 +J$,5-/)&'-fY!':),+&'*l
- 0 # -=<%< ' # E 'O$ A@ 2 +;5 E ' # 'D$ 7C:$S-$ 6 ':<%<-$ qt -/)&' 0 C # ' 0 ++ +&5' 0 0 4+&'
@ -RC*' A@ P 51$ P 7$ 0 # ':' # - P < FsEA0
p <4'O-/)&< F P 7$$ 43 P < F_r C A00 'DC*+&' # - 0 # -$ 6 ':<%<B-R$A J@ <%< =6 # r
);'DCN+;< F @ ) 3 +&5' # 8' 0 4+ 0 A@ E ' # 'O$ %CD$S- 0 # @ ) 3 +&5' @ -RC*+L+&5-/+L+&5' 0 P 1+
P < FVE0 P %$- @ );-=3g' \ +S)&':3U- 0 $+ C.5'OC&m P ) P '*),+ %'D$ I &
8 ) 0 +;'*) 0 -R<sY!'*),+ %C:'D$ @E ' (p , x, q) - 0 # E ' (t, x, p , r) 0 'SCD- 0 -/) E 1'
-$ 0 i ) P $ ?+ 0 s as ' 0 C*' 6 'Xi 5-DY!' + C 0 $ # '*)M+&5'oi Y!':),+ 7C*'O$ p r q i
- 0 # t A0 < F!
- 0 # 5' 0 C:' A@ P 5 7$
Z FUi ) $ ?+ 0 s7 ?+ 7$ -o$ 0E <%'Y!':),+&'*l A@
pi %3 < 4'O$ P P
-R$ 6 ':<%<S-R$ ; " $ p 7$k- )&' '*l Y!'*),+&':l 0 P ?+.$
P
0 C # ' 0 +$(' E 3g' 0 + @ ) 3 S 7$J$,-=+&1);-=+&' # i 0 P 2 F - 0 # 5' 0 C*' -R<%$ 5 < # $ 0 P ?+hC:- 00 + - PP 'O-/)b-R$ - 0 0 +&'*) ) Y!'*)&+&'*l
" $ - C A0 Y!'*l Y!'*),+;'*l A@
r
D
0 E ' (t, x, p , r) '*Y!' 0 @ 4+ 7$k- )&' '*l Y!'*),+&':l A@ +;5' 1- # ) %<%-=+&'*);-R<
ki 51$ - 'O-/);$ -R$ @ +&' 0 0 P -R$ ?+ # # 0 P - 0 #
(t, x, pi , r) 5 %C.5 %3 r <P%'DP $ 6
I &
P
]P r < ] P r
%$- 0 ' # E ' @ P 6 5 %C.5 43 P < %'D$ & " $ q
5' 0 C # ' 0 +$(' E 3g' 0 +
q
qt
7$S-k$ 0E <%'oY!':),+&'*l 0 P 2 +&5 - 0 # 5 < # +,) 4Y 7-=<%< F!
7$q- 0 ' # E ' @ P 6 5 %C.5 %3 P < %'D$ & >@
5' 0 C # ' 0 +k$(' E 3g' 0 +
t
qt
- 'D-/).$ + 6 %C:' 0 E ' (t, x, p , r) ?+ %$ - # 12<%' Y!'*),+&':l A@ P - 0 #
t PP <%< $ -R$
@ /6
0hi ) P $ 4+ A0 i V a 2d'DC:-R1$(' t - PP 'O-/);$ -R$- 0 0 +&'*) 0 -=<
a!
3.4. Dissection
Y!':),+&':l @ - E ' # 'O$ 7C +&5'*) 6 7$(' t 7$S- $ 0 E <4' Y!'*),+&':l A@
7$+,) 4Y %-R< \>0 2 +&5bCD-R$('O$ 7$ @ 1< <%<4' # P
- 0 # 3.4.3 Preserving Common Edges
_ @ -=) 6 'X5-DY!' -/) E 1' # 5 /6 + 3U- 0 +.- 0 - @ );-R3g' 0 C:-R$,' +&5-/+ 0 ' A@ +&5'
).- F $q$(5 + @ ) 3 -b);' ':l Y!'*)&+&'*l @ - # 7$,$('DC*+ A0 P < FsEA0 5 4+;$ - $(' E 3g' 0 +
+;5-/+f< 4'O$ 0 4+;$ 0 +&':) ) q=6 4+ %$+ %3g' + - ## )&'O$,$ 1)o$('DC 0 # C 0 C*':) 0
5 /6 + 3U- 0 +;- 0 - 0 -R<43 $ + 0 7C*' # %$&$('DC*+ 0
>0 # 'D' # +&5 7$ 7$ 0 + E ).- 0 +&' # 2 F 0/6X '*Y!' 0 @00 ' @ +&5' );- F $ # );- 60 5 ?+.$
- 0_F 0 +&':) ) $(' E 3 ' 0 + +;5' # 7$,$('OCN+ 0 +&5-/+f);'D$(1<4+;$ @ ) 3 )&' P ':+ 4+ ?Y!' $ P < ?+;$
-R< 0E );- F $ 0 ':' # 0 + @ 1< <%< i ) P '*),+ F -R$+&5'q'*l-=3 P <%' 0 8 E 1)&' $,5 /6 $ j ' 5-:Y!'H+ +;-=m!' 0 + -RCDC 1 0 +U+&5-/+ 6 5' 0 ':Y!'*) - );- F 5 4+;$ +&5'
2 1 0 # -/) F A@ +&5' C*1),);' 0 +o)&' E 0 - 0 # +&5V1$+&5' )&' E 0 %$X$ P < ?+ -=< 0E +&5 7$
).- F! +;5' ' # E ' 5 4+f3 E 5+M5-DY!' 2d':' 0 +&5'q<7-R$ +oC 3g3 0 ' # E ' @ P - 0 # 0 '
A@ +&5' 0 ' 6 < F C*)&'D-=+&' # );' E 0 $ ?=5 8( ? @D 5 D D 3(E(5 B "#%$&@ # ;650(
K( D 59(
Figure 27: K ? 45 $ B 5 D&@5 4 ?> ?=5 ( 7070 :D3=& E7 5 >0 ) # ':)o+ bE 1-=);- 0 +&':'g- 0 -=<%3 $(+ 0 7C*' # 7$,$('DC*+ A0 # 1) 0E +&5' 7$,$('DC r
+ 0 i 5-$(' 6 ' 0 +&) # 1C*'b- 0 +&5'*)k2-R$ 7C P ':);-/+ 0 );- Er # E ' +&5-/+ %$
- PP < 4' # 1 0 # '*)9C*':),+;- 0 C 4);C*13U$ +.- 0 C:'D$ 6 5' 0 +&5'o);- F 5 4+;$- C 3g3 0 ' # E '
A@ +&5' # %$,$,'DCN+ 0 P < FsEA0 D - 0 # +&5' @ ).-=3g' P 5'H'*l-RC*+ C 0 # 4+ A0 $
6 5' 0 +&5 7$ P ':);-/+ 0 %$ - PP < 4' # 6 4<%<2d' # %$,C:1$,$(' # 2d':< /6 - 0 # 6 ' 6 4<%<$(5 /6
5 /6 + ' 0 $(1)&'q- C 3g3 0 ' # E ' 2d'*+ 6 'D' 0 '*Y!':) F D - 0 # P +;5) 1 E 5 1+S+&5'
7$,$('OCN+ 0 i 5-R$(' a
Chapter 3. Hamiltonian Polygons
<& 785 Operation 4 (Drag-Edge(P, u, , pi, s , pk p` ))
S- @ );-=3g' 6 ?+;5 - 0 ) 4' 0 +.-/+ 0
- # 7$,$('OCN+ 0 P
u(P)
Y!'*),+&':l p @ $ 3g' D -q).- F s ':3U- 0 -/+ 0E @ ) 3 p i
i
5
4+
S
2
/
9
+
+
A
@
F
0
P
pk p`
∂D
x p k p`
s
A@ P - &) ' '*l
-0# -0 '#E'
7$ - C g
3 3 0 ' # E ' A@ P - 0 # D p 7$ - C A0 !Y '*l
Y!'*),+&':l A @ D 0 # s C*1+;$ ;+ 5' - 0E <%' ] p 0 + + 6 $ +,k) 7CN+&< F C A0 !Y '*l
D i
- 0E <%'D$ 705 B :D (
p
- k p`
2+.- 0
2 F )&' <%-C 0E &+ 5' ' # E ' p p 2 F +&5' P -/+;5
CD- 00 7C:-R<4< Fk `'*+ u( ) := −1
0 # u( ) := +1 @ )\-=<%< 0 +&'*) )
@) 3
P
E ' (p , x, p ) E ' P (p , x, pP) f 7$,$('DC*+ @ ) -=<%< k 0 +&':) i ) Y!':),+ 7C*'Oi$ A@E `' (p , x, p ) k
i
Y!'*),+ 7C*'O$ A@ E ' (p , x, p ) i
`
"507E; (P , u, )
s
p`
x
pk
p`
x
pk
pi
Figure 28:
pi
7E&
D&@5
(P, u, , pi , s , p k p` )
2$(':),Y!'h+&5-=+ );- Er # E 'HC*)&'D-=+&'D$U- 0 ' P EA0 -=< A@ +&5' @ ).-=3g' 0 CD-R$('
+;5-/+ p p 7$ -H$(' E 3g' 0 + @ ) 3 S ` %3 4<7-/)q+ ` l+;' 0 # r L' '*l +;5'*)&' -=)&' @ ' 6 $ 4+&k1-/+` 0 $ 0h6 5 7C.5 );- Er # E ' CN)&'O-/+&'O$C*1+ r ' # E 'D$ 6 4+&5 0 +&5' @ );-=3g' @ p 5-R$\- 0 ' E 5_2 ) w 0 D +&5-=+ 7$-M)&' '*l Y!':),+&'*l A@ D +&5' 0 w 3 E 5+
- PP 'D-=)i-R$- $('DC A0 # Y!'*),+;'*l A0 E ' (p , x, q) 5 %$ %$ +&5''*l-RCN+;< F +&5'[$,-=3g'
$ ?+&1-/+ 0 -R$ 0 l+&' 0 # r L' '*l $ 6 i ' # `0 + # %$,C:1$,$ 4+ 5':)&' - E - 0 n0
+;5' 0 -R<-=< E ) ?+;53 6 ' 6 4<%<':Y!' 0 -DY # $(1C.5 -o$ ?+&1-/+ 0 2d'OC:-=1$(' 4+JC:-R1$('D$
P ) 2<4'D3U$ @ ) +&5' 43 P < C:-=+ 0 i 5-$('[<%-=+&'*) Z\1+ @ ) +;5'93 3g' 0 + +&5 7$ %$
0 +- 0 %$&$(1' a
3.4. Dissection
" P =- ),+ @ ) 3 +&5'q1$(1-=< P $&$ ?2 4< ?+ F +&5-/+o- E ' # 'D$ 7C Y %$ 4+;$+ 6 Y!':),+ 7C*'O$
+;5-/+ @ )&3 - )&' ':lh+ 6 0 +&5':)&' %$- 0 +&5'*)f$ ?+;1-/+ 0 0 6 5 %C.5 ).- Er # E '
C*)&'D-=+&'D$ - C*1+ r ' # E ' 0 P @ +;5' +&5'*) = p k0 ' E 5_2 ) A@ p 0 +&5'
# %$,$,'DCN+ 0 P < FsEA0 D 1 0 # '*)JC A0 $ # '*);-=+ 0 7$ -[`)&' '*l Y!':),+&':l A@ Dk 4+ 3U- F
- PP 'D-=)-R$M- $,'DC 0 # Y!'*),+;'*l 0HE ' (p , x, p ) 0 ' P -/),+ 7C*1<%-=) # ' E ' 0 '*).-/+&'
0 $ +.- 0 C:' A@ +&5 7$ $ 4+&1-=+ A0 CDC*1).$ @ p k 7$ - # i -C*' 0 ++ p 0 D >0 +&5 7$ CD-R$('
+;5'k' # E ' p p 6 5 7C.5 7$o-=<4)&'D- # F - 0 k' # E ' A@ +&5' @ );-R3gi'k3U- F -=<7$ - PP 'D-/)
0 E ' (p , ix, kp ) -$ $(5 =60 0 8 E 1);' V 2$(':),Y!' +&5-/+q+&5 7$ $ 4+&1-=+ A0
CD- 00 + C:iC:1) @ k ) p @ 4+5- PP ' 0 $L+ 2d'X- );' ':l Y!'*)&+&'*l A@ D `
p`
p`
s
pk
pi
Figure 29:
7E&
pi
pk
D&@5 $ B 785E;5 B
5
D&@59( ;
pk
:D 7
p`
" $ @ )\+&5' +&5':)\2-R$ 7C P '*).-/+ 0 $ 6 ' P ) Y!'L+&5-=+ ).- Er # E ' P ) # 1C*'O$
- @ );-=3g' Proposition 3.17
7E&
P
?=5 3 D (E( 5 B D&@5 ( 7E
5
( 4% (8: B 5 ?=50 ?=5
1$ );$ +B-=) E 1' +;5-/+ P 7$B- P < FVE0 " $c+&5' ).- F s 5 4+;$ p p 0
0 +&'*) ) 2 +&5 -=)&' 0 +[C <%< 0 'D-=)[- 0 # +;5' + 6 gE ' # 'D$ %CD$Mk$(5`-/)&'
':ls-CN+&< F 0 ' Y!'*),+&':l p " $ p p %$ - 0 ' # E ' A@ P -R<4< ' # E 'D$ -=< 0E 2 +&5
E ' # 'O$ %CD$k-=)&' 0 C # ' 0 +ki + +&5' k 0 ` 0 ?+&' @ -RC:' A@ P 51$ P 7$ 0 # 'D' # P < FVE0
p <4'O-/)&< F P %$ $ 43 P < F_r C A00 'DC*+&' # + )&':3U- 0 $c+ C.5'OC&m P ) P '*),+ 4'O$ & - 0 # @ <%< /6 # 4)&'DC*+&< F @ ) 3 +&5' # 'E 0 ?+ 0 A@SE ' # 'D$ 7C:$
- 0 # @ ) 3 +&5' @ -CN+S+&5-/+S+&5' 0 P 1+ P < FVE0 P %$9- @ );-R3g' 8 ) 0 +&'*) 0 -=<_Y!'*),+ 7C*'O$ A@E ' (p , x, p ) - 0 # E ' (p , x, p ) 0 ' C:- 0 -/) E 1'
-$ 0 i ) P $ 4+ 0 s asq ' 0 C*' 6 k'k5-:Y!i' + C 0 $ # '*i)o+;5' `Y!':),+ 7C*'D$ p p i
k
Proof.
4+;$)&':<7-/+
4Y!'*' +
Chapter 3. Hamiltonian Polygons
-0#
0 < F
7$ -b)&' '*l $ 0E <4' Y!'*),+&':l 0 P 5':)&' @ )&'
Z F i ) P $ 4+ 0 V pi 4+;$
p
i
0 C # ' 0 + $(' E 3g' 0 + @ ) 3 S %$ $,-/+;1);-/+;' # 0 P - 0 # 5' 0 C*' 0 P 2 F
& " $- $ 0E <%'qY!':),+&'*l A@
Y!'*),+;'*l p - PP 'D-=);$+ 6 7C*' 0 P - 0 #
P
+;51
$ 5 < # $ -R$ 6 'D<4< @ <%< =6 $ i@ ) 3 +&5' @ -CN+ +&5-=+ s C*1+;$
+;5'o- 0E <%' ] p 0 + + 6 $ +,) 7CN+&< F C 0 Y!':l - 0E <%'D$ D i
- 'D-/).$o-R$ A@ +;' 0 0 P -$ 4+ # # 0 P - 0 # ] p <
p <%'D-=)&< F
pk
pk PP
P k
5
7C
;
5
43
<
4'
O
$
I & 6
P
]P p k
-=) E 1'X-$ @ )
p
p
p`
`
k
3.4.4 The direction of s
_ @ -/) ?+L3U- F $,':':3 C 0 Y!' 0 %' 0 + + $ %3 P < F -R< 6 - F $L- PP < F );- Er # E ' 6 5' 0
-`).- F 5 ?+.$ - C 3g3 0 ' # E ' A@ P - 0 # +&5' # 7$,$('OCN+ 0 P < FVE0 D 1 0 # ':)
C 0 $ # '*);-=+ A0c Z\1+-$ 6 ' 6 %<%< $(':' 0 +&5' # 7$,C:1$,$ 0 A@ +&5' %3 P < CD-/+ 0
i 5-$(' 6 'bC:- 00 + -R< 6 - F $ - PP < F );- Er # E ' @96 ' 6 - 0 +k+ C 0 $ +,)&1CN+ $ 43 P <%' P < FsEA0 @ ) 3 +;5' @ );-R3 'H- @ +&':) 6 -=) # $ 5' P ) 2<%':3U-/+ %C P -=),+ A@
);- Er # E ' %$ +&5-/+ 4+ C*)&'D-=+&'D$ - $('OC 0 # );' ':l CDC*1)&)&' 0 C:' A@ +&5'Y!'*)&+&'*l p i
5' )&'D-$ 0 + 0 +,) # 1C*' );- Er # E ' 6 -R$X+ ' 0 $(1);' i ) P ':),+ F &
+;5-/+ %$ -fC 3g3 A0 ' # E ' A@ 'O-RC.5 # %$,$,'DCN+ 0 P < FsEA0k6 ?+&5 +&5'S$(1),) 1 0 # 0E
@ );-=3g' >0 +;5' < E 5+ A@ +;5 %$ 4+ 7$ 0 + )&'D-R<4< F 0 'DC:'D$,$&-/) F + -R< 6 - F $ - PP < F
);- Er # E ' $(1 PP $,'M+;5-/+L-/+$ 3g' P 0 + 6 '[$(5 +-X);- F s @ ) 3 -X)&' '*l
Y!':),+&':l p A@ $ 3g' # 7$,$('OCN+ 0 P < FVE0 D - 0 # 2 +&5 ' # E 'D$ 0 C # ' 0 + + p -/)&'
C 3 3 0 i ' # E 'D$ A@ P - 0 # D @ 6 '+&5' 0 $ 43 P < F # 7$,$('DC*+ D -R< 0E s +&i5'*)&'
7$C:'*),+;- 0 < F $ + %<4< - C 3g3 0 ' # E ' 0 2 +&5h);'D$(1<4+ 0E # %$&$('DC*+ 0 P < FVE0 $ M0 +;5' +&5'*) 5- 0 # _6 '3U- F 5-DY!'9+ 2^'fCD-/)&' @ 1< 0 +&5' 0 ':l+ ?+&':);-/+ 0 1 PP $,'X+;5-/++&5' );- F s 5 4+;$- 0 ' # E ' 0 C # ' 0 ++ - 0 +;5'*))&' '*lbY!'*)&+&'*l
$(5 + @ ) 3 p 0
A@ D X @ );- ERr # E ' 7$ 0 +[- PP < 4' # +&5' 0 +&5' );- F p
q
k
+;5' 0 ':l+ ?+&':);-/+ 0 3U- F 5 ?+ 2-RC;m + s - 0 # +;5'*)&':2 F E ' 0 ':);-/+&'[- # %$&$('DC*k+ 0
P < FVE0 +&5-/+[5-R$ 0 C 3 3 0 ' # E ' 6 ?+&5 +&5'q$(1),) 1 0 # 0EU@ );-=3g' P o ':'
+;5'o'*l-=3 P <%' $(5 /60 0 8 E 1)&' As
P )&'*Y!' 0 + +&5 %$ $ ?+&1-/+ 0_6 '93U-=m!'91$(' A@ +&5' @ -RC*+ +&5-=+ $ @ -=) 6 ' # #
0 +$ P 'DC @ F P )&'OC 7$(':< F 0 6 5 7C;5 # 4)&'DC*+ 0 +&5'o).- F @ ) 3 -k)&' ':l Y!':),+&':l p
7$S$(5 + -$L< 0E -R$ 4+C*1+;$ +&5'[)&' ':l - 0E <%'o-/+ p 0 + + 6 $ +,) 7CN+&< F C A0 Y!'*l i
- 0E <4'O$ 9j ' 6 4<%<B$('D' +&5-/+ 4+ 7$M$,1 gC 4' 0 +[+ $,5 _i ++&5'X);- F 0 # 4)&'DC*+ 0À@
s
3.4. Dissection
pk
pk
q
pi
q
s
pi
s
?=5 (8? @D 5 DD (E( 5 B "#%$&' 3
Figure 30: 4 ?>?=5 E7 5 ? K( B 5
D&@5
+;5'[' # E '[+&5-=+ 6 -R$L5 ?+ 0 +&5' P )&'*Y 1$ 4+&':);-/+ 0 @ - 0_F $('D'X8 E 1)&' > 2
@K )- 0 *' l-=3 P <%' 3.4.5 A First Dissection Algorithm
-=2 '*+f Y!1' $fh$,13g +&3g' -=+&)5%e-/:+ 'q+&+&5 57$'
-=< EA ) 4+&53 @ )[+&5' 7$,$('DC*+ A0 i 5-R$('k-$ # 7$,C*1$,$(' #
7$ 1$(+ - P )&'D< %3 0 -=) F # 'O$,CN) P + 0 -=+&'*) +&5'g-=< r
E ) 4+&53 6 %<4< 2d' - # - P +;' # - 0 # )&E' 0 ' # + +&5' 0 'D' # $ A@ +&5' %3 P < CD-/+ 0
i 5-$('o+;5-/+ 7$+&5' $,12 'DC*+ A@ +&5' 0 '*lV+$('DC*+ A0c
?=5 # 4 3=&>=78 B 5
D 785> 3+3 ( E7 5
7
Algorithm
3.18
P
S
4F=& 4 ? 79 58; :D C4 8( B 5D (E( 5 B u
P
;+ -=),+ 6 4+&5 P ← C 0 Y (∂S) u +1 - 0 # ← {P} - 0 # );' P 'D-=+k+&5'
@ <4< /6 0E $ +&' P $U1 0 + 4<-/+ $ 3g' P 0 + -R<4< P < FsEA0 $ 0 -=)&' C A0 Y!'*l
- @ +&':) +&' P - .
- " PP < FH -/+;1);-/+;' (P, u) - 0 #b# %$,$,'DCN+ CD- 00 7C:-R<4< F
2 p 5 $(' n- P )&'OC*' # 0E 4+&'*);-=+ A0 3 E 5+5-:Y!'[3U- # '[+&5-/+SC.5 %C:'o-=< r
)&'O- # "F - )&' '*l Y!':),+&'*l p @ $ 3g' D - 0 # <%'*+ p r 2d' C 3g3 0 ' # E ' A@ P - 0 # Di $(1C.5 +&5-=+ ] r %$SC A0 Y!'*l p -Ri$ +9- );- F
@ ) 3 p +&5-/+9C:1+;$ ] p 0 + + 6 $ +,D) 7CN+&< F C 0 Y!':l - 0E <%'D$ s
i
D i
C @ s 5 ?+.$M-g$(' E 3g' 0 + qt S 6 4+&5 qt D +&5' 0 - PP < F lV+&' 0 # r
L' '*l (P, u, , pi, r, s ) - 0 # C.5 _ $(' q -R$S- 0 ':l+)&' '*l Y!':),+&':lU+ $(5 _ +9- );- F @ ) 3 # <7$(' @ s 5 4+;$S-kC 3 3 0 ' # E ' p p A@ P - 0 # D +&5' 0
k `
Chapter 3. Hamiltonian Polygons
@0 + 2 +&5k' # E 'D$ 0 C # ' 0 ++ p -/);'LC 3 3 0 ' # E 'D$ A@ P - 0 #
i
;
+
5
'
<
;
)
'
#
0
h
F
E
r
E
P
P
D
(P, u, , pi , s , p k p` )
) @ p 7$)&' '*l 0 D n-$,$(13g' 6 4+&5 1+S< $&$ A@ E ' 0 '*);-R< 4+ F +&5-=+
@ p` %$ )&' '*l +&5' 0 -R<%$ p 7$ )&' ':l +&5' 0 $(5 + +&5' 0 '*lV+
@ ).- F k@ ) 3 p 0 # 4)&'DC*+ A0bA`@ −−
pk p`
`
' <7$('X$ P < 4+ D -R< 0E s - 0 # 1 P # -/+&' -C:C ) # 0E < F
)
" ?< +;5 1 E 5 6 '5-DY!'$(5 /60 +;5-/+ 'D-C.5 $ 0E <%' P '*);-=+ A0 P )&'D$(':),Y!'D$+;5' @ );-R3 '
P ) P ':),+ %'D$ +&5':)&' -/)&' - @ ' 6 3 %$,$ 0E < 0 m$c2d'DC:-R1$('J$ 3g' A@ +;5' C ),)&'O$ P 0 # r
0E P ) P $ 4+ 0 $)&' 1 4)&' # +&5'XC:1),)&' 0 + # %$&$('DC*+ 0 + 2^' -=<%3 $ + 0 7C*' 8 )
-=+&1);-=+&' pJ ) <4<7-/) F V -R$&$('*),+.$ +&5-/+ +&5' 1+ P 1+ 7$g- @ );-=3g' 5' 0 '*lV+
P ) P $ 4+ 0 $ +;-=+&'D$M+&5-/+ -/+&1).-/+&'q-=<7$ )&'D$ P 'OCN+;$M+&5'qC:1),)&' 0 + # %$&$('DC*+ 0 0 -kC:'*),+.- 0 $(' 0 $,' 5
D&@50( B 8( 7 B 65
D 0$
; 7E;5 <
656 3
3
Proposition 3.19 &@79 ?' (8+$ 4 ?'< ?=5 D 3(E(5 B # $&@
? ;
4 K( D 5
D
D
3 ?=5 =785 B 5
D3=& 507E; 7 ?=5 7 (8 507E; 5 C 0 Y (∂S) D :=
8[
) +&5' 0 4+ 7-=<$ +&' P 6 5':)&' = {C A0 Y (∂S)} C*<%'D-/);< F -=<%< E ' # 'D$ 7C:$
Fq6 ?+&5 0 +&5' 0 < F # %$,$,'DCN+ 0 P < FsEA0 C 0 Y (∂S) \pJA0 $ # '*) - # %$,$,'DCN+ 0
P < FVE0 D +&5-/+ 6 -$ 3 # ' # 0 +&5' P );'*Y 1$ 4+&':);-/+ 0qA@ " < E ) ?+&53 s7D
- 0 # +&5' -/+&1).-/+&' $(+&' P -=+2d' E 0bA@ +&5' 0 '*lV+ ?+;'*);-=+ 0
p <4'O-/)&< F 00 ' A@ +&5' E ' # 'D$ %CD$C*) $,$('O$ - 0 ' # E ' A@ D +&5-/+ %$L-R<%$ P -/)&+
A@ - 0 ' # E ' A@ +&5' $(1),) 1 0 # 0EL@ );-R3 ' P 5' 0 < Ff +;5'*) P + 0 + <4'O-DY!' D
7$2 F CN) $,$ 0E - 0 ' # E ' e @ D +&5-/+JC ),);'D$ P 0 # $+ -[);- F +&5-/+ 6 -R$ $,5 +
a
@ ) 3 $ 3 ' )&' '*l Y!'*)&+&'*l 0 - P )&':Y 1$$ +&' P pJ0 $ # '*)+&5' ).$ + Z\1 4< # r p - P
P '*).-/+ 0 0 +&5' -/+&1).-/+&'g$ +&' P 6 5':)&'k+;5' C A0 $ +,)&1C*+&' # E ' # 'O$ 7C C*) $,$('D$
- 0 # <4':+ p q 2d' +&5' $(' E 3 ' 0 +f$,-=+&1);-=+&' # 2 F +&5 7$ P '*).-/+ 0 6 5'*)&' p %$
e-R<?);'Di
# F - Y!':i),+&':l A@ +&5' @ ).-=3g' -/+S+;5-/+ P 0 +- 0 # <%'*+ r = Y (p ) u i
8 ?).$ + 0 +&'+&5-/+c+&5'\).- F +;5-/+ E ' 0 ':);-/+;' # e 6 -$ 0 + $(5 + @ ) 3 p 2d'DC:-R1$('
2 F i ) P '*),+ F -=<%< )&' '*l Y!':),+ 7C*'O$ A@ - @ );-R3g' -/)&' $,-=+&1);-=+&' # i " $X+&5'
0 +&'*) ) @ +&5' P < FVE0 E ' (q, p , r) (p , q) %$ # 7$3 0 + @ ) 3 ∂S - 0 # -=<%<
0 +&'*) ) Y!'*)&+ %C:'D$ @E ' (q, p , r) i-/)&' )&' i'*l 0 +&5 7$ P < FVE0 +&5' ' # E ' e
C*) $,$ 0EE ' (q, p , r) 3 1$ + 0 i+&'*);$,'DCN+ ' ?+;5'*) p q ) p r \p <4'O-/)&< F e # 'O$ 0 +
0 +&'*);$,'DCN+ p q @ i- );- F 5 4+;$S- $,' E 3g' 0 + 0 +&5' i 0 +&'*) i ) A@ D +&5' 0b lV+&' 0 # r
S' ':l %$ - iPP < %' # Z F -$,$(13 P + 0 +&5'kC 0 $ +&)&1CN+ 0 A@ E ' (q, pi, r) 6 -R$
+;5' );$ +9+ 43g'X+&5-/+[- # %$&$('DC*+ 0 ' # E ' 6 -$MC*) $,$(' # 2 F P +&5V1$ e # 'D$ 0 +
C*) $,$ p r ' 4+&5'*) Q )&' Y!':) @ e 0 +&':);$('DC*+;$ p r +&5' 0 +&5' );- F 3 1$(+
a
i
i
5-:Y!' 5 4+ p r Z\1+k+&5 %$ %$ 43 P $,$ 42<%'h$ 0 C:' +&5' ' # E ' %$ 0 C # ' 0 +k+ - 0
i
Proof.
$(+;-
3.4. Dissection
1 0 ,$ -/+;1);-/+;' # ($ ' E 3 ' 0 +9- 0 # +;5'f);- F $(5 _ + 0E 7$9- PP < %' # A0 < F + @ );-R3 'O$ 0
6 5 %C.5b-=<%<$(' E 3 ' 0 +.$ -=)&' $,-=+&1);-=+&' # \
5V1$ VE ' (q, p , r) # 'D$ 0 + 0 +&'*);$,'DCN+
i+&'
=
%
<
<
'
D
'
$
%C
D
$
C
$
&
+
&
)
1
N
C
;
+
'
&
+
5
7$
/
&
+
1
.
)
/
&
+
'
(
$
#
#
#
0
E
0
0
P $(+;- F 6 ?+;5 0 D e
+g)&':3U- 0 $ + - 0 -R< F e:'b+;5' P ) P '*),+ %'D$ A@ +&5' # 7$,$('OCN+ 0 0 " < EAr
) ?+&53 s7Ds
3( 4 4 $#( 7E 5 D ( 4
'4 $;( 24% (8 B 5
Proposition
3.20
P
D (E( 5 B E507
5
3
&'79 '
? 3
P
5 ' $ +;-=+&':3g' 0 + 7$+,)&1' @K )f+;5'A);$ + 4+&':);-/+ 0 A@ " < EA ) 4+&53 V D
FHi P $ 4+ A0 V - 0 # pJ ) <%<%-=) F s SpJA0 $ # '*)M- 0 -/)&2 ?+&);-/) F 4+&'*).-/+ 0
A@ " < E ) 4+&53 V D - 0 # $(1 PP $,' +&5'X$ +;-=+&':3g' 0 +5 < # $- @ +&'*) +&' P - . ':+
# ' 0 +&'o+&5' # 7$,$('OCN+ 0 P < FsEA0 3 # ' #b# 1) 0E +;5 %$ ?+&':);-/+ 0
D
@ +;5' @ );-R3 ' 7$3 # ' # 0 +&5 %$ 4+&'*);-=+ A0 +&5' 0 ' ?+;5'*) l+;' 0 # r L' '*l
) );- Er # E ' 7$S- PP < %' # " $ +&5' C:1),)&' 0 + # 7$,$('OCN+ 0 %$S-R<43 $ + 0 7C*' 2 +&5
P '*).-/+ 0 $ 1+ P 1+o- @ ).-=3g' -RCDC ) # 0E + i ) P $ ?+ 0 s7Da - 0 # i ) P $ r
+ 0 s Og6 5 7C.5 7$ P )&'O$('*)&Y!' # 2 F +&5' @K <%< /6 0E -=+&1);-=+&' +&' P 2 F pJ ) < r
<7-/) F s @ +&5' @ );-=3g' 7$ 0 +\3 # ' # +&5' 0 +&5':)&'S-/)&' 0X0 ' 6 1 0 $&-/+&1).-/+&' #
$,' E 3g' 0 +;$o- 0 # 5' 0 C*' 4+ 7$ 0 + 3 # ' # 0 +&5' @ <%< /6 0E` -=+&1);-=+&' +&' P ' ?+;5'*) \
5V1$ P 7$$ + %<4< - @ ).-=3g' - @ +&'*) +;' P n- @ +&5' @ <4< /6 0E ?+&':);-/+ 0
+ );':3U- 0 $ + $(5 =6 +&5-/+ $ +;- F $ - 0 -R<43 $ + 0 7C*' # 7$,$('OCN+ 0Bj ' 5-DY!'
+ C.5'DC;m i ) P ':),+ %'D$ & & ; & - 0 # ;
7$- 0b 2VY 1$LC 0 $,' 1' 0 C:' A@ +&5'oC:- 00 7C:-=< # 7$,$('OCN+ 0 2d'DC:-R1$('
+;5' @ ).-=3g' 7$ 00sr $ %3 P <%'X-/+ # ' E )&'D' @ 1)SY!':),+ 7C*'O$9- 0 # C*1+ r ' # E 'D$ A0 < F! " 0
' # E ' E ' 0 '*);-=+&' # 2 F - );- F P ) P '*);< F C*1+;$ D 0 + + 6 $ 43 P <%' P < FsEA0 $2d' r
CD-=1$,'o+&5'o);- F $ + P $L-$$ _0 -$ ?+5 ?+;$ ∂D 8 ) o 2$('*),Y!' +&5-=+ -H);' ':l Y!':),+&'*l A@ $ 3g' D %$ CN)&'O-/+&' #
2 F ' ?+;5'*)fZ\1 %< # r p - P ) l+&' 0 # r L' '*l @ -q).- F 5 4+;$- 0 ' # E 'f+;5'o)&'O$(1<4+ r
0E Y!':),+&':l 7$ $ +,) 7CN+&< F C 0 Y!'*l 0 2 +&5 $ # 'O$ - 0 # );- Er # E ' # 'DC*)&'D-$('D$
+;5'b- 0E <%'D$g-/+g2 +&5 Y!':),+ 7C*'O$ A@ +&5'h' # E 'h+;5-/+ %$ 5 4+g2 F +;5' );- F Z +;5
lV+&' 0 # r )&' ':l - 0 # ).- Er # E ' ) %' 0 +B+&5' Y!'*)&+ %C:'D$ -=< A0E +&5' 4) C 0 $ +,);1CN+&' #
E ' # 'O$ %CD$9$(1C;5h+;5-/++;5'*)&' %$ 0 -R<?+;'*) 0 -=+ A0 0 +;5'f)&'O$(1<4+ 0E ) %' 0 +;-=+ 0
L'DCD-=<%< +&5-=+ A0 < F ' 0 # P 0 +;$ A@ 1 0 $,-=+&1);-=+&' # $,' E 3g' 0 +;$LC 1 0 +L-$- 0 -=<4+&':) r
0 -/+ 0 " <%$ 00 ' @ +&5'oY!':),+ 7C*'O$ 0 C # ' 0 +9+ - 0 1 0 $,-=+&1);-=+&' # $(' E 3g' 0 +
7$ ) %' 0 +&' # + /6 -/) # $-q)&' ':l Y!'*)&+&'*l A@ - 0F # 7$,$('OCN+ 0 P < FsEA0c ' 0 C:'f2 F
i ) P $ 4+ 0 s D -=+&1);-=+&' # 'D$ 0 + CN);'D-/+;'- 0_F )&' '*l + 6 0 0 - 0_F # 7$,$('DC r
+ 0 P < FVE0b
5' A0 < F P '*);-=+ A0 +&5-/+ 3g- F C*)&'D-=+&' $(1C.5 -h)&' '*l + 6 0
7$ lV+&' 0 # r S' ':l Z1+ +;5' 0 $(+;-/),+ 0E`@ ) 3 +&5' 0 '*lV+ 4+&'*);-=+ A0 @ " < EAr
Proof.
2 )
Chapter 3. Hamiltonian Polygons
) ?+&53 s :s0 ' A@ +&5'SY!':),+ 7C*'D$ 0 +;5'S+ 6 0 7$ 0 +\)&' '*l 0 - 0_F # 7$,$('DC*+ A0
P < FVE0 - 0F 3 )&'k2d'DC:-R1$('g+&5' 0 ':l+X);- F %$ $(5 + @ ) 3 +;5 %$oY!'*),+;'*l U
5 7$
P ) Y!'D$ ;
L' E -/) # 0E \0 +&'[+&5-=+-=<%< ' # E 'O$ A@ # %$,$,'DCN+ 0 P < FsEA0 $-=)&' C 3 r
3 0 6 ?+&5 +;5' @ );-R3 ' 2d' @ )&' +&5'A).$ +[);- F %$f$,5 + " )&' '*l Y!'*),+&':l a C:- 0
0 < F < $('U- 0 0 C # ' 0 + C 3g3 0 ' # E ' A@ 4+;$ # 7$,$('OCN+ 0 P < FsE0 D 6 ?+;5
+;5' @ );-=3g' @ 4+ %$5 4+M2 F - );- F s $(5 + @ ) 3 $ 3g' +&5'*)9);' ':lbY!'*),+&':l @
Z1+f+&5' 0 ' ?+;5'*) );- Er # E ' %$o- PP < 4' # 6 5 7C.5 0 C*'k- E - 0 CN)&'O-/+&'O$ - 0
D
C 3 3 0 ' # E ' A@ D - 0 # P 0 C # ' 0 ++ a )M+&5' 0 '*lV+9);- F 7$$,5 + @ ) 3 a
- 0 # a 7$ 0 + - )&' ':l Y!':),+&':l A@ - 0F # %$,$,'DCN+ 0 P < FsE0 - 0F 3 )&'k-=+o+&5'
' 0 # A@ +;5' 0 ':l+ ?+&':);-/+ 0>0 +;5'M3g'O- 0 + 43g'9+&5' +&5'*)J' # E ' 0 C # ' 0 + + a
>+&5' 0 'M+&5-=+ 6 -R$ 0 +J5 4+J2 F %$$ + %<%<^- C 3g3 0 ' # E ' A@
- 0 # D Z F
P
s
0 +\2 +;5gY!'*),+ 7C*'O$ A@ +&5'' # E '5 4+ 2 F -/)&'9)&' ':l 0
5'*);' @ )&'
C A0 + 0 1'D$+ 5 < # D
s
%$ C*'*)&+;- 0 < F +,);1'H2d' @K );'H+&5' );$ + ).- F 7$ $(5 + j 5' 0 ':Y!'*) - 0
P '*).-/+ 0 l+;' 0 # r L' '*l ) );- Er # E ' 7$ - PP < 4' # 0 - 0 ?+&':);-/+ 0 -=<%<
);'D$(1<4+ 0E # %$,$,'DCN+ 0 P < FsEA0 $ 5-DY!'oC 3g3 0 ' # E 'O$ 6 ?+&5 +&5' @ );-=3g'o-=< 0E
+;5'XC A0 $ +,)&1C*+&' # E ' # 'O$ 7C:$ ' 0 C:' $,1 PP $(' +;5-/+ 0 P '*).-/+ 0 7$9- PP < %' #
0 - 0 4+&':);-/+ 0 - 0 # +&5' # 7$,$('OCN+ 0 P < FVE0 D %$ 1$ + # 7$,$('OCN+&' # -R< 0E ).- F s $(5 + @ ) 3 $ 3g'X);' ':l Y!':),+&':l p @ D M @ 2 +;5 ' # E 'O$ 0 C # ' 0 ++ i C:<4'O-/)&<
/
&
)
'
C
3
3
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$
&
+
5
'
#
#
0
E
A
@
0
0
F 2 +&5 )&'D$,1<?+ 0E # 7$,$('DC*+ A0
pi
P
D
P < FVE0 $5-DY!' - C 3g3 0 ' # E ' 6 4+&5h+&5' @ );-=3g' 0 -R3 'D< F 0 C # ' 0 +S+ pi +&5':) 6 %$,' - 0 ' # E ' 0 C # ' 0 + + pi 3 1$ +q5-DY!'g2d':' 0 5 ?+q2 F - );- F a
+;5-/+ 6 -R$ $(5 + @ ) 3 - 0 +;5'*) );' ':lXY!':),+&'*l b 0 +&5' 43g3g' # 7-/+;':< F P );'DC*' # 0E
4+&':);-/+ 0 j 5' 0 ':Y!'*) - 0 ' # E ' 0 C # ' 0 + + -o)&' ':l Y!':),+&':l 7$ 5 4+ +&5' 0 '*lV+
).- F %$f$(5 + @ ) 3 +&5-/+)&' '*lHY!':),+&':l 2$(':),Y!' +&5-=+2 +&5 ' # E 'O$ 0 C # ' 0 +
+ b 3k1$ + 2^' C 3g3 0 ' # E 'O$ @ P - 0 # +&5' C )&)&'D$ P 0 # 0E # %$&$('DC*+ 0
P < FVE0 $2d'DCD-=1$(' +&5'*) 6 %$(' );- Er # E ' 6 1< # 5-DY!'q2d':' 0 - PP < %' # + b 0 '*+ # 4)bc&'OCN+ # '00 A+&' @ +&+&55' 'h' C # E 3g' 3 +;5A0-/+ ' 6 # -RE $' 5 A?+@ P2 F - 0 # D CD" - 0$k0+& 5+X' 5 );?+- F 2-RsC;m`7$ + $,5 + Q );' Y!':) s # 'O$ 0 + 5 ?+ bc 2d'DCD-=1$('+&5' 0 a );-sERr # E ' 6 1< # 5-DY!'M2^'Da' 0
- PP < 4' # + p H
5V1$ 6 5' 0 D 7$ # %$&$('DC*+&' # -R< 0E s +;5'*)&' -=)&' C 3g3 0
' # E 'D$ 6 4+&5 Pi 0 2 +&5 )&'D$,1<?+ 0E # 7$,$('DC*+ A0 P < FVE0 $ M0b0 'o$ # ' bc A0
+;5' +&5'*)U$ # ' +;5'h' # E ' 0 C # ' 0 +k+ p +&5-/+ 6 -$ 0 + 5 4+ 2 F 5 %$
a
i
P ) Y!'D$ &
%$ - E - 0 - C A0 $(' 1' 0 C*' A@ +&5' CD- 00 7C:-R< # 7$,$('DC*+ A0 + E '*+&5':)
6 4+&5 +&5' @ -RC*+ +&5-=+ l+;' 0 # r L' '*l 7$ C:-R<4<%' # 6 5' 0 '*Y!':) - );- F CN) $,$('O$ - 0_F
$,' E 3g' 0 + 0 +&5' 0 +&'*) ) A@ +&5' @ ).-=3g' A
3.5. Simplification
2$(':),Y!'g+&5-=+ -=+X+&5'U' 0 # A@ " < E ) 4+&53 s : -R<4< # 7$,$('DC*+ A 0 P < FsE0 $
-=)&' C A0 Y!'*l - 0 # 5' 0 C*' +&5' # %$,$,'DCN+ 0
%$ 0 %C:' 3.5 Simplification
n0 +&5 7$o$('OCN+ 0 6 ' # %$&C*1$,$o+&5' $ 43 P < C:-=+ 0 $(+&' P 0 i 5-$(' k>0 ) # ':)
+ C 0 $ +&)&1CN+o-U$ %3 P <%' P < FVE0 @ ) 3 - @ ).-=3g' P 6 ' 5-DY!' + )&'D3 Y!' 0 '
C:C:1),)&' 0 C*' A@ ':Y!'*) F # 12<4'Y!':),+&'*l @ ) 3 P 1)\$(+,);-/+;' EAF %$ + ':< 43 0 -/+&'
-q)&' ':l C:C*1),)&' 0 C*' 6 5 $,'o'*l %$(+&' 0 C:' 7$ E 1-/);- 0 +;':' # 2 Fhi ) P $ 4+ 0 s pJA0 $ # '*)S- )&' ':l CDC*1),);' 0 C:' p 0 P A@ - # 12<4'Y!'*)&+&'*l 5' 2VY r
1$ 6 - F + )&' # 1C:' p + - $ 0E <4'qY!i ':),+&':l @ P 7$+ $ %3 P < F 3 4+ ?+ +&5-=+
7$ + )&' P <%-C*' (p , pi , p ) 2 F (p , p ) 0 P [
5 %$[C:- 0 2^' # 0 '
@ (p , p , p i) 1@ )&3Ui $ 6 i 51-=+ 6 ' C:-=i<% < 1- 4 i5
D1&'5k+&5-=+ %$ # E' 0 ' # 2d':< /6X
i 1
i
i 1
7
(E D 5072 ( ";?
B Definition
3.21
k
U = (p, q1 ,( . . . ,? qk? , r)
4 ?=50785
785!785 5 7 ; '507 B 59(!
;
B
P
qi 1
i 79 k( '5 # $&@ B
C := (r, qk , . . . , q1 , p, r)
?=50 ?=5A( 5 50 B 5
( B 4 5
D
(
(C)
\
∂U)
∂S
=
(q
,
.
.
.
,
q
)
1
k
3
?=587 4 3( 5
3( B 4 5
D <
cap
P
(q1 , . . . , qk )
anti-cap
P
7
4 5 ( 43%$ A?=5 "7858 ?=59( 50( D B 4 -?=5 '587 5 k=1
B C7 B q1
( ";?
3( B 43F5
D D $ q
,
.
.
.
,
q
,
r)
P
wedge
3( D F5 (p,
1
k
'587 5 7 4 :D
3(
q i P
1 i k
(q1 , . . . , qk )
B
q
r
p
s
Figure 31:
P
?=5 785 5 > BEB
797858 B 5 ' 587 5 3
79 ( B q
P
( 4 5
D&@5 ( B 8 5 B ( 5
(p,
q, r)
6587 ( 5 B ( ?=5 79 =&# 5
2$('*)&Y!'X+;5-/+ @ )
C
r
(r, q, s)
+ g@ )&3 -gC A0 Y!'*l P < FsEA0 0
pq
:D
3
'E 0 ?+ 0 s Vo0 ' A@
a
Chapter 3. Hamiltonian Polygons
+;5' Y!':),+ 7C*'D$ q 1 i k 3k1$ +o2d' )&' ':l 0 P 8 E 1)&' $(5 /6 $ - 0
':ls-R3 P <4' @ ).-=3gi' 6 ?+&5 2 +&5 CD- P $g- 0 # - 0 + r CD- P $ @ ) %<%<41$(+,);-/+ 0 @ 0 E 1)&' 6 ' 6 - 0 ++ ':3 P 5-R$ %e:'S+;5-/+ -)&' '*l Y!':),+&':l %$ - 0 - 0 + r CD- P +&5' 0 4+ %$
$,5 /60 2 F - $ 1-=)&' # + @ 6 'MCD- 0 ' 0 $(1)&'+&5-/+ '*Y!'*) F # 12<%'Y!':),+&'*l @ P
- PP 'D-=);$f-/+[<4'O-R$ + 0 C:' 0 - 6 ' # E ' 4+ %$'D-$ F + C*)&'D-=+&' -U$ %3 P <4' P < FVE0
@ ) 3 - @ ).-=3g'o2 F 3g'D- 0 $ A@ +&5' @ <%< =6 0E P '*);-=+ 0
<& 785 Operation 5 (Chop-Wedges(P))
- @ );-=3g' P
"507E; " $ < A0E -R$X+&5':)&' %$ - 6 ' # E '
)&' P <7-RC:'g+&5'
(p,
q
,
.
.
.
,
q
,
r)
1
k
P -/+&5 (p, q1, . . . , qk, r) 0 P 2 F +&5' $ 0E <%'o' # E ' (p, r) P
q
q
r
−→
P
p
r
p
s
P
s
Figure 32:
-?= @=3=& ?=5 4 5
D&@5
2$(':),Y!'q+&5-=+ p 5 P r>j ' # E 'D$ # 'D$ 0 +oC.5- 0E '
(p, q, r)
>0 @ -CN+ d6 ' 6 %<%<
o
0 +fC.5 P RT 6 ' # E 'D$f-R$f-=),2 4+,);-=) 4< F -$+&5' # 'O$,CN) P + 0 -/2 Y!'q3 E 5_+f$(1 Er
E 'O$ + >0 $(+&'D- # @ )o'D-RC.5 # 12<4' Y!':),+&':l 6 ' 6 4<%<\<%-=2^'D< 0 ' C:C:1),)&' 0 C*' -R$o6 ' # E ' +;5-/+ %$ -$ (+ 2d' );':3 Y!' # 2 F`p 5 P r>j ' # E 'D$ 2Y 1$,< F 4+95-R$
+ 2d' ' 0 $(1)&' # +&5-=+S+&5 7$ P -/),+ %C:1<%-=) CDC*1)&)&' 0 C:' 7$ 0 # ':' # P -/)&+ @ - 6 ' # E '
0 P " $ @ )[+&5' +&5':) P ':);-/+ 0 $ 6 ' < 4m!'q+ P ) Y!' +&5-/+ -=<7$ +&5' )&'O$(1<4+ A@
p 5 P r j ' # E 'D$ - PP < %' # + - @ ).-=3g' F %':< # $ - E - 0 - @ );-R3 ' h
5' @ <4< /6 0E
P ) P $ 4+ 0 P ) Y # 'O$S- ).$ +$ +&' P 0 +&5-/+ # 4)&'DC*+ 0
V(P)
?=5! F Proposition
3.22
# $K&' ? ; ( ; (
j ' 5 -DY!'+ @ P # 'D$ 0 +
' 0 +&'f+&5' 1+ P 1+
Proof.
) 3
59(
-?= (
3
%$ B 5 B 65
D
$(5 / 6 +&5-=+cC.5 PP
0 Y-=< # -/+&'`- 0F
P < FsEA0hA@\p 5 5
D&@50( ( 0 ET
A@ +&5'
P r>j ' #
- 6 ' # E ' U = (p, q , . . . , q , r)
@ );-=3g' P ) P ':),+ %'D$U13g' 0 + 0k ' # E 'D$2 F P A
3.5. Simplification
5 < # $ 2^'OC:-R1$(' (P ) (P) p <4'O-/)&< Fg0 Y!':),+&'*l - PP 'D-=);$ 3 )&'
A@ +;' 0 0 P -R$ 4+o- PP 'O-/);$ 0 P o
5'*)&' @ )&' i ) P ':),+ F %$f$&-/+ 7$ ' #
-$ 6 'D<4< Z F # '8 0 4+ A0 A@ 6 ' # E ' ?+ 7$
':+ K :=s 0 ((p,
q1 , . . . , qk , r, p)) \ ∂U
P -/),+ %C:1<%-=) pr
∂S = 6 5 7C.5 %3 P < %'D$ &
K ∂S =
j ' C:<%- 43 +&5-/+ 0 ' # E ' 0 C # ' 0 +X+ H0 ' A@ q 0 P 1 i k CD- 0
0 +&'*);$,'DCN+ K Z F # 8' 0 4+ A0 A@L6 ' # E 'U+&5 7$ %$qC:'*),+.i- 0 < F +,)&1' @ )q$,' E 3g' 0 +
' # E 'D$ 21+L+&5':)&' C 1< # 2d' - Y 7$ 42 4< 4+ F ' # E ' A@ P 0 C # ' 0 +S+ k0 ' @ +;5'D$('
Y!':),+ 7C*'O$ 1 PP $(' $(1C.5 - 0 ' # E 'b':l 7$ +;$ @ 4+ %$U- C:1+ r ' # E ' A@ P +&5' 0
3 1$ +9+,);-DY!'*);$,' +&5 %$C:1+ r ' # E ' - @ +&'*) q 0 $ +&'D- # A@ P ) C*'D' # 0E -R< 0E U P
<7$(' +&5':)&' 7$ - 0 ' # E ' 0 C # ' 0 +k+ q - i0 # 0 +&'*).$('DC*+ 0E K +;5-/+ %$ 0 + C:1+ r ' # E ' A@ P Z\1+g+&5' 0 P %$ 0 +U$ i43 P < Fr C 00 'DC*+&' # 0 C 0 +,).- # 7CN+ 0
+ +;5' -$,$(13 P + 0 +;5-/+ P 7$ - @ );-R3 ' ' 0 C*' +&5'gC*<7- %3 5 < # $X- 0 # -$XC 0 $(' 1' 0 C*' P %$S$ 43 P < F_r C A00 'DC*+&' # 3.5.1 Labeling Wedges
" $ @ -/)-R$L+&5' 0 # 1CN+ ?Y!'f2) # E 0E $ +&' P %$C 0 C*':) 0 ' # +&5'o3 $(+ %3 P )&+;- 0 +
P ) P ':),+ F A@ +&5' # %$&$('DC*+ 0 %$ 6 5 %C.5 ' 0 $(1)&'D$\-[C 3g3 A0 ' # E ' @ 'D-C.5
# %$,$,'DCN+ 0 P < FsEA0 D 6 4+&5 +;5'q$(1),) 1 0 # 0EU@ );-R3 ' " < A0E +&5' < 0 '
A@Si ) P $ 4+ A0 s 6 ' ) 1 E 5< F m 0=6 5 /6 + 3U- 0 +;- 0 - 0 -=<%3 $(+ 0 7C*'
# %$,$,'DCN+ 0 # 1) 0E +;5' -=+&1);-=+ 0 - 0 # 7$,$('OCN+ 0 i 5-R$,' Z1+ 0/6 6 '-=<7$ 5-:Y!'9+ +;-/m!' 0 + -RCDC 1 0 + +&5' %3 P < C:-=+ A0 i 5-R$(' 6 5':)&'$ 3g'' # E 'O$ A@
+;5' @ );-R3 ' # %$,$&- PP 'D-=)9-R$+&5' F -=)&' P -/)&+ A@ - 6 ' # E 'f+&5-=+ 7$SC.5 PP ' # T 0
p 5 P r j ' # E 'D$ " $ -R<?)&'O- # F 0 # 7C:-/+;' # 6 5' 0hp 5 P r>j ' # E 'O$ 6 -$ 0 +&) # 1C*' # 6 ' # 0 +
C.5 P Tb6 ' # E 'D$ -/),2 4+,);-=) %< F! 21+ 0 $ +;'D- # 6 ' 85 0 ' C:C:1),)&' 0 C*' A@ 'D-C.5
# 12<4' Y!'*)&+&'*l 0 P -$ - 6 ' # E ' " Y!'*)&+&'*lqC:- 0 2d'DC 3 '- # 12<%'JY!':),+&'*l A@
+;5' @ ).-=3g' 0 +;5)&':' # T ':)&' 0 + 6 - F $ - 0 # @K ) 'D-RC.5 @ +&5'O$('CD-R$('O$ 6 ' # 'D$,C*) ?2d'
2d'D< /6 6 5 7C.5 CDC*1)&)&' 0 C:' A@ +&5'fY!':),+&':l %$S+ 2d' <7-/2d'D<4' # -R$- 6 ' # E ' j 5 ' 0 '*Y!'*)J-fY!'*),+&':l 7$)&'*Y %$ ?+&' # 0 - E ' # O' $ 7C !6 '<7-/2d':<+&5' ) E 0 -=<
CDC*1)&)&' 0 C:' A@ +&5'[Y!':),+&':l -R$ P -/,) + A@ - 6 ' # E ' )
)
> 0 +&5' C:-R$,' 6 5'*);' l+&' 0 # r L' '*l E ' 0 '*);-=+&'D$[3U- 0F C*1+ r ' # E 'D$A 8 E r
1)&' ao0 i - E ' aa; 6 '9<7-/2d'D<+;5'SY!'*)&+&'*l C:C:1),)&' 0 C*'O$ +&5-/+J-/);'MC*< $('*)
+ t +;5- 0 + r -R$ P -/),+ A@ - 6 ' # E ' Chapter 3. Hamiltonian Polygons
j 5 ' 0 *' Y!'*) ).- Er # E ' %$J- P P < %' # + -o)&' ':l Y!':),+&':l p 6 '9<7-/2d':<+&5'
i
) E 0 -R< C:C*1),)&' 0 C*' A @ p R- $ P /- ),+ A@ - 6 ' # E ' i
" 0 ' # E ' 0 P 7$JCD-=<%<4' # - 4 5ED&'5 5
D&@5 @ - 0 # 0 < F @0 ' A@ ?+.$ ' 0 # P 0 +.$
7$S<7-/2d':<%' # -R$- 6 ' # E ' 2Y 1$(< F 4+5-$ + 2^'\' 0 $(1)&' # +;5-/++&5'<%-=2^'D<4' # C:C:1),)&' 0 C*' A@ - Y!':),+&'*l
0 # 'D' # C )&)&'D$ P 0 # $+ - 6 ' # E '-R$ # E' 0 ' # 0 8' 0 4+ 0 s s 5 7$ 7$ 1)
E -=< 0 +&5' @ <4< /6 0E 5-/+L3 'O- 0 $ 6 '[3 1$ + 0 ':Y!'*)<%-=2^'D<- 0 - 0 + r C:- P -$6 ' # E ' )9C*)&'D-=+&'X- 0 - 0 + r CD- P -=+9- Y!'*),+;'*l +&5-/+ 7$S<%-=2d':<%' # -$- 6 ' # E ' " $ - ).$ +J$ +&' P <%'*+ 1$ C;5-/);-CN+&':) 4eD'C:'*),+.- 0 C*<7-R$,$('O$ A@ Y!'*),+ %C:'D$+&5-=+ 3U- F
0 '*Y!'*)L2d'DC 3 'X- 6 ' # E ' # 1) 0E " < E ) ?+;53 s :s
)
Definition
'3.23
587 5
P (
D 507 7E 5
4 ? 4
(8- B 5 D3(E(5 B P
3
( B 4
F5
D
:
D $ v
P
safe
) v 3((8 79 B %$ B 45 < P
) 7 v ( 50:D < ( 5+& 50 D&@:4
) 7 v ( 785 5 (
3=&# 5 '507 65 P ? ;3( 3 B D 50:6 D3(E(5 B 5
D&@5 ? ; ( K 5
D&@5 :D ? ; B
(
< A4 (870 B $
P
]P v
B '5 &;F50(
7
)
( D v
F5 '
587 5
P
? ; 3( K 8 5 5
D K( 4 5
D&@5
3
(
:D $ ; 5EK(8 5 705 5 4 3
(a,
.
.
.
,
b)
P
safe
a
3( ( 5 <
'587 5 7 785 5 4 < 3
? ;- (A K ( 5 3(
b
P P
705 5879785
D K
(
unsafe
7
" $ - 0 ':ls-R3 P <4' @K ) +;5'S<%-$ + + F P ' A@ $,- @ 'Y!'*)&+ %C:'D$ 0 '3U- F +&5 0 m A@ +&5'
$,'DC 0 # C:C*1),)&' 0 C*' A@ -qY!'*),+&':l E ' 0 '*).-/+&' # 2 F );- Er # E ' V6 5'*)&'f+;5' ) Er
0 -R< C:C*1),)&' 0 C*' 6 -R$9<%-=2^'D<4' # -$- 6 ' # E ' -C:C ) # 0E + 1)<7-/2d':< 0E )&1<%'D$ 5' @ <4< /6 0E P ) P $ 4+ 0 <%' E ?+ 43U-/+;'D$ 6 5 F +;5 $('LY!'*),+ %C:'D$\-/)&'MC:-=<%<%' # $,- @ ' ( 5 4507 5 A 7E 5 ( (870 B $ B '5 < 4
D (
Proposition
3.24
(5 B "#%$&'( 4 ?=50785 @"5E79(
( 5 '507 65 4 5 4507 85 B 5
@
& 79 ?' ( 5 7 85 85 F
5 D K( 4 5 D&@532 ?=5 B 7 ( 5 1) 0E " < E ) 4+&53 s D +&5' @ );-R3 ' P - 0 #
# 2 F +&5'X+;5)&':' 2-R$ 7C P '*);-=+ 0 $ Z\1 4< # r p - P
);- E r # E ' 0 < F ) 0 ' # 7$,$('DC*+ A0 P < FsE0 7$ $ 43
Proof.
3 # '
?+;$ # %$&$('DC*+ 0 l+;' 0 # r L' '*l P < F $ P < ?+q-=< 0E -=)&'
-0#
);- F
R
3.5. Simplification
'D3U- 0 -=+ 0E @ ) 3 0 ' A@ ?+.$9);' ':lbY!'*),+ 7C*'O$ pJ0 $ # ':)o-U$,- @ ' Y!'*),+&':l p 0
i
P
@ p %$X$ +,) 7CN+;< F C 0 Y!'*l 0 P +&5' 0 0hE ' # 'D$ 7C 6 %<%< );'*Y %$ 4+ 4+ " <7$ 0 ' 4+&5'*) i l+;' 0 # r L' '*l 0 ) );- Er # E 'gCD- 0 2d'U- PP < %' # + p " <%<\2-R$ %C
P '*).-/+ 0 $ # 'DC*)&'D-$('+&5'- 0E <%'D$ -/+ 2 +&5 Y!'*)&+ %C:'D$ A@ +&5'S' # E '+&i5' F );' P <7-RC:' ' 0 C:' p )&':3U- 0 $S$ +&) %C*+&< F C 0 Y!'*l +;5) 1 E 5 1+ i
@ p 7$ - 0 ' 0 # P 0 + A@ -[$(' E 3g' 0 + # %- EA0 -R<s+&5' 0 4+ 7$ -$ 0E <4'Y!'*)&+&'*l A@
+&5-=+ i7$M$(+,) 7CN+&< F C 0 Y!'*l 0 2 +&5 # 7$,$('OCN+ 0 P < FVE0 $ 6 5'*)&' 4+[- PP 'D-=);$ P 5'*)&' )&' 'qC:- -=) 1'q-R$
@ 6
0 E
@ )+&5'q$ +,) 7CN+;< F C A0 Y!'*lhY!':),+ 7C*'D$-/2 Y!' o 43 r
%<7-/)&< F @ p 7$9-k)&' '*lb$ 0E <%' Y!'*),+;'*l A@ P +&5-=+ %$ 0 C # ' 0 ++ - # 7$,$('DC*+ A0
' # E 'S+&5-=+ i%$ 0 + - 0 ' # E ' A@ P - 0 # +&5-/+ C*1+.$ ] v 0 + + 6 $ +,) 7CN+&< F C A0 Y!'*l
P
- 0E <4'O$ +o)&'D3g- 0 $[+ C 0 $ # '*)X+&5'kCD-R$(' +&5-=+ v %$o- # 12<%'qY!':),+&'*l A@ P +&5-=+
7$ 0 +b<%-=2d':<%' # -$b- 6 ' # E ' 5' 0 2 F 1)b<%-=2d':< 0E $,C.5'D3g' +&5' +&5':)
C:C:1),)&' 0 C*' A@ v 0 P %$h<7-/2d'D<4' # -R$b- 6 ' # E ' " CDC ) # 0E + v
- PP 'D-=);$9-$-k$ +&) %C*+&< F C A0 Y!'*l Y!'*),+&':l 0 2 +&5 # 7$,$('DC*+ A0 P < FVE0 $S- 0 # 6 '
CD- 0b0 C:' - E - 0 -=) E 1' -R$S2^' @ )&' " $ -fC 0 $(' 1' 0 C*' A@ i ) P $ 4+ 0 s [6 ' # o0 +\5-DY!'+ [6 ),) F -=2 1+
- 0 + r C:- P $CN)&'O-/+&' # -/+\$,- @ ' Y!':),+ 7C*'O$ A@ +&5' @ );-R3g'J2d'DCD-=1$,'L+&5' F 6 %<4< 0 ':Y!'*) 2^'
<7-/2d'D<4' # -R$9- 6 ' # E ' # 1) 0E +&5' -=< E ) ?+&53 ' 0 C:' 1) 0 +;'*)&'O$ + 7$ @ C:1$(' #
-=+1 0 $,- @ 'X- 0 + r C:- P $ 0 +&5' @K <%< /6 0E
3.5.2 Anti-Cap Control
n0 &+ 5 %$L$('OCN+ 0 6 ' 6 4<%<- 0 -=< F eD'[+&5'[2-$ 7C P ':);-/+ 0 $L- 0 # $(5 =6 +&5-/++&5' F
# 0 + C*)&'D-=+&' + _ 3U- 0F 1 0 $,- @ 'g- 0 + r C:- P $ j ' CD- 00 +X-DY # +&5-/+ $ 3 '
1 0 $,- @ ' - 0 + r CD- P $ -/);' CN)&'O-/+&' # 0 +&5' C 1).$(' A@ +&5'\-R< E ) 4+&53 21+ 6 '\2^':+,+&':)
3U-=m!'$(1)&'S+;5-/+ +&5' F # 0 +\2d'DC 3g' # 12<4'LY!':),+ 7C*'D$ A@ +&5' @ );-R3 ' " $ @ )
+;5' # 7$,$('DC*+ A0 $ 0 7C*'[Y!'*).$(1$S-=<%3 $ + 0 %C:' 4++;1) 0 $ 1+L+&5-=+ 6 'o-/)&' 0 ' @
+;5'*)&' ':l 7$ +;$f-=+3 $ + A0 'q1 0 $,- @ ' - 0 + r CD- P -=+f- + 43g' [
5' )&'D-$ 0 7$M+&5-=+
6 ' CD- 0 %3 3g' # %-=+&':< F # 'D-R< 6 4+&5 - $ 0E <%' - 0 + r C:- P 2 F $(5 + 0E +&5' 0 '*lV+
).- F @ ) 3 +;5'*)&' j ' $,- F +;5-/+X- 0 P '*);-=+ 0 B 785
;650( - 0 - 0 + r C:- P @ - 0 # 0 < F @ 0 +&5'
1+ P 1+ @ ).-=3g' P +;5'*)&' - PP 'D-/).$ - 0 - 0 + r C:- P +&5-/+ 6 -$ 0 +h-=<4)&'D- # F - 0
- 0 + r C:- P 0 +&5' 0 P 1+ @ );-R3 ' P 2$('*),Y!'X+&5-=+ @ - 0 P '*);-=+ 0 CN)&'O-/+&'O$M- 0
- 0 + r C:- P +;5' 0 +&5 7$ - 0 + r C:- P - PP 'D-=);$ < C:-=<%< F -/+ 0 ' @ +&5' Y!'*)&+ %C:'D$ @ )
6 5 %C.5H- 0 0 C # ' 0 +' # E ' C.5- 0E 'D$ # 1) 0E +&5' P ':);-/+ 0
Proposition
3.25
K B +3 $
( ; 78;5
D>( 5+& 4 7EDK( ( 5
?=58 D 59( K B
$ ( 5785 5 Chapter 3. Hamiltonian Polygons
(
D 507 7E 5
4 ? 7950+;
? ; D 59(
P
u
465879:; 2785E '587 ( @"( 5> ? ; ?=58785 ( 4 ?=(5 50:D < <
( 795065
D 50
S
s
V(P)
785 5 '587 5 3
; 7E;65A3( @ 5
D
3
P
(P, u)
705
;5 $ ( 5 B :D D 59( K &'50 507E;5
4 3 pJ0 $ # ':) -S$ 0E <%' P '*);-=+ 0 Z1 4< # r p - P - 0 # # ' 0 +&'+&5'\Y!'*),+ 7C*'O$ -R$
0 i P $ ?+ 0 s a 2 F p q r - 0 # E ' (q, p , r) = (q = q , . . . , q = r) @K )$ 3 ' k B ' 0 +&'Ji +;5' 0 P 1+ @ );-=3g' A@ +&i5 7$Z\1 4< # r p - P 0 2 F Q - k0 # +&5'
1+ P 1+ @ );-=3g'2 F Q " $,$(13g' 6 4+&5 1+ < $,$ A@VE ' 0 '*);-R< 4+ F +&5-/+ u(p ) = +1 i
Z1 %< # rp - P P ) # 1C:'D$U':ls-CN+&< F A0 ' 0 ' 6 )&' '*l Y!':),+&':l q '*+ C :=
6 5':)&' ` 7$LC;5 $,' 0 3U-/l 43U-=<%< F $(1C.5 +;5-/+ ` < k - 0 # q %$ -=+
(q0 , . . . , q`) ) -R<4<
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i `
A@ E ' # 'D$ %C i `+1
Z F p %$B-9C 0 Y!'*l $ 0E <%' Y!':),+&':l A@ Q - 0 # 5' 0 C*'-M$(+,) 7CN+&< F C A0 Y!'*l
$ 0E <%' Y!'*)&+&'*l Ai@ Q ' 0 C*' @ p %$ P -/),+ A@ -C:- P (. . . , t, p ) 0 Q %3 P < 4'O$
+;5-/+ p %$ -/+ 0 Q +&5' 0 (. .i. , t) %$9- C:- P 0 Q +&5-/i+' 0 # $S-/+ t i
8 ) +&5'o$(1C:C:'D$,$ ) r A@ p 0 Q +&5'f$ 4+&1-=+ 0 %$ P +&' 0 + 7-=<%< F # T '*);' 0 +
2d'OC:-=1$(' 4+;$ P )&' # 'DC*'O$,$ ) C.5i- 0E 'O$ @ ) 3 p + q 1 PP $,'h+&5-/+ r %$
P -/),+ A@ -hC:- P
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-kC:- P 0 Q +&5':) 6 %$,' r 7$L)&' '*l 0 Q @ r 6 -$k-R<?)&'O- # F P );'D$(' 0 + 0 P +;5' 0 r 7$k$&- @ ' 0 P 1 PP $(' r %$
1 0 $,- @ ' 0 P " $ p %$ ) 4' 0 +;' # + =6 -/) # $- 0 1 0 $,- @ 'X)&' ':l Y!':),+&':l 0 Q i
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@K <%< /6 $ 0 # 1C*+ 4Y!':< F +;5-/+ +;5'*)&' 7$ - Y!'*),+;'*l 0 P +&5-/+ 7$ ) 4' 0 +&' # + /6 -/) # $
r 0 C 0 +&);- # 7CN+ 0 + +;5'k-R$&$(13 P + 0 +;5-/+ 0 P 0 Y!'*),+&':l %$ ) 4' 0 +&' #
+ /6 -=) # $f- 0 1 0 $,- @ ' )&' '*l Y!'*),+&':l q ' 0 C:' @ r 6 -$o-=<4)&'D- # F P )&'D$,' 0 + 0 P
+;5' 0 r %$M$,- @ ' 0 P - 0 # - 0_F - 0 + r C:- P ))&' '*l + 6 0 C*)&'D-=+&' # -=+ r 7$9$&- @ '
0 Q - 0 # 5' 0 C*'o2 F i ) P $ 4+ 0 s 0 P +&5':) 6 %$,'k+&5'qY!'*),+;'*l r 5-R$[2d'D' 0 C*)&'D-=+&' # 0 +;5 %$ -/+&1).-/+&' $ +&' P 0 P )&'DC:' # 0E Z1 %< # rp - P P ':);-/+ 0 6 5':)&' - $,' E 3g' 0 + rs @ ) 3 S 5-R$U2^'D' 0
Proof.
)
3.5. Simplification
&$ -/+&1).-/+&' # - 0 # u(s) = −1 Z1+M+&5' 0 s - 0 # 4+;$ P )&' # 'DC:'D$,$ ) 0 +&5' @ );-R3g'
-=+ +&5-=+ P 0 + @ )&3 - 0 -=<4+&':) 0 -/+ 0 " $Z\1 %< # r p - P P )&'O$('*)&Y!'D$ ) %' 0 +;-/+ 0 $ +;5-/+-=<4+&':) 0 -/+ 0 3 1$(+L-R<%$ '*l 7$ + 0 P 0 C 0 +,).- # 7CN+ 0 + k 1)-R$&$(13 P r
+ 0 +&5-=+ u # 'D$ 0 +[5-:Y!' - 0F -=<4+&':) 0 -/+ 0 0 P 5'*)&' @ )&' r - PP 'O-/);$
-R<?);'D- # F -$S-qY!'*)&+&'*l 0 P - 0 # 6 'XC:- 0 -/) E 1' -$-/2 Y!' '*lV+ o6 ' 6 1< # < 4m!' + P ) Y!' - 0 - 0 -=< AE + i ) P $ 4+ A0 s @ )
lV+&' 0 # r L' ':l W 0@ ),+&1 0 -/+&'D< F l+&' 0 # r L' '*l C:- 0 CN);'D-/+;' 1 0 $,- @ ' - 0 + r
CD- P $M21+ @ ),+;1 0 -/+;':< F
-=+f3 $ + 0 ' -$< 0E -R$ 6 ' CD- 0 ' 0 $,1)&' +&5-=+
+;5'Y!'*),+&':l r +;5-/+ - PP 'O-/);$ -R3 0E +&5' P -/).-=3g'*+&':);$ %$\C A0 Y!'*l ) $,- @ ' 0 P >0 +&5' P )&'DC 0 # 4+ 0 $ A@c lV+&' 0 # r L' ':l
7$
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?=5 '507 65 3 658:D 5 5 3 ( B 45 Proposition
3.26
r
?=58 50:D 5 5 B 785
;650( $> 5 ( 5 07 5 5 4 3 :
5 %$
P
D ; (0 5 ( 5 B ;
5 7 (q, t)
q
t
pJ0 $ # '*)J- 0 P ':);-/+ 0 l+&' 0 # r L ' '*l - 0 #U# ' 0 +&'S+&5'LY!':),+ 7C*'O$ -R$
Proof.
0 +&5' # 'D$,C*) P + A0 @ lV+&' 0 # r L' ':l 2 F p r q - 0 # t - 0 # <%'*+ x 2d'f+&5'
P 0 + 6 5'*);' +&5' );- F s 5 ?+.$ qt L 'OC:-R<4< +;5-/+ x qt -R$[- P );'DC 0 # ?+ 0
A@ l+&' 0 # r L ' '*l 0 P -/),+ %C:1<%-=) s - 0 # qt -=)&' 0 +qC <%< 0 'D-=) " $ 1$(1-R<
i
# ' 0 +&'[+&5' 0 P 1+ @ );-R3g' A@ +&5' P '*);-=+ A0 2 F P - 0 # +&5' 1+ P 1+ @ );-R3g'f2 F
" $,$(13 ' 6 4+&5 1+L< $,$ A@E ' 0 '*);-R< 4+ F r = p Bj 'f5-DY!'M+ Y!'*) @ F +&5-=+
P
i 1
CD- P $ 0 P );':3U- 0 C:- P $ 0 P p <4'O-/)&< F p 7$- $ +&) %C*+&< F C 0 Y!':l $ 0E <4'XY!'*)&+&'*l 0 P S ' 0 C:' - P $&$ ?2<4'
CD- P (. . . , p i , p ) 0 P );':3U- 0 $- CD- P (. . . , p ) 0 P i
i 1
i 1
@ r 7$ C 0 Y!'*l 0 P +;5' 0 4+ 7$ $ +,) 7CN+;< F C 0 Y!'*l 0 P ' 0 C*' - P $&$ ?2<4'
CD- P (r = p , p , . . .) );':3U- 0 $-UC:- P (p , . . .) 0 P M>@ r %$M)&' '*l
+;5' 0 4+ 7$$,i- @ '1 0 i P2 2 F -R$&$(13 P + 0 - 0 # 5i' 0 2C*' # 1'X+ i ) P $ ?+ 0 s -R<%$ $&- @ ' 0 P " <%< 0 +&':) )LY!'*)&+ %C:'D$ 0 +&5'[+ 6 E ' # 'D$ %CD$S-/)&' C 0 Y!':l - 0 # E 1-/) # ' # 0 P 2 F +;5' $,- @ 'qY!'*),+ %C:'D$ p ) r XpJ 3 P -/);' # + P +&5':)&' -/)&' + 6 0 ' 6
);' ':lkY!'*)&+ %C:'D$ 0 P q - 0 # ti p <4'O-/)&< F +;5' F @ )&3 - 0 1 0 $,- @ 'M)&' '*l + 6 0
j ' 6 4<%<c$,5 /6 +&5-=+9-/+<%'D-R$(+ 0 ' A@ +&5'D3 7$- C:- P 0 P ' :+ q 2^' +&5' );$ +9$(+,) 7CN+&< F C 0 Y!':l Y!':),+&'*l A@BE ' (q, x, p ) >':l 7$ +;$2^' r
CD-=1$,' p 7$0$(+,) 7CN+&< F C 0 Y!'*l 0 P . - 0 # <4':+ t 2d'L+&5-' );$ + $ +,i) 7CN+;< F C A0 Y!'*l
Y!':),+&':l Ai@ E ' (t, x, p , r) ) r @B0 $(1C.5bY!':),+&'*0l '*l %$(+;$ pJ0 $ # ':)+&5' );- F $
i
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−
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tt
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i
+;5 %$c+&) %- 0E <4' - 0 # iE ' (t, x, p , r) $ +.- F $c+ +&5' <%' @ + A@ −
6 4+&5 0 +&5' +&) %- 0E <4'
tt0
i
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0 +;'*);$('OCN+ -=+X- P 0 +
− - 0 # −
y
tt
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CD- 00 + 0 +;'*);$('OCN+9$ 0 iC:' 0 ' $ +.- F $ 6 4+&5 0 +&5'f+,) %- 0E <4' (p0 , x, q) -00 # +&5'
+&5'*) $ +.- F $ 6 4+&5 0 +&5' 1- # ) 4<7-/+&':);-=< (t, x, p , r) - 0 # +&5i ' 0 +&':);$('DC*+ A0
i
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i
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+;5' 0 t %$S- C:- P 8 E 1)&' 2 ':<70$(' q 7$S-kCD- 0P 8 E 1)&' n- 0 P 0
t
x
q
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y
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r
](x, t, r)
5 5
5
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t
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?=5( 5+& 5058:D
q
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t
r
3( B 3
50:D
@ t - PP 'D-=);$M+ 6 7C*' 0 E ' (t, x, p , r) 6 ' 5-DY!' + 3U-=m!' $(1)&' +&5-/+M+&5'
);' ':l C:C:1),)&' 0 C*' A@ t 7$g- C:- P A@ iP +&5-/+ CD- 0 2d'HC;5 PP ' # T <7-/+;'*) n0bE ' 0 '*).-=< t CD- 0 2d' - 0 - 0 + r C:- P 21+S+&5' @K <%< /6 0E C ) <4<7-/) F A@i ) P $ r
+ 0 s a $,1 EE 'O$ +;$S- 6 - F + ' 0 $,1)&'o+;5-/+ t %$S-R< 6 - F $- C:- P 0 +;5-/+C:-$(' :D
@ 5
79(
3 50:D 5 5 E ' Corollary
3.27
(p
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x,
q)
p
q
t
i
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4 B 5 E ' ?=58 ?=5 705 5 BEB 797850 B 5 3
( (t,
x,
p
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r)
t
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i
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i
C 0 Y!':l Y!'*),+&':l A@ (p , r, t, x) 5V1$ +&5' ).- F −
0
−
= −
qq
qp
i
0
i
i P 4+ 0 s a CD- 00 + 0 +&':);$('DC*+ tt 0 +;5' 0 +;'*) ) A@ (p , r, t, x) - 0 #
i
5' 0 C:'f+&5'f);' ':l CDC*1)&)&' 0 C:' A@ t 7$- 0CD- P 0 P >@ t - PP 'D-=);$ + 6 7C*' 0 E ' (t, x, p , r) +&5' 0 +&5'`);- F s CD- 0 2d' $,< E 5_+&< F
) +.-/+&' # -/) 1 0 # p + /6 -=) # $ q + i @ )&3 - );- F +&5-/+ $(+ 4<%< C:1+;$ ] p
s
Proof.
%$bpi)
$
i
0
D i
3.5. Simplification
0 + + 6 $(+,) 7CN+&< F C 0 Y!'*l - 0E <4'O$ JQ )&' P )&'OC 7$(':< F!6 '[) +;-/+&' s -=) 1 0 # p
i
+ /6 -=) # $ q 1 0 + 4< ?+5 ?+;$ 0 ' @
- 0 # t 7$ - CD- P 2 F pJ ) < r
- +&5' 0 E ' 8 E 1);'
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(p
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$,1 PP $(' 0 $ +&'O- # +&5-=+S+&5'[);- F s 5 ?+.$S- P 0 + A0 +&5' $(' E 3g' 0 +L+&5-=+L5-R$$,1 gC 4' 0 +;< F $(3U-=<%< 21+ P $ 4+ 4Y!' # 7$ +;- 0 C*'o+ +&5'f3 ' 0 + 0 ' # ' 0 # P 0 + x
s
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6587 ?=5 87 K; Proposition 3.28 36 4 (8 79 B %$ B 45
B (
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s
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r
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' *+ w - 0 # r # ' 0 +&'Ui+&5' Y!'*)&+ %C:'D$ A@ D - # (-RC:' 0 +q+ p $,1C.5
+;5-/+ 6 ?+&5 1+< $,$ A@ E ' 0 ':);-=< ?+ F ](w, p , r) 7$$(+,) 7CN+&< F C 0 Y!':l Z' @ i)&'+&5'
i
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+
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rp
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- 0 # P
D
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Chapter 3. Hamiltonian Polygons
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F C 0 Y!':l - 0 E <%'D$ ] p
D i
" $ lV+&' 0 # r L' '*l P $,$ ?2< F CN);'D-/+;'D$g- 0 1 0 $,- @ 'b- 0 + r CD- P 6 'h5-DY!' + 3U-=m!'g$,1)&'g+&5-=+X+;5 %$ - 0 + r C:- P # 'D$ 0 +q- PP 'O-/) 0 - E ' # 'O$ 7C 0 0 ' A@
+;5' @ <4< /6 0E $ +;' P $ i -/)&+&< F +&5 7$qC:- 0 2d'U+;-=m!' 0 C:-=)&' A@ 2 F $(5 _ + 0E +&5'
0 '*lV+ );- F @ ) 3 +&5'h- 0 + r CD- P Z\1+ 2^' @ )&' +&5' 0 '*lV+ );- F %$ $(5 + - 0 +&5':)
$&-/+&1).-/+ 0 $ +&' P +.-/m!'D$ P <%-C*' (
D 507 Proposition 3.29
(5 & 50 3 658:D 5 5
F 7E 5 50:D ?=5 # 4 3=& B 4 6
; 3
? ; 3(
@"5
7 ( K( E50
q
?=5 4 !
50:D "3 (
:D
q
58 K5 9$
?=5 < P
5 5 5 D 58 K65 ?=5
Q
7E;65 ?=58
; 7E;65 D 3
3
K( @"5E79( 3
Q
?=5 350797
t
:D 9$
?=5
P
F 78 5 95 ( K 785 (
q
P
" %< <1 0 $,-=+&1);-=+&' # $(' E 3g' 0 +;$ 0 P -/)&' 0 C # ' 0 + + ' ?+;5'*) E ' (p , x, q)
E (r, p , x, t) pJ0 $ # '*) 0 < F +;5'k$(' E 3g' 0 +;$ 0 C # ' 0 + + E ' (pi, x, q)
@K ) +;5' 3 3 i ' 0 + -=+&1);-=+ 0E +;5':3 3 E 5+ 0 +,) # 1C*' 0 ' 6 1 0 $,-=+&1);-=+&' i# $(' Er
3g' 0 +;$X-=< 0E +&5' E ' # 'O$ %CD$ C 0 $ +,);1CN+&' # 2 F Z1 4< # r p - P H
5'D$('U-=)&'g+&5' 0
$&-/+&1).-/+&' # -$ 6 ':<%< - 0 # +;5 %$ P ) C*'O$,$LC 0 + 0 1'D$ 1 0 + %< 00 ' A@ +&5' E ' # 'D$ 7C:$
CD-/+;C.5'O$- 0 ' 6 $(' E 3 ' 0 + ':+ a 2d' - Y!'*),+;'*l +;5-/+ 7$SY %$ ?+&' # 2 F Z1 4< # r p - P
-=+$ 3g' P 0 + # 1) 0E +&5 7$ P ) C*' # 1)&' L'DCD-=<%<^+&5-/+9Z1 %< # r p - P )&'*+;- 0 $ +&5'
) %' 0 +;-=+ 0 - 0 # P ) P - E -/+&'O$ 4+ -=<%< -=< 0E +&5'MC 0 $ +,)&1CN+&' # E ' # 'D$ 7C \ ' 0 C*'
4+ 7$ u(a) = −1 8' 0 'h- P -/+&5
@ ) 3 a + p -R$ @ <4< /6 $ E`@ ) 3 a -=< 0E +&5'
ρ(a)
i -=+&1);-=+&'H$ +&'
);$ + E ' # 'D$ 7Ch+&5-=+ Y 7$ ?+;' #
&
+
5
%$
0
P 0 # 4)&'DC*+ A0 A@ +&5'
a
C 0 Y!':l ' 0 # P 0 + c A@ +;5' E ' # 'O$ %C! +&5-/+ 7$ - 6 - F @ ) 3 +&5' ' 0 # P 0 +
A@ +&5'H$,' E 3g' 0 + +;5-/+U5-$ 2^'D' 0 $,-=+&1);-=+&' # 0 +&5 7$ P -=),+ 7C*1<7-/)hZ\1 4< # r p - P
P '*).-/+ =0 . 5' 0 @ ) 3 c P ) C*'D' # -R< 0E ρ(c) 5' 2-R$,' CD-R$(' @ ) +&5 %$
);'DC*1);$ 4Y!' # 8' 0 4+ A0 %$ P ) Y # ' # 2 F +;5' 0 +;'*) )oY!'*),+ %C:'D$ A@E ' (q, x, p )
@ ) 3 6 5':)&' 6 ' $ %3 P < F 6 -R<?mq-=< 0E[E ' (q, x, p ) + p 8 E 1)&' ! # $(5 =6 i $
- 0 '*l-=3 P <%' @ )9- Y!':),+&'*l a - 0 # 4+;$C )&)&'D$ P 0 i# 0E P i-/+;5 ρ(a) " $o-=) E 1' # -=2 Y!' 4+ %$ u(v) = −1 @ ) -=<%< v V(ρ(a)) 5 7$ %3 P < %'D$
+;5-/+ ρ(a) = (a = a , . . . , a = p ) ` 7$ - ) E 5+ r +&1) 0c +;5-/+ 7$ %$S)&1' ':l ) ` -=+ V@ i )9-=<%< 1 < k < ` 8s1)&+&5'*);3 )&' ρ(a)
](ak+1 , ak , ak−1 )
7$ 00sr C*) $,$ 0E 2^'OC:-R1$('L'*Y!':) F ' # E ' A@ -9C 0 $ +,)&1CN+&' # E ' # 'D$ 7C 2d'OC 3g'D$ - 0
' # E ' @ +&5' @ );-R3g' %3g3g' # 7-/+&'D< F +&5-/+ 6 4<%< 0 +92d'qCN) $&$(' # 2 F - 0_Fb@ 1),+&5':)
E ' # 'O$ %Ck-RC:C ) # 0E + i ) P $ ?+ 0 V as 8 0 -R<4< F 2$(':),Y!' +&5-=+ ∂(ρ(a))
s - 0 # -=+ 'O-RC.5 0 +&'*) ) Y!'*),+&':l +&5' 0 C # ' 0 +
(P ) ∂(ρ(a))
Q =
Proof.
) '
3.5. Simplification
q
q
t
t
pi
!
q
pi
!
$ $%)$ (pi , x, q)
q
t
t
a
*) pi
!
$%11
01
Figure 35: 3
ρ(a)
#"&%0" pρ(a) ; E7 ;65 D 50( K 785 (
i
q
a
Chapter 3. Hamiltonian Polygons
,$ ' E 3g' 0 + @ ) 3 S < 4'O$ 6 4+&5 0 +&5'fC 0 Y!'*l - 0E <%' +;5-/+ %$ + +&5'M) E 5+ A@ )
0= ρ(a) 1 PP $('\+&5-=+ q %$Y %$ ?+&' # 2 F -=+&1);-=+&' - 0 # <%'*+ b 2d'\+&5'\' 0 # P 0 + A@ +&5'
$,' E 3g' 0 ++&5-=+ %$$,-=+&1);-=+&' # 0 +&5 7$ P '*);-=+ 0 $(1C.5H+&5-/+ b 7$ 0 +M-kY!'*)&+&'*l
A@ +;5' @ ).-=3g' -=++&5-=+ P 0 + L'OC:-=<%<c+&5-=++&5' $(' E 3g' 0 + 0 C # ' 0 +9+ q qt 7$J-R<?);'D- # F P -/),+ @ +&5' @ );-=3g'M2d' @ )&'M+&5'C:-=<%<^+ -=+&1);-=+&' X
5' 0 +&5' P -/+;5
0 # 1C:'D$- 00sr CN) $,$ 0E ) E 5_+ r +;1) 0 P -=+&5 R 6 4+&5 0 (P ) @ ) 3 q + ρ(b)
$(1C;5 +&5-=+ q 7$J- )&' ':l ) -/+ Y!':),+&':l A@ +&5' P -=+&5 R := E ' (p , x, q) R
p
i
- 0 # qt 7$X+ +&5'g) E 5+ A@ ) =0 R Z1+ -$ +&5'UC 0 Y!'*l - 0E <%i' -/+ q %$
':l+;'*) ) + P +&5' P -=+&5 R # 'D$,C*) 42^'O$U- C*< $(' # C*1),Y!' 6 ?+&5 0 P 6 5 7C.5
7$ 0 + 0 1<4< r 5 3 + P 7C 0 C A0 +,);- # 7CN+ 0 + P -R$ - @ );-R3 ' 2d' 0E $ %3 P < Fr
C 00 'DC*+&' # 5'-=) E 13g' 0 + @ )B+;5'JY!':),+ 7C*'O$ Y %$ ?+&' # 2 F -fZ\1 4< # r p - P P '*);-=+ A0k6 ?+;5
P $ 4+ 4Y!' ) 4' 0 +;-=+ A0c +&5-/+ 7$ 0 4+ 7-/+&' # 2 F $ 3g' 1 0 $&-/+&1).-/+&' # $(' E 3g' 0 +
0 C # ' 0 +S+ kE ' (r, p , x, t) %$S$ F 3g3g'*+,) 7C 8 )9- Y!':),+&':l a Y 7$ 4+&' # 2 F $(1C.5
- 0 P '*);-=+ A0 6 'gC:- i0 # E' 0 'g- P -/+;5 σ(a) $ %3 %<7-/) + ρ(a) +&5-=+ C A00 'DC*+;$
+ 5' -/+&5
7$k-H<%' @ + r +&1) 0 P -/+&5 6 4+&5 0 (P ) - 0 # -=+k'D-C;5
a +&'*) r ) Y!'*)&+&'*l P +&5' σ(a)
0 0 C # ' 0 +$(' E 3 ' 0 + @ ) 3 S < %'D$ 6 ?+;5 0 +&5'C 0 Y!'*lq- 0E <%' +;5-/+ %$ + +&5' <4' @ + A@ ) =0 σ(a) S>0 +&5' $,-=3g' 6 - F -R$[-=2 Y!' 6 ' E ':+C 0 +,).- # 7CN+ 0 + +;5' @ -CN+ +&5-/+ P 7$ $ %3 P < Fr C 00 'OCN+&' # @ $(1C.5 -[Z\1 4< # r p - P
P '*).-/+ 0 )&'*Y %$ ?+;$ q >0 $(13g3U-/) F 6 'XC 0 C*<%1 # 'f+;5-/+ -=+&1);-=+&' # 'D$ 0 +L)&':Y 7$ 4+ q >0 C 0 +,);-$ +S+ q +&5'f+&5' +&5'*)9$,' E 3g' 0 +' 0 # P 0 + t 3U- F 2d'o)&':Y 7$ 4+&' #
# 1) 0E +&5'gC:-=<%<\+ ` -=+&1);-=+&' # ?)&'OCN+&< F - @ +;'*) +;5' lV+&' 0 # r L' ':l P '*).-/+ 0
-$ +&5'f'*l-=3 P <%'o$(5 =60 0 8 E 1)&' Aa # ':3 0 $ +&);-/+&'O$ Z1++&5 7$LC:- 0 5- PP ' 0
0 < F @ t 7$k-H)&' '*l Y!'*),+&':l A@ +;5' 1- # ) 4<7-/+;'*);-R< (t, x, p , r) @ t 7$kC 0 Y!':l`Y!':),+&':l @ (t, x, p , r) 6 ' C:- 0 -=) E 1' -R$ 0 i ) P i$ 4+ 0 s + E ':++;5' @ <%< /6 0E $ +;-=+&':3g' 0 i+ (E D 5072?=5 4 50:D "3 (
:D
?=5
Corollary
3.30
q
t
(5 & 50 < 50:D 5 5 :D( @"(5 ? ;
( B '5 t
?=5 @D79 ;507E4
58 K65 9$ ?=5 3 :D
(t,
x,
p
P
i , r) F 7E 5 50:D 5 5 5 D 58 K65 ?=5 F Q ; 7E;65 D 59( K
?=5 #3F 4 <=& B 4 ; 7E;5 ?=50
3
3
? ; 3(
@ 5
7 ( K( E50 3
K( @ 5
7 ( 3
t
Q
P
3 50797
'507 65>
9$
?=5
P
78 5 785 (
t
5 ' C ) <%<%-=) F 2d':< /6 ($ 13g3U-/) %e:'O$ +;5' # %$&C*1$,$ 0 /- 2 1+q1 0 &$ - @ ' - 0 + r
DC - P $ P +&' 0 + %-R<4< F C*)&'D-=+&' # 2 F l+;' 0 # r L' '*l 3.5. Simplification
q
q
x
pi
s
pi
t
r
Figure 36:
t
587 5
t
50:D 5
Corollary
3.31
B 3( K 785 3 (
5ED D
$ 8 5 785 (
5
D D
793=&
3
r
!
#
; E7 ;5
5 B 87 5
;59( ( 5 B ?=50
703=& ?=5 # 4 3 =& B 43 :
; E7 ;65 ?' (
3
" $ # 7$,C*1$,$(' # 0 i ) P $ ?+ 0 s as lV+&' 0 # r S' ':l C:- 0 C*)&'D-=+&'
0 0 @ 'h- 0 + r CD- P -=+ 0 ' @ q ) t 0 < F @ 4+ C*)&'D-=+&'D$ - 0 - 0 + r C:- P -/+
+&5' 0 q 7$ 0 + )&'*Y 7$ ?+;' # 0 +&5' @ <4< /6 0E -/+&1);-/+&'b$ +;' P -C:C ) # 0E + q)
i P $ 4+ 0 s s @ lV+&' 0 # r S' ':lkCN)&'O-/+&'O$ - 0 - 0 + r CD- P -=+ t +&5' 0 ' ?+;5'*) t
7$ -C 0 Y!'*l Y!':),+&':l A@ +&5' 1- # ) %<%-=+&'*).-=< (t, x, p , r) - 0 # +&5' C*<7- %3 @ <%< /6 $
i
2 F pJ ) <%<%-=) F s s ) t 7$ - )&' '*l`Y!'*),+&':l A@ (t,
- 0 # +&5' 0 # 1'
x,
p
i , r)
+ +&5' ) +;-=+ 0 A@ s # 'O$,CN) 42d' # 0 i ) P $ 4+ 0 s t 7$U- C:- P 0 +&5'
);'D$(1<4+ 0E @ );-=3g'f2 FbpJ ) <%<%-=) F s A
':+1$C 0 C:<41 # 'f+&5'X- 0 -R< F $ 7$ A@ - 0 + r CD- P C*)&'D-=+ A0 $L2 F < m 0E -/+S+&5'
<7-R$(+)&':3U- 0 0E P ':);-/+ 0 \ );- Er # E ' Proof.
- 1 $&-
(
Proposition
3.32
6587 (
:D
P
p
p
k
`
D
( B 45 7A(
pk
( 5 785 5 4 3 p`
Proof.
'D$,C*)
P+
;) - Er #
;+ 5' 1+ P
#
D 507! "507E;
78& D&@5!:D ( "7E 5
( K I7 ( 5 785 5 4 < 3
p
P
`
53
?=50
E7 & D&'5 D 50( K B 785
;65 $
(0 5 ( 5 B ;
P
D B 705
;
59( ; ' 0 +&'+&5'SY!':),+&'*l @ ) 3 6 5 %C.5U+;5'S);- F s 7$J$(5 + 2 F p -R$ 0 +&5'
0qA@ );- Er # E ' L'OC:-R<4<+&5-=+ s 5 ?+;$ p p -R$ - P )&'DC Ai 0 # ?+ 0 A@
E ' 0 P -=),+ 7C*1<7-/) s - 0 # p p -/)&' 0 +\C k <%< `0 'D-=) " $ 1$(1-R< # ' 0 +;'
1+ @ );-=3g' A@ );- ERr # E 'o2 F k P` " $S- P );'DC 0 # ?+ 0bA@ ).- Er # E '
Chapter 3. Hamiltonian Polygons
+;5'b$(' E 3g' 0 + p p 7$g- 0 ' # E ' A@ P -R$&$(13g' 6 ?+;5 1+ < $,$ @[E ' 0 ':);-=< 4+ F
k 5-D
` Y!'[+
X
'
j
Y!':) @ F +&5-=+9C:- P $ 0 P )&':3U- 0 CD- P $ 0 P p` = pk 1
5' C:C*1),)&' 0 C*' @ p 0 P %$ 0 +M3 # ' # - 0 # +&5V1$)&'D3U- 0 $9- C:- P
0 P @ 4+ 6 -$ P -=),+ @i - C:- P 0 P pJ 3 P -/);' # + P +;5' Y!'*),+;'*l p
- PP 'D-=);$ - $('OC 0 # + %3g' 0 P -$J- C 3g3 0 ' 0 # P 0 + A@ 2 +&5 E ' # 'D$ 7C:$i
C 0 $ +,)&1CN+&' # 0 );- ERr # E ' ZF # '8 0 4+ A0hA@ E ' # 'D$ 7C[+&5 %$$,'DC 0 # C:C:1) r
);' 0 C:' %$ - C:- P kQ )&' Y!'*) -RC:C ) # 0E + 1) 6 ' # E ' <7-/2d':< 0E $&C;5':3g'q+&5'
) E 0 -=< C:C:1),)&' 0 C*' A@ p %$q<7-/2d'D<4' # -R$ - 6 ' # E ' - 0 # 5' 0 C*' +&5' $('DC A0 #
C:C:1),)&' 0 C*' 7$$,- @ ' 0 P i
@ p %$UC A0 Y!'*l 0 P +&5' 0 ?+ 7$U$ +,) 7CN+&< F C A0 Y!'*l 0 P ' 0 C*' P $,$ 42<%' k CD- P (. . . , pk 1, pk) )&':3U- 0 $f- C:- P (. . . , pk 1 ) 0 P q <7$(' pk
7$f$,- @ ' 0 P 2 F -$,$(13 P + 0 - 0 # 5' 0 C*' 4+ %$f-R<%$ $,- @ ' 0 P -RCDC ) # 0E
+ i ) P $ ?+ 0 s @ p %$JC 0 Y!':l 0 P +&5' 0g6 'M-=) E 1'9-$ -/2 Y!' @K ) p <7$(' p 7$\)&' '*l
0 P - `0 # 3g- F 2d'DC 3 'q- 0 1 0 $,- @ ' - 0 + r C:- P 0 P +k %$ 0 + P `-/),+ A@ - 0
1 0 $,- @ 'L)&' ':lq+ 6 0 0 P A0 0 '$ # ' -=<%<Y!'*)&+ %C:'D$\-=< 0E +&5'C 0 $ +,);1CN+&' #
E ' # 'O$ %CX-/)&' C 0 Y!'*l - 0 # +&5' +;5'*)S' 0 # P 0 + @ +;5' E ' # 'D$ 7C p 7$$,- @ ' 0 +&5' +&5':)$ # ' +;5' $,1C:C:'D$,$ ) A@ p` # # 0 +MC.5- 0E ' @ ) 3 Pi + P " $ p 6 -R$ 0 + P -=),+ A@ - 0 1 0 $,- @ 'S)&' '*lk+ 6 0 0 P 4+ %$+&5V1$ 0 ' ?+&5'*) P -/)&+
A@ - 0 ` 1 0 $,- @ 'o)&' '*l + 6 0 0 P " <%< 0 +&':) )LY!'*)&+ %C:'D$ 0 +&5'[+ 6 E ' # 'D$ %CD$S-/)&' C 0 Y!':l - 0 # E 1-/) # ' # 0 P 2 F +&5'f$,- @ ' C:C:1),)&' 0 C*' A@ p - 0 # ' 4+&5':) +&5'[$&- @ 'Y!'*)&+&'*l p )J+&5'
i
k
1 0 $,- @ 'fY!'*),+;'*l p `
" $ # %$,C:1$,$(' # 0hi ) P $ 4+ A0 s ! +&5'[$('DC 0 # 0 ' 6 )&' ':l C:C:1),)&' 0 C*'
A@ p 0 P <%'*+ 1$ )&' @ '*) + ?+ -R$ p %$ - $,- @ 'hC:- P 5' ) %' 0 +;-/+ 0
-R< 0iE +;5' E ' # 'D$ %CD$ 7$9C.5 $(' 0 0 $(1C.5`m- 6 - F +&5-/++&5'*)&' 3U- F 2d' Y!':),+ 7C*'O$
0 C # ' 0 +f+ 1 0 $,-/+;1);-/+;' # $(' E 3g' 0 +;$f+&5-/+ -=)&' ) %' 0 +&' # + /6 -=) # $ p " $oC 0 $(' 1' 0 C*' p 3U- F 2d'OC 3g' - 0 - 0 + r C:- P n-$q$(5 /60 0 8 E 1)&m' !f )
':Y!' 0 - C 0 Y!'*l`mY!'*),+;'*l @ P # 1) 0E +;5' -=+&1);-=+&' $(+&' P @ <4< /6 0E );- Er
# E ' 5'*)&' @ )&' 4+ 7$[);-/+;5'*) %3 P ),+.- 0 +f+&5-/+ p 7$ 0 # ':' # $,- @ ' 0 P m
" J$ - C 0 $,' 1' 0 C:' 6 '[3 1$ + 0 +- PP < F ;) - Er # E ' 6 5 ' 0 p 7$ - 0 - 0 + r
CD- P 0 +&5' @ ).-=3g' 8 )X+;5' P $&$ ?2< F" 1 0 $,- @ ' - 0 + r C:- P p 6 ' i CD- 0 /- ) E 1'
-$ @ ) l+;' 0 # r L' '*l 0hi ) P $ 4+ 0 s +&5=- + p 7$ 0 + &) `'*Y 7$ ?+;' # # 1 ) 0E
`
+;5' @ <%< /6 0E C:-R<4<+ U -=+&1);-=+&' Corollary
3.33
3 :D 9$
E( D 05 7 ?=5 '587 5 3
7E& D&'5
p
`
?=5 F E7 5 E7 & D &'5 P
5
58 K65 9$
?=5
P
D 58 K5 ?=5
Q
3.6. Preparations for Bridging
pk
pk
s
?=5
Figure 37: ;
p`
p`
x
pi
pi
; 87 ;5 (8656 6587
3
pi = pm
7E&
D&@5 $ B 785EK65 B F E7 5 ?=5 #
4 3 =& B 43 7E;65 ?=58
;
78;5 D 59(
;
3
3
K 87 5 (
? ;
(
@"5
7 ( K
( 658 <
K( @ 5
7 ( <
p
p
Q
`
`
P
3.6 Preparations for Bridging
Z\' @ )&' 6 'o$(13g3U-/) %e:'o+&5'f-=< E ) ?+&53 @ )+&5' -/+;1);-/+ 0 7$,$('DC*+ A0c - 0 #
%3 P < CD-/+ 0 i 5-R$('O$- 0 # - 0 -R< F e:' +&5'f);'D$(1<4+ 0E # %$&$('DC*+ 0 $ 6 'o5-DY!'f+ - ## A0 ' 0 -=<^2 4+ 0 - 0 + %C P -/+ 0 A@ +&5' 0 # 1CN+ ?Y!'M2) # E 0E $ +&' P +&5-/+ @K < r
< /6 $ " $ A@ +&' 0 0 0 # 1C*+ ?Y!' P ) _A@ $ 6 ' P ) Y!' -$(< E 5+&< F $ +,) A0E ':) $ +;-/+;':3g' 0 +
+;5-/+f' 0 -/2<%'D$[1$+ P 1+[+ E '*+;5'*)+&5' 0 # 1C*+ 4Y!':< F C 3 P 1+&' # $ <%1+ 0 $+ # T '*)&' 0 + $(12 r P ) 2<4'D3g$ oj ' 6 - 0 +X+ l - 0 -/),2 ?+,).-/) F ' # E ' {y, z} A@ +&5'
0 4+ 7-=< @ );-R3 ' P = C 0 Y (∂S) +&5-=+ 3 1$(+ $ +;- F - 0 ' # E ' A@ +&5' @ );-=3g'`-=<%<
+;5) 1 E 5`+&5' -R< E ) 4+&53 0 P -/),+ %C:1<%-=) ?+ %$X- 0 ' # E ' A@ +&5' -=3 %<?+ A0 %- 0
P < FVE0 + 2d'XC 0 $ +,)&1CN+&' # " $,$(13 ' 6 4+&5 1+B< $,$ A@E ' 0 '*);-R< 4+ F +&5-/+ z @K <%< /6 $ y 0 P @ {y, z} %$Y 7$ ?2 4< ?+ F ' # E 'S- 0 # +;5'$,' E 3g' 0 + @ ) 3 S 0 C # ' 0 ++ y %$ 1 0 $,-=+&1);-=+&' # 2 F P
+;5' 0 +&5'' # E ' {y, z} 6 4<%<2d'L)&' P <%-C*' # 2 F - E ' # 'O$ 7C # 1) 0E +&5' -=+&1);-=+ 0
i 5-$('X-R<?);'D- # F ' 0 C:' 6 'o5-:Y!'[+ $(< E 5+&< F - # - P +S+&5' ) 4' 0 +;-=+ A0hA@ P 2 F
$,'*+,+ 0E
7$(' 5 %$ $(3U-=<%<^C.5- 0E 'C 3 P -/)&' # + +&5'1 0 @ )&3 ) 4' 0 +.-/+ 0 %$ =- <4)&'D- # F ($ 1 r
uy (v) :=
−1 , v = y,
+&5'*) 6
+1 ,
Chapter 3. Hamiltonian Polygons
C 4' 0 ++ ' 0 $(1);'[+&5-/+
-R< E ) 4+&53 {y, z}
)&':3U- 0 $L- 0 ' # E ' @ +&5' @ ).-=3g'[+&5) 1 E 5 1+J+&5'
5 ( 795065
D 0$
:D 7E&
?=5 3 4 7E Proposition
3.34
u
y
3( K @ %5
D 4 ?=50 C
7E$ '
? ( ?=5 5
D&@5
?=50
(
D&@5
{y, z}
{y, z}
5
D&@5 ?=5 7E 5 4 ?'87 &? ?=5
7E;
;
3(8( 5 B :D
3
<
B ;
? K(50(
3
5' @ );-=3g' 7$f3 # ' # 1$ 0E +;5'q2-$ %C P ':);-/+ 0 $ 0 < F!q ' 0 C*'
gC 4' 0 +L+ P ) Y!'M+&5-=+ 0A0 ' A@ +&5'O$(' P '*);-=+ 0 $ )&' P <7-RC*'O$ {y, z} " $
C 0 Y!':l 5V1<4< Y!'*)&+ %C:'D$ 2 +&5 y - 0 # z -/);' C A0 Y!'*l Y!'*),+ %C:'D$ 0 ':Y!'*) F @ );-=3g'
@K ) S [j ' 6 %<%< -/) E 1' +;5-/+ 00 ' A@ +&5' 2-$ %C P ':);-/+ 0 $[CN)&'O-/+&'O$f-g)&' '*l
Y!':),+&':l -/+ y ) z Z1 %< # rp - P CN);'D-/+;'D$ -)&' ':l Y!':),+&'*l A0 < F -/+ - 0 ' 0 # P 0 + q A@ - 0 1 0 $,-/+;1 r
).-/+&' # $,' E 3g' 0 + 6 5':)&' q P lV+&' 0 # r L' ':l CN)&'O-/+&'O$)&' ':lUY!'*),+ %C:'D$ 0 < F
-=+9' 0 # P 0 +;$ A@ - $(' E 3g' 0 + s P );- Er # E ' # 'O$ 0 +9C*)&'D-=+&' - 0_Fb0 ' 6
);' ':lqY!'*),+ 7C*'O$21+ 1$(+ - 0 ' 6 CDC*1),);' 0 C:' A@ - 0 '*l %$ + 0E )&' '*lqY!'*)&+&'*l " $
C 0 Y!':l 51<4< Y!'*),+ %C:'D$[2 +&5 y - 0 # z CD- 00 + - PP 'D-=) -R$ 0 +&'*) )oY!':),+&'*l A0
0 ' A@ +&5' E ' # 'O$ 7C:$ ' ?+&5'*) 51$[2 +;5 y - 0 # z );':3U- 0 C A0 Y!'*l`$ 0E <%'
Y!':),+ 7C*'O$ A@ +&5' @ );-=3g'M+&5) 1 E 5 1+ +;5' -=+&1);-=+ A0 - 0 # %$&$('DC*+ 0bi 5-R$('O$ 2$(':),Y!'+&5-/+ lV+&' 0 # r S' ':l 0 < F )&' P <7-RC*'O$J' # E 'D$ +&5-/+-/)&' 0 C # ' 0 +J+ - );' ':lbY!'*),+&':l f );- Er # E ' # 'D$ 0 +)&' P <7-RC:' yz 2 F -R$,$,13 P + A0cXp 5 P r
j ' # E 'D$ 0 < F )&' P <%-C*'O$q' # E 'O$ 0 C # ' 0 +q+ - # 12<%'gY!'*),+&':l A@ +&5' @ );-=3g' 8 0 -R<4< F! Z1 %< # rp - P 0 < F )&' P <7-RC*'O$q' # E 'O$ @ ) 6 5 7C.5 0 ' ' 0 # P 0 + %$ ) r
' 0 +;' # + =6 -/) # $ +&5' +&5':) " $ +;5' ) %' 0 +;-=+ 0 A@ y - 0 # z 5-R$J2d':' 0 C.5 $(' 0
$,1C.5b+&5-=+2 +;5H-/)&' ) 4' 0 +;' # - 6 - F @ ) 3 'D-RC.5 +;5'*) Z1 %< # r p - P # 'D$ 0 +
);' P <7-RC:' {y, z} ' ?+;5'*) - 0 # +;5' C*<7- %3 @ <4< /6 $ Proof.
4+ %$L$(1
p 5- 0E 0E + - 00sr 1 0 @ )&3 ) %' 0 +;-/+ 0 5-$ C A0 $(' 1' 0 C*'D$ @ )$ 3g' A@
+;5'9C*<7- %3U$\+&5-=+ 5-DY!'S2d':' 0 3U- # ' 0 +&5' P )&'OC*' # 0E $('OCN+ 0 $ >0 P -=),+ 7C*1<7-/)
6 'MCD- 00 + 1$,' i ) P $ ?+ 0 s + C:<%- 43 +&5-=+ +&5' 0 4+ 7-=<^C:-R<4<+ k -/+&1);-/+&'
# 'D$ 0 +MCN);'D-/+;' - 0_F 1 0 $,- @ ' - 0 + r CD- P L>0 # ':' # +&5'X$ 43 P <%'X':ls-R3 P <4' E ?Y!' 0
0 8 E 1)&' $,5 /6 $+&5-=+ -/+&1).-/+&' 3U- F C*)&'D-=+&' - 0 1 0 $,- @ 'k- 0 + r C:- P @ +&5'
) %' 0 +;-=+ 0 A@ +;5' @ ).-=3g'MC A0 +;- 0 $J- 0 -R<?+;'*) 0 -=+ A0c Z\1+ -$ +&5' ) %' 0 +;-/+ 0
7$M-R<43 $ +M1 0 @K );3 +&5' %3 P -RC*+ 7$ 0 + # );-R3g-=+ 7C Lj ' 6 %<4< # %$&C*1$,$9+&5'
u
y
C 0 $(' 1' 0 C*'O$ 0 +;5' 0 '*lV+9$('OCN+ 0
3.6. Preparations for Bridging
a
*) 65879:; ( 3 ?=5 70 58 + ; $ 5
@D Figure 38: B 4507 5 3 & 705 D 3 ; E7 ;65 3
a
( 5
!
Chapter 3. Hamiltonian Polygons
3.6.1 Wedge Control
S'DC:-R<4<+&5-/+
0 ) # ':) + 3U- 0 +.- 0 - 0 n-=<%3 $(+ o0 7C*' # %$&$('DC*+ 0 6 ' 0 ':' #
+ ' 0 $(1);'g-hC 3g3 0 ' # E ' A@ 'D-C.5 # 7$,$('OCN+ 0 P < FsE0 D 6 ?+;5 +&5'
$,1),) 1 0 # 0EM@ );-R3g' pJ 3 P -/);' # + " < E ) ?+&53 s : +&5':)&' -=)&' 0=6 + 6M0 ' 6
C 3 P < 7C:-=+ 0 $ );$ + +&5' @ 1+&1);' # %$&- PP 'D-=);- 0 C*' @6 ' # E 'O$ $(5 1< # 2^'J+;-/m!' 0
0 + -C:C 1 0 + - 0 # $('OC 0 # +&5'o$ P 'DC %-R< ' # E ' {y, z} $(5 1< # 0 +L2d' C 1 0 +&' #
-$f- C 3g3 0 ' # E ' A@ +;5' # 7$,$('DC*+ A0 P < FVE0 ?+f2 1 0 # $[- 0 # +&5' @ );-=3g' 5V1$ 6 'H);- %$,'H+&5'H)&' 1 4)&':3g' 0 +;$ @ )U+&5' # 7$,$('DC*+ A0 + 2d' 3U- 0 +;- 0 ' #
C ),)&'D$ P 0 # 0E < F
(
D 507 E7 5
7 ( 5 D ( < A
( 5 & 50 (
Definition 3.35
P
S
D>
'5
D 5
D&@5
oC A0 Y
4 (8 B 5D3(8( 5 B {y,
z}
(∂S)
3( B 4 5
D
:D %$ P
beautiful
C5'5070$
"#%$&@
? K(
D
K 4 5
D&'5 5
D&@5 7> 3(>
5
D&'5 ( B 4 5ED 7
good edge
?=50785 3( ; (8 5 ( 8K ?
( 5 B >3
P
?=5 785 5 4 3
7
B 5
D&@5 4 ?
? ; 3(
P
? 5 +( "5 B 45
D&@5
=
( B ? {y,
z}
< ?=5 # 4 3=&
D
5 785 5 4 3 :D!; (8A 5
5 (8 ?=58 ?=5 B 3( "7 5 '5879$ 785 5 '587 5 ( 5
?=58785 ( 23 B r
D
D 58 5
D&'5
? ; ( &@ D 7
:D ( B ?>? ;
( B 45 rq
D
q
7( 5 '507 65 <
3
( B ? 5
D &@5 ( B 4 5ED P
perfect edge
?=5 # 4 3=& L + %C:'b+&5-=+ 43 P < %'D$ & 43 P < 4'O$ ; - 0 # %3 P < 4'O$ &
1) EA -R< 7$X+ 3U- 0 +.- 0 - 2d'D-=1+ @ 1< # 7$,$('DC*+ A0 A@ +;5' @ );-=3g' # 1) 0E
+;5'f-=< EA ) 4+&53 S/6 '*Y!':) -=<4)&'D- # F 6 5' 0 -=+&1);-=+&' %$L- PP < 4' # + +&5' @ );-R3 '
C 0 Y (∂S) +&5':)&' 3 E 5_+f2d' 3g- 0_F`6 ' # E ' r ' # E 'O$ 0 +&5'k)&'D$,1<?+ 0Eh@ );-=3g' @
-=+&1);-=+&' E ' 0 ':);-/+;'D$h- )&' '*l + 6 0 -/+h+&5'`+ 6 -R<?+;'*) 0 -=+ A0 Y!'*)&+ %C:'D$h- 0 #
0 ' A@ +;5' Y!':),+ 7C*'O$ A@ +;5 %$k)&' '*l + 6 0 %$k)&':Y 7$ 4+&' # 2 F - E ' # 'D$ 7C +&5' 0
0 ' ' # E ' 0 C # ' 0 ++ +&5' $('OC 0 # )&' '*lbY!':),+&'*l p A@ +&5' )&' ':lb+ 6 0 %$f6 ' # E ' r ' # E ' Y!' 0 @ +&5'gY!'*) F );$ +q).- F %$q$(5 + i@ ) 3 p 4+q3U- F CN)&'O-/+&'
- # 7$,$('OCN+ 0 P < FsEA0 +;5-/+ 5-R$ 0 E # ' # E ' -R$o+&5' '*li-=3 P <%' $(5 /60 0
8 E 1)&' A # ':3 0 $ +&);-/+&'O$ Q )&' Y!':) -R$ 6 ' 5-:Y!' $(':' 0 0 8 E 1)&' AsM -=+&1);-=+&'H3U- F C*)&'D-=+&' - 0
1 0 $,- @ 'k- 0 + r CD- P - 0 # 6 ' 5-DY!' + ' 0 $(1)&'q+&5-=+[+&5 7$M1 0 $,- @ 'k- 0 + r CD- P 7$ 0 +
3.6. Preparations for Bridging
z
z
y
y
z
*) z
y
y
z
z
pi
* y
y
; 7E;<=& ( 5+& 50( <27 9 7E79$ 7ED 507A $ 5
@D D (
Figure 39: 3( 5 > # $K&' ? ; ? K( &' D 5 D&@5 4 B
B 5
D&@50(
?=5 7E 5!:D2?=5 (E? @D 5
D 705 & <
<& 87 5 785
4 5
D&@5 5ED&'59(
Chapter 3. Hamiltonian Polygons
);'*Y %$ 4+&' # 2 F - E ' # 'D$ %C 2$,'*),Y!' +&5-/+q+;5'D$(' P ) 2<%':3U$ CD- 0 C:C*1) 0 < F
# 1) 0E +&5' Y!'*) F );$ +fC:-=<%< + b -/+&1).-/+&' -R$ 0 < F -=+M+;5-/+ P 0 +9+&5':)&' %$[- 0
-R<?+;'*) 0 -=+ A0 0 +&5' @ );-=3g' ) %' 0 +;-=+ 0
5'q$ <41+ 0 %$M);-=+&5'*)o$ +&);- E 5_+ @K ) 6 -/) # $ @ -/) 6 'q3U- # ' 0 -R$&$(13 P r
+ 0À0 +&5' ) # '*) 0`6 5 7C.5 1 0 $,-=+&1);-=+&' # $(' E 3g' 0 +;$-/);' $,-/+&1);-/+&' # # 1) 0E
-=+&1);-=+&' 5 %$ C;5- 0E 'D$ 0/6 Mj ' # ':3U- 0 # +;5-/+f+&5 $,' 1 0 $,-=+&1);-=+&' # $(' Er
3g' 0 +;$ 2d'DC 3 'X$,-/+&1);-/+&' # <7-R$ + 6 5 $(' Y!'*),+ %C:'D$ 0 +&5' @ ).-=3g'
)
)
P =- ),+ 7C
$ 4+ A0
)9-/)&'
P -/+&' 0 - 0 -R<?+&':) 0 =- + 0X@ +&5' ) 4' 0 ;+ -=+ A0 n- 0 # +&5V1$^2 Ffi ) P r
s D P +&' 0 + 7-=<%< F E ' 0 '*).-/+&' - )&' '*l + 6 0=
) 4' 0 +;' # + /6 -/) # $S- )&' '*l Y!':),+&'*l @ ;+ 5' @ );R- 3g' Q );' Y!':) @0 ' A@ +&5'LY!'*)&+ %C:'D$ +&5-=+ P +&' 0 + 7-=<%< F 2^'OC 3g'D$ P -=),+ @ -[)&' '*l
+ 6 0 %$o)&'*Y 7$ ?+;' # 6 'gC.5- 0E 'k+&5' ) 4' 0 +;-=+ A0 @K ) +&5' );':3U- 0 0E 1 0 $,-=+&1 r
).-/+&' # Y!':),+ 7C*'O$J+ /6 -=) # $ +&5'oC A0 Y!'*lU);' rn CDC*1),);' 0 C:' A@ +;5-/+LY!'*)&+&'*l Bn0 $,1C.5
- 6 - Fg6 'fCD- 0 $ 43 P < F -:Y # C*)&'D-=+ 0E -o)&' '*lg+ 6 0 0 +&5 7$ CD-R$(' @V0 +&5'
+&5'*)o5- 0 # +&5 7$[Y!':),+&'*l %$ 0 +f)&'*Y 7$ ?+;' # 1 0 + %<B+;5' Y!':) F ' 0 # +&5' 0 6 ' # 0 +S3 0 # CN)&'O-/+ 0E - 0 - 0 + r C:- P +&5':)&' -R$- $ 0E <%'oY!':),+&':l A@ +&5' @ ).-=3g' 6 '
m 0/6 5 /6 + # 'O-=< 6 4+&5 4+ 5'`-=2 Y!' # 7$,C:1$,$ 0 7$ $(13 3U-=) 4eD' # 2^'D< =6 0 - );E' 0 ' # Y-=) %-=+ A0
A@ -/+&1);-/+&' +;5-/+ 6 ' C:-R<4< 785 3 ; 7E;5 p <4'O-/)&< F -=<%<kC:<%- 43U$ +&5-=+
5-:Y!' 2d':' 0 3U- # ' -/2 1+ -/+;1);-/+;' 0 +&5' P )&'OC*' # 0E $,'DCN+ 0 $M-=<7$ 5 < # @ )
p -=)&' @ 1< r -=+&1);-=+&' 2d'OC:-=1$(' ?+ 1$ +MC.5- 0E 'D$+&5' ) # '*) 0b6 5 7C.5 Z\1 4< # r p - P
7$o- PP < 4' # + 1 0 $,-/+;1);-/+;' # Y!'*)&+ %C:'D$o- 0 # +&5' P $,$ ?2<%' );' rn ) 4' 0 +.-/+ 0 # 'O$
0 + 0 # 1C:'X- 0_F -R<?+;'*) 0 -=+ A0 ) ) %' 0 +;-/+ 0 $+ =6 -/) # $S-q)&' '*l Y!'*),+&':l Operation 6 (Careful-Saturate(P, u))
9- @ ).-=3g' - 0 # - 0b ) %' 0 +;-/+ 0
@
P
u(P) = uy (P)
"507E; )L$ 3g'
y V(P)
" $ < 0E -R$ +;5'*)&' 7$ - 0 1 0 $&-/+&1).-/+&' # Y!'*)&+&'*l 0 V(P) \ {y} $('D<4'OCN+ p -R$
+&5'A);$(+1 0 $,-/+&1);-/+&' # Y!'*),+;'*lb+&5-=+[- PP 'D-=);$o- @ +&'*) 0 + 0 C*<%1 # i0E4
0 P - 0 # <4':+ (P, u) ← Z\1 %< # r p - P (P, u, p ) y
i
' *+ x # ' 0 +&'f+;5' P )&' # 'DC:'D$,$ ) A@ y 0 P " $L< 0E -R$L+;5'*)&' %$- 0 1 0 $,-=+&1);-=+&' # Y!':),+&':l 0 V(P) ) $,':<%'DCN+ p -$+&5'<7-R$ + 1 0 $,-=+&1);-=+&' # Y!':),+&'*lk+&5-/+J- PP 'D-=);$\2d' @ )&'
i
- 0 # <%'*+ (P, u) ← Z1 %< # r p - P (P, u, p ) 0
x
P
i
!
3.6. Preparations for Bridging
>@ +;5' P &) 'OC*' # 0E Z1 4< # r p - P P '*);-=+ A0 &) ':Y 7$ 4+&' # x +;5' 0 ($ ':+
@ )[-R<4< v +&5-/+- PP D' -=)M2d'*+ 6 'D' 0 +&5 ' ) E 0 -=< - 0 #
u(v)
:=
+1
+;5' 0 ' 6 C:C:1),)&' 0 C*' @ x 0 P (P, u)
)
8 E 1);' ! %<4<%1$ +,).-/+&'O$ p -/)&' @ 1< r -/+&1).-/+&'q);1 00 0E 0 - $,3g-R<4< '*l-=3 P <%' j ' 5-:Y!' + P ) Y!' +&5-/+ p -=)&' @ 1< r -/+;1);-/+;' 0 # ':' # E 1-/);- 0 +;':'D$ E # ' # E 'O$
0 C # ' 0 +9+ -=<%< )&' ':l Y!':),+ 7C*'O$ +&5-/+ 0 ++ _ 3U- 0F - 0 + r CD- P $-/)&' CN);'D-/+;' # - 0 # +&5-=+ 0 - 0 + r C:- P 7$- # 12<%'fY!':),+&'*l 0 +;5'[)&'D$,1<?+ 0E @ ).-=3g' (
D 507 ?=5 7E 5
C
Proposition 3.36
P=
DC ?=5 7950; @ $
P
u
y
D 50 K65 ?=5 7050( <& 78 5 9$
=? 58
P
?=# D ?=50785 3( ; (8 5 785 5 4 3 3
)
B 0 Y
4 ?! 5
D&@5
(∂S)
{y, z}
785 ; 7E;65!6
:D
P
3
?=5 # F 4 3=& (8;50 50 (
P
:D; (0 5 B <
3(
B @"5E7 $!; 705 5 4507 5 ? ;
P
( B C<
07 5 5 4 < 2785 =785 B 3(5 %$ a
?=58 =
? 58705 ( 785 5 4 3
P
(a, . . . , b)
8K ? < B D 58 5
D&@59( 87 5
7 5 45070$ 785 5 C(
3=&;F5 4507 65 )
P
&@ D 7 ?=5 B 797859(+" D<=& D (E( 5 B # $K&' )
"7
)
Proof.
+;'*l
70 ( 785 5
(a, . . . , b)
(
3=&# 5 4507 B 50( P
4 <2<
P
?=50
8K ?
a
:D
b
785
'*+&+5-=1+ $ %$);0$ +J+ -/P) E -=1),+'SA+&5@ -=-[+ )&2' '*+&5Ul +' 6 # E 0 'D$ -=)&0 ' C E# _' 0 # + @ + )\-o+&5)&'' C ': l ),);$'D$ 0P E <%0'# Y!'*0) E r
A @ P
# %$,$,'DCN+ 0 P < FsE0
pJA0 $ # '*)k-h)&' '*l $ 0E <4'UY!':),+&':l v 0 P +&5-/+ %$ 0 + P -/),+ A@ -b)&' '*l
+ 6 0 " $S-q$ 0E <4'oY!'*),+&':l 4+ 7$ 0 + P -/),+ A@ - 6 ' # E 'f2 F # 'E 0 ?+ 0 0 C*' v
7$ 0 + P -=),+ A@ -S);' ':lX+ 6 0 2 +&5 0 ' E 52 );$ -/)&'C 0 Y!'*l Y!':),+ 7C*'O$ 0 P " $
0kE ' # 'D$ 7CoCD- 0 )&':Y 7$ 4+9- -/+SY!':),+&':l 6 ?+&5 1+S-R<%$ );'*Y %$ 4+ 0E - 0 - # (-RC:' 0 +
# ?);'DCN+;< F )9Y 7- +&5':) -/+9Y!'*),+ 7C*'O$ )&' '*l Y!'*),+;'*l +&5' 0 ' E 5_2 );$ A@
-/)&'
v
$ 0E <%'gY!'*),+ %C:'D$ 0 P - 0 # 5' 0 C*' 0 + P -/),+ A@ - 6 ' # E ' ' 4+&5':) 8 0 -=<%< F
2 F i ) P $ ?+ 0 V 2 +&5 y - 0 # z )&'D3U- 0 C 0 Y!':l $ 0E <%'qY!':),+ 7C*'D$ A@ +&5'
@ );-=3g' +&5) 1 E 5 1+ - 0 # +;51$ yz 7$ 0 + 0 C # ' 0 + + - 0F );' ':l Y!'*)&+&'*l # 1) 0E
p -=)&' @ 1< r -=+&1);-=+&' 0 P -=),+ 7C*1<7-/) 0 ++ v \
51$ 2 +&5h' # E 'O$ 0 C # ' 0 +S+ v
-=)&' E_ # @ )S+&5' C ),)&'D$ P 0 # 0E # 7$,$('DC*+ A0 P < FVE0 -$C*<7- %3g' # a
Chapter 3. Hamiltonian Polygons
y
y
x
y
y
*) x
x
y
* x
x
y
785 ;
E7 ;5 ?=5 785 7950+; ' K
D ( B 785E; 3=& Figure 40: 3
B ; 4 5
D&'5>:D & 7E5850( &@ D 5 D&'59( ;785 5 '587 B 95 (
!
3.6. Preparations for Bridging
p '*),+;- 0 < F 0 Y!'*),+;'*l 0 P 7$ ) %' 0 +&' # + /6 -=) # $ - )&' ':l Y!'*)&+&'*l 0 P
2d'OC:-=1$('b+&5':)&'b-/)&' 0 )&' '*l Y!':),+ 7C*'D$ 0 C 0 Y (∂S) " <%$ +&5':)&' %$ 0 < F
0 'b-R<?+&':) 0 -=+ 0 0 -R3g':< F -/+ y 51$ 2 F i ) P $ ?+ 0 V D +&5':)&' %$g-=+
3 $ + A0 '9)&' '*lU+ 6 0 0 P Q );' P )&'DC %$('D< F! -$ 0bi ) P $ 4+ A0 s7D - 0 #
i ) P $ 4+ 0 s A0 )&' '*lq+ 6 0 - 0 # 0 - 0 + r C:- P 7$\CN)&'O-/+&' # 2^' @ )&'MZ1 %< # r
p - P %$ - PP < 4' # + y 0 +&5' $('OC 0 # < _ P A@Mp -=)&' @ 1< r -=+&1);-=+&' ` +q3 E 5+
2d' +&5-=+o+&5'g$(' E 3g' 0 + yy @ ) 3 S 0 C # ' 0 + + y 7$ -R<?)&'O- # F $&-/+&1).-/+&' # -/+
+;5-/+ P 0 + n0 +&5 7$ CD-R$(0'`+&5':)&' %$ 0 +&5 0E 3 )&' + $(5 =6X 5'*)&' @ )&' $,1 PP $(' +&5-=+XZ\1 %< # r p - P %$f- PP < 4' # + y -/+[2d' E 0 @ +&5' $,'DC 0 # < _ P 0
p -=)&' @ 1< r -=+&1);-=+&' @ x 7$L-qC 0 Y!'*lgY!'*),+;'*l A@ +;5' @ );-=3g'o-=+ +&5' P 0 + 6 5':)&' Z1 %< # r p - P %$
- PP < 4' # + y +&5' 0 x %$ - $(+,) 7CN+&< F C 0 Y!'*l`Y!'*),+&':l A@ +&5' )&'D$(1<?+ 0Eb@ );-=3g' 5' 0 0 )&' '*l + 6 0 6 4<%<S2d'bCN)&'O-/+&' # - 0 # 6 'bC:- 0 A0 C*'b- E - 0 -/) E 1'b-R$
2d' @ )&' M
51$M$(1 PP $,'X+;5-/+ x %$- )&' '*lhY!'*),+;'*l A@ +&5' @ );-=3g' 2d' @ )&'q- 0 #
- @ +&':)gZ1 %< # rp - P %$k- PP < %' # + y Z F +&5' ) # '*) 0 6 5 %C.5 Y!':),+ 7C*'D$ 6 '*)&'
$&-/+&1).-/+&' # # 1) 0E +&5A
' );$(+o< P 0 p -/);' @ 1< r -/+&1);-/+&' x %$+&5' <%-$ +o)&' '*l
Y!':),+&':lUC*)&'D-=+&' # 0 +&5' );$ +L< _ P @ p -/);' @ 1< r -/+&1);-/+&' 0 P -/),+ 7C*1<%-=) x 7$$ 0E <%'oY!':),+&'*l @ +&5' @ );-R3 'X-/+S+&5-=+ P 0 + @E ' (y , y, x) y x - 0 # +&5':)&'U-/)&' 0 1 0 $,-=+&1);-=+&' # Y!'*)&+ %C:'D$ -=< 0E
E ' (y , y, x)0 +;5' 0 y - 0 0# x 3U- Ff@ )&3 -)&' '*lf+ 6 0 0 +&5' );'D$(1<4+ 0E@ );-=3g' n0 +&5 7$0 CD-R$(' 2 +&5 -=0)&'U$ 0E <%' Y!'*),+ %C:'D$ A@ +;5' @ );-=3g'g- 0 # y 7$X- C:- P " $
+;5'*)&' -=)&' 0 3 )&'g1 0 $,-/+&1);-/+&' # $(' E 3g' 0 +;$ \p -/)&' @ 1< r -=+&1);0-=+&'U+&':)&3 0 -/+&'O$
%3g3g' # 7-/+&'D< F - 0 # +;5'*)&' %$ 0 +&5 0E 3 )&'[+ $,5 /6X
<7$(' @ +&5'*)&' 7$ -=+ <%'D-R$(+ 0 'L$ +,) %C*+&< F C A0 Y!'*l Y!'*)&+&'*l 21+ 0 1 0 $&-/+&1).-/+&' #
Y!':),+ 7C*'O$ -=< 0EHE ' (y , y, x) +&5' 0 0 );' ':l`+ 6 0 7$ C*)&'D-=+&' # - 0 # +&5':)&' %$
0 +;5 0E 3 )&'[+ $(5 /06X
+&5':) 6 %$,'U+&5':)&' -/);' $ 3g'U1 0 $,-=+&1);-=+&' # Y!'*)&+ %C:'D$q-R< 0E E ' (y0, y, x) 5' $('DC A0 # < _ P A@ p -=)&' @ 1< r -=+&1);-=+&' 7$ # 'D$ E0 ' # + $,-/+&1);-/+&' +;5'D$(' Y!':) r
+ %C:'D$ 0 - 0 ) # '*)9$(1C;5H+&5-/+':Y!'*) F + %3 ' 2d' @ )&' Z\1 %< # r p - P 7$- PP < %' # + Y!':),+&':l p
i
)
)
-=<%<c1 0 $,-=+&1);-=+&' # !Y ':),+ 7C*'D$9-/)&' 6 4+&5 0 +&5' $(12 6 -R<?m (x, . . . , p ) A@ P
i
- 0 # x %$S+&5' 0 < F &) ' '*l Y!'*),+&':l 0 +&5' $(12 6 -=<4m (x, . . . , p ) A@ P i
1 P P $('U+&5-=+ 0 $ 3g' Z\1 4< # r p - P P ':);-/+ 0 +&5' E ' # 'D$ %C )&'*Y %$ ?+;$ x
- 0 # +&5' Y!'*)&+ %C:'D$X2d'*+ 6 'D' 0 +&5' ) E 0 -R<J- 0 # +&5' 0 ' 6 CDC*1),);' 0 C:' @ x 0
<%'*+X1$XCD-=<%< ?+ x -/)&' )&' r> ) %' 0 +&' # " $ x %$ +&5' 0 < F )&' '*l Y!':),+&'*l 0
P
+;5' $(12 6 -R<?m (x, . . . , x ) A@ P 0 Y!'*)&+&'*l 7$ ) %' 0 +&' # + =6 -/) # $U- )&' '*l
Chapter 3. Hamiltonian Polygons
Y!':),+&':l 0 +&5' @ ).-=3g' - @ +;'*)h+&5 7$ )&' rn ) %' 0 +;-/+ 0 Q )&' Y!'*) +;5'*)&' 7$ 0
-R<?+;'*) 0 -=+ A0 0 +&5' ) 4' 0 +.-/+ 0 2d'DCD-=1$,'q2 +;5 x - 0 # x -/)&'k$&-/+&1).-/+&' # -=<%<
1 0 $,-/+;1);-/+;' # Y!'*),+ %C:'D$ 6 4+&5 0 (x, . . . , x ) -=)&' ) %' 0 +&' # 1 0 @ )&3g< F +1 - 0 #
-R<4<S1 0 $,-/+;1);-/+;' # Y!'*),+ 7C*'O$ 6 4+&5 0 (x , . . . , x) -=)&' ) 4' 0 +&' # 1 0 @ )&3g< F −1 5'*)&' @ )&' 6 'f3U- F C 0 C*<%1 # '[+&5-/++&5')&':3U- 0 0E 4+&':);-/+ 0 $ # 0 +LCN)&'O-/+&'
- 0_F )&' '*l + 6 0H )9- 0 + r CD- P 0 +;5'X$,-R3g' 6 - F -R$ 0 i ) P $ ?+ 0 s : - 0 #
i ) P $ 4+ 0 s A
+S)&':3U- 0 $+ C 0 $ # ':)S+&5' CD-R$(' +&5-=+ 0 Z\1 4< # r p - P P ':);-/+ 0 # 1) 0E
+;5'9$('OC 0 # < _ P 0 p -/)&' @ 1< r -=+&1);-=+&');'*Y %$ 4+;$ x B
5' 0 x 7$ - $ 0E <%'Y!':),+&'*l
A@ P 0 < F +&5'[Y!':) F <7-R$ +[Z\1 4< # r p - P P '*).-/+ 0 0 p -/);' @ 1< r -/+&1);-/+&'X3U- F
C*)&'D-=+&'g- )&' '*l`+ 6 0c0 -=3g':< F -=+ x - 0 # +&5'k' 0 # P 0 + p @ +&5' $(' E 3g' 0 +
0
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0
) x -=)&' E_ # @ )L+&5' 4)MC 3g3 0 # %$,$,'DCN+ 0 P < FsE0
3.7 Algorithm Summary
5 %$ $,'DCN+ 0 # 'O$,CN) ?2d'D$B+&5' 0 -R<V-R< E ) 4+&53 @ ) +&5' -/+&1).-/+ 0 %$,$,'DCN+ 0
- 0 # 43 P < C:-=+ 0 i 5-R$('O$q- 0 # P ) Y!'D$o+&5-/+ 4+ 3U- 0 +;- 0 $ - @ );-R3g' P + r
E ':+&5'*) 6 ?+&5h- 2^'O-=1+ @ 1< # %$,$,'DCN+ 0 @ P \ $&$(' 0 + 7-=<%< F! ?+ %$- )&'E 0 ':3g' 0 +
A@ " < E ) 4+&53 s : - # - P +&' # + +;5' 0 'D' # $ A@ +&5' 43 P < C:-/+ 0 i 5-R$,' Algorithm 3.37
- $,'*+
S
4 ; P
u
D
pi
r
w
&'79 ?'
←
←
←
←
←
←
←
A@ # %$ 0 +S< 0 ' $(' E 3g' 0 +;$L- 0 # - 0 ' # E '
C 0 Y
uy
{P}
s
s
s
s
(∂S)
{y, z}
@ C 0 Y
!"
!
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4(57689
25;<5,=
=
r
pi
D
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p -=)&' @ 1< r -=+&1);-=+&' (P, u) - 0 #H# 7$,$('DC*+
:
C:- 0A0 %CD-=<%< F!
17
pi
:
D
P
D
(∂S)
3.7. Algorithm Summary
>@ p /- );' @ 1< r -/+&1);-/+&'SC*)&'D-=+&' # -9);' ':lX+ 6 0 (a , . . . , a ) k 0 P +&5' 0
<4':+ (p , r, w) ← (a , b, a ) 6 5'*)&' b %$+&05' +;5'*k) = a J0 ' E 52 ) A@
i
0
1
1
0
a0
P
S' P 'D-=+1 0 + 4<'*Y!'*) F D %$C 0 Y!':l 0` +;' P - 2d':< /6X
- @ ':Y!'*) F D 7$C A0 Y!'*l +&5' 0 p 5 P r>j ' # E 'O$ (P) - 0 # '*l ?+ 2 @ p = +;5' 0 <%'*+ (p , r, w) ← (b, a, c) 6 5'*)&' b 7$U- )&' '*l
Y!':),+&i'*l A@ $ 3g' # 7$,$('OCN+ i0 P < FsE0 D @ ) 3 - 0 # a - 0 # c -=)&'
+&5' 0 ' E 5_2 );$ A@ b 0 D C '*+ s 2d'k+&5' ).- F ':3U- 0 -/+ 0E @ ) 3 p 0 # 4)&'DC*+ A0 A@ −
− - 0 #
wp
i
i
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# @ s 5 4+;$ - $,' E 3g' 0 + qt D -/+U- P 0 + x qt
- 0 # t %$
0 +;'*) )9+ +&5' +,) %- 0E <4' (p , r, x) +&5' 0 ) +;-=+&' s + /6 -=) # $ q -R$
# 'O$,CN) ?2d' # 0 i ) P $ ?+ 0 s is
' @ s 5 ?+.$S-k$(' E 3g' 0 + qt D +&5' 0
) (P, u, ) ← l+&' 0 # r L' '*l (P, u, , p , r, s)
i
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t
i
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A@ D p k p`
s
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1 0 $,- k@ '[`);' ':lU+ 6 0 - 0 # 0 i+ 2 +&5 p r - 0 # p w -/)&' E # @ )
+;5' 0 (P , u, ) ← );- Er # E ' (P,iu, , p ,is , p p ) D
k `
<7$(' # 7$,$('OCN+ D -=< A0E s - 0 # <%'*+ P ← P i
l +&5' 0 <4':+ (p , r, w) ← (p , a, c) 6 5':)&' a
) @ ] p %$)&' '*
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7$ +&5' +&5':) = p B0 ' E 52 ) Ai@ p 0 P - 0 `# c 7$ +&5' +&5':)
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#
(P, )
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Chapter 3. Hamiltonian Polygons
" 0 ':ls-R3 P <4' 4<%<%1$ +,);-=+ 0E +&5' # T '*);' 0 +$ +&' P $ A@ " < E ) 4+&53 s ! %$ P ) r
Y # ' # 0 8 E 1)&' 2d'D< /6X
" $ - );$ +J$ +;' P + =6 -/) # $ P ) Y 0E +;5'9C ),);'DCN+ 0 'D$,$ A@ " < EA ) 4+&53 s ! 4< ':+
1$S-/) E 1'f+;5-/+ 4+ %$ 0 # ':' # - 0 -=< E ) ?+;53 &'79 ?' 50703:;59(
Proposition 3.38 2$(':),Y!' );$ + +&5-/+ 0 2-$ 7C P ':);-/+ 0 ':Y!'*))&':3 Y!'O$ -MY!'*),+&':l @ ) 3
@ P 7$ C;5- 0E ' #U# 1) 0Eq -=+&1);-=+&' 0 +;' P E +;5' 0 |V(P)| 0 CN)&'O-R$('O$\2 F
+ 0 ' 2^'OC:-R1$(' 0 ' ' 0 # P 0 + A@ - 0 1 0 $,-/+&1);-/+&' # $,' E 3g' 0 + 7$- ## ' #
+ V(P) Q );' 0 ' 6 Y!'*),+ %C:'D$9C 1< # - PP 'D-=)-R< 0E +&5' E ' # 'D$ 7C:$ U ' 0 C*'
$,1C.5HC;5- 0E 'D$SCD- 0H C:C:1) 0 - 0 4+&' 0 13q2d'*) A@ ?+;'*);-=+ 0 $ A0 < F!
+&5':) 6 %$,' ' ?+&5'*) +&' P ' ) +;' P @ %$g':lV'OC*1+;' # 0 ':Y!'*) F ?+&':);-/+ 0
n0 +&' P ' |V(P)| 0 C*)&'D-$('D$k2 F -/+k<%'D-$ + + 6 2^'OC:-R1$(' +;5' ' 0 # P 0 +;$ A@ $,' E 3g' 0 +q+&5-/+ 6 -R$ 0 P -=)&' 0 C:<41 # ' # -R$qY!':),+ 7C*'O$ 0 + P >0 +&' P @ -=+
<%'D-$ + 0 '[)&' '*l Y!'*),+;'*l A@ -q)&' E 0 D @ ) 3 7$L)&' 0 # '*);' # C 0 Y!'*l - 0 # 0
0 ' 6 )&' '*l Y!'*),+ %C:'D$ A@ -f);' E 0U@ ) 3 -/)&'[CN);'D-/+;' # +&'+&5-/+ 0 ' 6 )&' '*l
Y!':),+ 7C*'O$
3U- F - PP 'D-=) 0
0 C 3 P -=) 7$ 0 + +&5' P )&'*Y 1$ 4+&':);-/+ 0
+;5' @ <%< =6 0E CD-=<%<^+ -/+&1).-/+&' 0H +&' P E 21+ -R$ # %$&C*1$,$,' # -/2 Y!'+;5 %$ C:- 0
5- PP ' 0 - 0 4+&' 0 13q2d'*) A@ + %3 'O$ A0 < F!q ' 0 C*' - @ +&'*)X- 0 4+&' 0 13q2d'*) A@
4+&':);-/+ 0 $L'*Y!':) F D 7$C 0 Y!'*l - 0 # +&5' -R< E ) 4+&53 +&'*)&3 0 -/+;'D$ " $-U$('DC A0 # $ +;' P 6 'q$(5 /6 +&5-=+$(5 + 0E +&5' );- Fb@ ) 3 - P ) 2<%':3 r
-=+ 7C Y!'*),+&':l 3U-/m!'O$+&5 7$Y!'*),+;'*l $,- @ ' " $ 6 'fm 0/6 @ ) 3 i ) P $ ?+ 0 s +;5-/+ $&- @ ' Y!'*),+ %C:'D$g)&'D3U- 0 $,- @ ' +&5 7$U3g'D- 0 $ 0 # ':' # +&5-=+ - 0 + r C:- P $ - 0 #
1 0 $,- @ 'f)&' '*l + 6 0 $SC:- 0 2d'f)&'D$ <4Y!' # 0 'o2 F A0 ' Proof.
V(P)
-=+<%'D-$
?=50 5 4507 7E$ ((E?=K 78 87 5 5>(E<=&# 5 4507 5 Proposition 3.39
?=5 7E 5 < ( 5A 507E;C
&@79 '
? ( B ? ? ;
3(
pi p
i
7
( I7 A ( 5 705 5 4 3 ?=5 4507 65
3(
B
p
p
i
i
( 5 ; ?=5 50:D> ? ; 65878; L +&'f+&5-=+ );- Er # E ' 7$ 0 +9- PP < %' # + - Y!'*),+;'*l +;5-/+ 7$- 0 - 0 + r
CD- P ) P -/)&+ A@ - 0 1 0 $,- @ 'X)&' ':l + 6 0L
5'*);' @ )&' ' 4+&5':) l+;' 0 # r L' '*l %$
- PP < 4' # + p - 0 # +&5' 0 p 7$ - $(+,) 7CN+&< F C A0 Y!'*l Y!'*),+&':l A@ +&5'U);'D$(1<4+ 0E
@ );-=3g' - 0 # 5' i0 C*'X$&- @ ' ) i6 ' $ %3 P < F # %$&$('DC*+M-=< 0E +;5'f);- F $,5 + @ ) 3 p
i
- 0 # 0 C*'X- E - 0 p %$S- $,- @ ' Y!'*),+;'*l A@ +&5'f);'D$(1<4+ 0E @ );-=3g' i
' );$ + # %$,$,'DCN+ 0 -R< E ) 4+&53 # 'O$,CN) ?2d' # 0 'DC*+ 0 V 6 '
" $ @K ) +&5
$,5 /6 +&5-=+ +&5' 43 P ),+;- 0 + P ) P '*)&+ 4'O$ A@ +&5' P < FsEA0 P - 0 # +&5' # 7$,$('DC*+ A0
-/);' 0 Y-=) %- 0 +.$ A@ " < EA ) 4+&53 s !
Proof.
D
3.7. Algorithm Summary
y
z
y
z
y
z
y
z
y
z
y
z
&@79 '
? <=&
5 # 5 4 5 D&'59( 705
Figure 41: E( ? @
D 5ED D@7 D ?=5 < ( 78 4 ?' B ? E7 $ ( 8( ?=K 785
7 #5
D B 37 B 5 D 05 K650( B 4 ?' 5 ( 785 (8:
DK( 7
B D!
Chapter 3. Hamiltonian Polygons
y
z
y
z
y
z
y
z
y
z
y
z
<=& &@79 ?' 54 5 B 3 5 D Figure 42: =
? 5 K(8 (8656 4 04 5
D&@59( 785 B ?= @"5
D :D4 5 + 3 D (E( 5 B 36 B '5 "#%$&@ (
P
D
3.7. Algorithm Summary
3( 4
'4 $;( E7 5 :D
Proposition
3.40
P
(5 B 85 07 5 5 < &@70 '? P
3
3=&< #79 ( ?=# D $ 3 5 E507
5
3
5 (8( '507 65
r
7
( 5705 5 4 3
( B '
5
3
7 ( 53
P
3(4'4 $;( 85E :D (
785E '507 ?=5 # 4
3 &'79 ?' ?=587855 (8( ( 5 B 3 ?=50 3 (
P
p
i
?=50
( "7 pi
?=50785
P
'587 B 95 ( <
4
? ;
785 87 5 5 2< ( 5
P
D
D 7 ?=5 4507 B 50( ( 5 87 5 5 4 3
B 5 7
p
i
3 B D 50 5E
D &'59( 78
5 85 7 5 B 0K5? K
8K? 5ED&'59( 3 B D 50 3
785 "507 5 B ?=50 50 ?=587
p
P
i
( "7 ( 5 785 5 4 < 3
7 5 D&'5 < B D 58 pi 4 K( ?' 9$ ?=5 7E$ 4F=& 4 ?' ? ?=P5
D3(8( 5 # $&@ 4 K(
B
B p
i
(+ % < ?=5 =705 ( 507E; 7$ R- < 6 - F $U- @ );-=3g'H- 0 # - 0 # 5 < # @K )g+&5'H$,-=3g'
A0
0 i ) P $ ?+ 0 s Rs\ +S);':3U- 0 $L+ $(5 =6 +&5-/+ I 5 < #
- 0_F + 43g' - @ +&'*) +;' P 2 - 0 # +&5-/+ . ; - 0 # 5 < # - 0_F
+ 43g'f2d' @K );' +&' P - 0 " < E ) 4+&53 s !_
':+o1$ -=) E 1' +&5-/+f+&5' $ +;-=+&':3g' 0 +;$[5 < # @ )o+&5' );$ + ?+&':);-/+ 0 A@ +&5'
< _ P 0 " < E ) ?+&53 s !
7$ - # 4)&'DC*+ C A0 $(' 1' 0 C*' @ i ) P $ 4+ A0 s as Z F i ) P $ r
+ 0 V q0 # 12<4'Y!':),+&'*l A@ +&5' @ ).-=3g' %$ )&' '*l 0 - 0F # 7$,$('DC*+ A0 P < Fr
E0B
51$B'*Y!':) F )&' '*l Y!'*),+;'*l A@ - 0_F P < FsE0 D 7$\-f$ 0E <%'Y!'*),+;'*l A@
+;5' @ );-R3g'o- 0 # @ <4< /6 $ @ ) 3 i ) P $ 4+ A0 s aV >0h@ -CN+ -RCDC ) # 0E
+ i ) P $ 4+ 0 s Aa +&5'*);' 7$L-/+S3 $ + 0 '[' # E ' 0 C # ' 0 + + -q$ 0E <%'f)&' '*l
Y!':),+&':l A@ P +&5-/+ 7$ 0 + P '*) @ 'DC*+ +;5'' # E '92d'*+ 6 ':' 0 +&5'9+ 6 Y!':),+ 7C*'O$\+&5-=+
@K );3 - )&' '*l + 6 0 @ ':l 7$ +&' 0 + 5 7$ %3 P < %'D$ ;
& & - 0 # -/)&'q-UC A0 $(' 1' 0 C*' A@ +&5' Y!':),+&'*lb$('D<4'OCN+ 0 - @ +&':)
p -=)&' @ 1< r -=+&1);-=+&'o+ E '*+;5'*) 6 4+&5 i ) P $ 4+ 0 V as
C*<%'D-=)&< F 5 < # $o2^' @ )&' +;5' ).$ +o);- F %$X$(5 + Z\' @ )&'k+&5' C:-=<%<\+ p -=)&' @ 1< r -=+&1);-=+&' '*Y!'*) F # %$,$,'DCN+ 0 P < FVE0 5-R$-/+[<4'O-R$ ++&5)&'D' ' # E 'D$- 0 #
$,1)&':< F A0 ' A@ +&5'D3 7$ # T '*)&' 0 + @ ) 3 yz @ +&5' 0 - E ' # 'O$ %C )&':Y 7$ 4+;$kY!':),+&':l -R<4<L)&'O$(1<4+ 0E # 7$,$('DC*+ A0 P < FsEA0 $ 5-DY!' -`C 3g3 A0 ' # E ' 6 4+&5 P
-R< 0E +&5' E ' # 'D$ %C - 0 # +&5 7$S' # E ' 7$ 0 ' ?+&5'*)- 6 ' # E ' r ' # E ' )&'OC:-=<%< +&5-=+ @
Y!':),+ 7C*'O$S-/)&' )&'*Y %$ ?+&' # +&5' 0 +&5' ?) ) E 0 -=< C:C:1),)&' 0 C*' %$<%-=2^'D<4' # - 6 ' # E ' Proof.
);'D-R$
$-R$P
D
Chapter 3. Hamiltonian Polygons
-$ -/) E 1' # 0 i ) P $ 4+ 0 s
-0# z
0 ) ' 1-R< + yz 2^'OC:-R1$('
y
);':3U- 0 C 0 Y!'*l $ 0E <%'oY!'*)&+ %C:'D$ A@ +&5' @ );-=3g'f+&5) 1 E 5 1+ " < E ) 4+&53 V !
51$ 0 # 'D' # -=<%<XC:<%- 43g' # 0 Y-/) %- 0 +;$h5 < # @ )h+&5' ).$ + ?+&':);-/+ 0 A@
" < E ) 4+&53 V !B '*lV+ 6 ' 6 4<%<$(5 /6 +&5-=+B- 0 -=),2 4+,);-=) F 4+&'*).-/+ 0 3U- 0 +;- 0 $
-R<4< 0 Y-=) 7- 0 +;$ P ) Y # ' # +&5-/+S+;5' F 5 < # 0 +&5' 4+&'*).-/+ 0 2^' @ )&' >0 ':Y!'*) F 4+&'*).-/+ 0 A@ " < E ) 4+&53 s ! ':ls-CN+&< F 0 ' A@ +&5' @ <4< /6 0E
+;5)&':'U':Y!' 0 +;$ CDC*1);$ ' ?+&5'*) l+;' 0 # r L' '*l 7$q- PP < %' # B ) );- Er # E ' %$
- PP < 4' # )q- # %$,$,'DCN+ 0 P < FsEA0 7$ $ P < ?+k-R< 0E -h);- Fbj ' 6 4<%<L- 0 -R< F e:'
+;5'D$(' +&5);':' C:-$('D$ $(' P -/);-=+&':< F - 0 # -/) E 1' +;5-/++&5' 0 Y-/) %- 0 +;$c-/);'3U- 0 +;- 0 ' #
@K )S'O-RC.5 A@ +&5'D3 Z1+ );$ +9<%'*+1$S3U-/m!' - E ' 0 '*);-R<);':3U-/),m )&' E -=) # 0E - P +&' 0 + 7-=< 1 0 $,- @ '
- 0 + r C:- P )J- 0 1 0 $&- @ 'S);' ':l + 6 0 +&5-=+ '*l %$ +.$ 0 +&5' @ );-R3 '-=+ 2d' E 0UA@ +&5'
4+&':);-/+ 0
" C:C ) # 0E + +&5'*);' %$g-/+U3 $ + A0 'H$(1C.5 C 0 E 1);-=+ A0 - 0 #
2 F p 7$ P -/),+ A@ 4+ " $[+&5'q).- F s 0 +&5 7$ 4+&'*).-/+ 0 %$X$(5 + @ ) 3
iC:<41
'
C
# '[2 F i ) P $ ?+ 0 s +&5-/+- 0F 1 0 $&- @ 'o- 0 + r C:- P )S1 0 $,- @ '
V
6
0
p
i
);' ':l + 6 0 +;5-/+ '*l %$(+&' # -/+ 2d' E 0 A@ +&5' 4+&'*);-=+ A0 7$ $&- @ ' -=+ +;5'g' 0 # A@
+;5' 4+&'*);-=+ A0c
1 PP $(' +&5-/+ lV+&' 0 # r L' ':l 7$ - PP < 4' # 0 - 0 ?+;'*);-=+ 0 A@
Case
1:
" < E ) 4+&53 V !
5 < # $ -/+ +&5' ' 0 # A@ +&5 7$ 4+&':);-/+ 0 2d'OC:-=1$(' -=<%< )&'O$(1<4+ 0E # 7$ r
$,'DCN+ 0 P < FVE0 $S5-:Y!' - E_ # ' # E 'q-=< 0E +&5'qC A0 $ +,)&1C*+&' # E ' # 'D$ 7C:
$ 0
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A@ - 0 # i ) P $ ?+ 0 s Ras\pJA0 $ # '*)J+;5' @ <%< =6 0E CD-=<%<^+ -/+&1);-/+&' 0
+;' P E " $ +;5' ) %' 0 +;-=+ 0 @ +&5' @ );-R3g' # 'D$ 0 +C A0 +;- 0 - 0F -=<4+&'*) 0 -/+ 0
- 0 # +&5'MY!'*),+ 7C*'O$ -R< 0E +&5' E ' # 'O$ 7C:$L-/)&' ) %' 0 +&' # + /6 -/) # $J+&5'[C 0 Y!'*l )
$&- @ ' Y!':),+ 7C*'D$ p - 0 # r -/+;1);-/+;' # 'O$ 0 +[CN)&'O-/+&' - 0_F 1 0 $,- @ ' - 0 + r CD- P )
- 0_F 1 0 $&- @ 'k)&' i':l + 6 0 2 F i ) P $ 4+ 0 s q
5V1$ 5 < # $o-=+o+&5'
' 0 # A@ +&5 7$ ?+;'*);-=+ 0
Q )&' Y!':) Vi ) P $ 4+ A0 V a +;':<%<%$J1$\+&5-=+ @ l+;' 0 # r L' '*lgCN)&'O-/+&' # - 0
- 0 + r C:- P +&5' 0 ?+ 7$ -/+ A0 ' A@ q ) t 5'9$('D<4'OCN+ 0 A@ p 0h +&' P '' 0 $,1)&'D$
+;5-/+ p 7$M$,'*+9+ +;5 %$9- 0 + r CD- P @ 4+9'*l 7$ +;$ " $ p %$M$('*i+M+ ' 4+&5':) q ) t 4+ %$ P -/i)&+ A@ +&5' 0 < F 1 0 $,- @ ')&' '*lU+ 6 0 (q, t) B
i 5'*)&' @ )&' - 0 # 5 < # -R$ 6 ':<%< " <7$ X 2$,'*),Y!'L+&5-/+ @ p 7$ $('*+ + t 0 +;' P 'S+&5' 0 r %$ $,'*+ + -fY!'*)&+&'*l a
0HE ' (t, x, p , r) +&5-/+ %$MiC 0 Y!':l )M$,- @ ' ':Y!' 0 ) );-/+;5'*) 0 P -=),+ 7C*1<7-/) i
D!
3.7. Algorithm Summary
@ 4+ 7$9+&5' $,-=3g' r -R$ 0 +&5' P )&':Y 1$ 4+&':);-/+ 0M %3 4<7-/)&< F! @ p %$M$('*++ i'
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7$C A0 Y!'*l )9$,- @ ' 0 P +&i5-=+ 7$ 5 < # $-/+S+;5'o' 0 # @ +&5 %$ 4+&':);-/+ 0
L' E -/) # 0E - 0 # $(1 PP $('o+&5-/+- P ':) @ 'DC*+S' # E ' ac 0 C # ' 0 +
+ -h)&' '*l Y!'*),+;'*l a A@ $ 3g' # %$&$('DC*+ 0 P < FVE0 A 7$ 0 + P '*) @ 'DCN+
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0 - 0F # 7$,$('OCN+ 0 P < FsEA0 - 0F 3 )&' 2 Fì ) P $ ?+ 0 s7 0 +&5' +&5':)
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A@ +;5' @ ).-=3g'- @ +&':) l+&' 0 # r L' '*l B
5'*i )&' @ )&' -=<%<d' # E 'O$ 0 C # ' 0 +J+ )&' '*l
Y!':),+ 7C*'O$ A@ D );':3U- 0 P ':) @ 'OCN+ -$ < 0E -$J+;5'9Y!':),+ 7C*'O$J);':3U- 0 );' ':l 0 D j ' 3g- F -=) E 1'o$ %3 4<7-/)&< F @K )L+&5' @ <4< /6 0E CD-=<%< + -/+&1);-/+&' 0 +&' P E
0 + 0E +&5-/+ -=+k+&5-=+ P 0 + +&5' 0 < F 1 0 $,- @ ' );' ':l + 6 0 0 +&5' @ );-R3 ' %$
" $+&5' F -/)&'qC Y!'*)&' # 2 F +&5' '*lC*' P + A0 0 , 5 < # $-/+M+&5'
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t)
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' 0 +&' +&5 7$ @ );-=3g' 2 F Q L'DCD-=<%< +&5-/+[2 F i ) P $ 4+ 0 s q 7$f-U$ 0E <%'
);' ':l Y!':),+&'*l A@ Q Q )&' Y!'*) +;5' +&5':) = t B0 ' E 52 ) q A@ q 0 Q %$
0 +&5' E ' # 'O$ 7Cf+ /6 -/) # $ p P $,$ 42< F q = p - 0 # ) %' 0 0+&' # - 6 - F @ ) 3
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7$ - # 12<%'LY!'*)&+&'*l A@ Q +&5' 0 ?+ 7$ 0 +\)&' '*l i 0 - 0F # 7$,$('O0CN+ 0 P < FsEA0 2 F
i ) P $ 4+ 0 s - 0 # 5' 0 C:' +&5'*)&' %$ 0 +&5 0E 3 );'o+ $(5 =6X +&5':) 6 %$,' 7$o- $ 0E <%'kY!':),+&'*l @ Q - 0 # 6 ' CD- 0 -=) E 1' -$o-/2 Y!'q+;5-/+f+&5' ' # E ' tt
t7$ '*) 'DC*+
P @ s6 5':)&' t0 7$L+&5' +&5'*) = q 0 ' E 52 ) A@ t 0 Q 5 7$+ %3 0'
+;5'SY!'*)&+ %C:'D$ 0 +;5' E ' # 'D$ %CM-/)&' ) 4' 0 +;' # - 6 - F @ ) 3 t - 0 # E 1-/) # ' # 2 F
+;5'X$ +&) %C*+&< F C 0 Y!'*l )$,- @ 'XY!'*)&+&'*l r 0 Q k
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Case
2:
) ?+&53 s !
5 < # $f-/+o+;5' ' 0 # A@ +&5 7$ ?+;'*);-=+ 0 2d'OC:-=1$(' +&5' $('OC 0 # )&' '*l
C:C:1),)&' 0 C*' A@ p C*)&'D-=+&' # 0 );- ERr # E ' %$ $,- @ ' 5':)&' @K );' -R<4< );'D$(1<4+ 0E
# %$,$,'DCN+ 0 P < FVEi0 $o5-:Y!'g- EA # ' # E 'g-=< A0E +&5'gC 0 $ +,);1CN+&' # E ' # 'D$ 7C:$
Da
Chapter 3. Hamiltonian Polygons
0 2 +&5 );- Er # E 'X- 0 # +&5' @ <%< =6 0E C:-R<4<+ -/+&1).-/+&' .
L' E -/) # 0E 0 +;'9+&5-/+J2 F - 0 # +&5'MY!'*),+ 7C*'O$ pk - 0 #
0 +&' @ # k0 + @ )&3 - 0 1 0 $,- @ '[)&' ':l + 6 0
5V1$ 6 ?+;5 1+< $,$ A@
p' ` '*);-R< 4+ P
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F p 7$ 0 + P -/),+ @ - 0 1 0 $,- @ 'M);' ':l + 6 0 - 0 # p 7$JC A0 Y!'*l )
$&- @ ' 0 P \
5' `0 +&5'oC 0 # ?+ 0 $ @ ) 3 i ) P $ 4+ 0 s ! -=)&k' @ 1< <%<%' # - 0 #
5' 0 C:' );- ERr # E ' # 'O$ 0 + C*)&'D-=+&'S- 0F 1 0 $,- @ 'J)&' '*l + 6 0 - 0 # -/+ 3 $ + 0 '
- 0 + r C:- P V0 -R3 'D< F -=+ p " $ +&5'MY!':),+ 7C*'O$ -R< 0E +&5' E ' # 'O$ %CD$LC 0 $ +,);1CN+&' #
0 );- Er # E 'h-=)&' ) %'` 0 +&' # - 6 - F @ ) 3 p - 0 # p - 0 # +&5' ) %' 0 +;-/+ 0
# 'D$ 0 + C A0 +;- 0 - 0F -R<?+;'*) 0 -=+ A0c6 'fC 0 C:k<41 # '92 F ì ) P $ 4+ A0 s +&5-=+
+;5' @ <4< /6 0E CD-=<%<+ -=+&1);-=+&' 0 +&' P E # 'O$ 0 +\C*)&'D-=+&'- 0F 1 0 $,- @ 'S)&' '*l
+ 6 0b0 )9- 0F 1 0 $,- @ 'X- 0 + r C:- P \
5 7$ P ) Y!'D$ .
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6 -R$ -R<%$ ` )&' '*l 0 +;5' @ );-R3 '2^' @ )&' );- ERr # E ' " $ -=) E 1' # -=2 Y!' p 6 -R`$
0 + P -/)&+ A@ - 0 1 0 $,- @ '[)&' '*lU+ 6 0 2d' @ )&' );- Er # E ' 5'*)&' @ )&' ?+;$ ` +&5':)
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P ) Y!'D$ 0 CD-R$(' +&5-/+ r 7$S$(':+ # 1) 0E +&' P @
<7$(' p 7$ -=+ 0 D - 0 # 5' 0 C:' 4+ 6 -R$ );' ':l 0 D - 0 # 2 F i ) P $ r
+ 0 s -R`<%$ 0 P 2^' @ )&' );- Er # E ' " $ -/2 Y!' p 6 -$ 0 + P -/),+ A@ - 0
1 0 $,- @ ')&' '*l + 6 0 2d' @ )&' );- Er # E ' " $,$,13 0EX6 4+&5` 1+ < $&$ A@ E ' 0 ':);-=< r
4+ F +&5-/+ p = p ?+ @ <4< /6 $ +&5-=+ +&5':)&' %$c-L$(12 P -=+&5 (p , a = a , . . . , a )
0 P @ )f` $ 3gk' m1 6 5'*)&' a 7$ -/+ @K )f-=<%< 1 ì < m 1- 0 # a m%$
$&- @ ' M0 +&5' +&5':)o$ # ' p 7$ E 1i -=) # ' # 2 F +&5' $('OC 0 # - 0 # $,- @ 'q)&' m'*l
C:C:1),)&' 0 C*' A@ p +&5-/+ 6 -$LCN`)&'O-/+&' # 0H ).- Er # E ' 51$ p %$ 0 + P -/),+ A@
`
- 0_F - 0 + r CD- P @ i?+ 7$ -=+ 0 D >0 $(13g3U-/) F 6 ' 5-DY!' $(5 /60 +&5-/+ @ );- ERr # E ' C*)&'D-=+&' # - 0 1 0 $,- @ '
- 0 + r C:- P +&5' 0 p 7$S$('*+9+ +&5 %$S- 0 + r C:- P 0` +;' P @\
5 7$ P ) Y!'D$ ,
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7$q$(':+ 0 +&' P 2 -=+q2d' E 0 A@ +&5' 0 '*lV+ 4+&'*).-/+ 0 " $ +&5':)&' %$ 0 1 0 $,- @ '
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5'
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);' ':l + 6 0 0 +&5' @ );-R3g' -/+M-R<4< 5' ) E 0 -R< C:C:1),)&' 0 C*' A@ p 7$<7-/2d':<%' #
-$[- 6 ' # E ' 21+ 4+ # 'D$ 0 +f- PP 'O-/)o-R$[- )&' '*lHY!':),+&':l A@ - 0F i # 7$,$('DC*+ A0
P < FVE0 - 0_F 3 )&' @ p` 7$S$ + %<4<)&' '*l 0 D +&5' 0 pi %$9$('*+L+ p` 0` +&' P @
- 0 # 2 +&5 ' # E 'D$ 0 C # ' 0 + + p -/)&' P '*) @ 'OCN+ @ ) p a 6 ' 5-:Y!' -=<4)&'D- # F
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D!
3.7. Algorithm Summary
-=) E 1' # -=2 Y!' +&5-=+ ?+ # # 0 +oC.5- 0E ' # 1) 0E +;5 %$ 4+&':);-/+ 0 - 0 # @ )[+&5'
+&5'*)g' # E 'H-=< 0E +&5' E ' # 'O$ 7C + /6 -=) # $k+&5'H$,- @ ' CDC*1)&)&' 0 C:' A@ p +&5 7$
7$q' 0 $(1)&' # 2 F ) 4' 0 + 0E +&5' 0 ' E 5_2 ) A@ p - 6 - F @ ) 3 p " E - 0 i +&5 7$
0 ' E 52 ) A@ p 7$ C A0 Y!'*l )L$&- @ ' @ 4+ %$ p ` 0 +&5' @ );-=3g'f- `0 # E 1-/) # ' # 2 F +&5' $,- @ ' CD`C*1)&)&' 0 C:' A@ p \
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Case
3:
-R< 0E -g);- F s $(5 + @ ) 3 $ 3g' )&' '*l Y!'*),+&':l p @ D 0 - 0 ?+&':);-/+ 0 A@
i
" < E ) 4+&53 V !
@ ':+2 1+&$ 5 ' );#$ +E 'OC $ A0 0$ C # '*# ) ' 0 + + -/)&' E_ # @ ) +&5' 0 C*<%'D-=)&< F +;5' F -/)&'
E # ' # E 'O$ @ )L2 +&5h);'D$(1<4+ 0E p# i%$&$('DC*+ 0 P < FVED0 $ M0 +;5' +;5'*)[5- 0 # 2 F -=+<%'D-$ + A0 ' A@ +&5' ' # E 'O$ 0 C # ' 0 ++ 7$ EA # @ ) D \Q )&' Y!'*) +;':<%<%$ 1$+&5-/+ @ 0 +\2 +&5U' # E 'O$ 0 C # ' 0 +
p
i
+ p -=)&' E # @ ) D +&5' 0 ' 4+&5'*) 0 ' @ +&5'o' # E 'O$ 6 -R$5 ?+S2 F +&5'[);- F 0
+;5' P i )&':Y 1$ 4+&'*);-=+ A0b ) p %$ P -/),+ A@ - 0 1 0 $,- @ 'f)&' '*l + 6 0 0 P i
' :+c1$ );$(+ C A0 $ # '*) +&5' C:-$(' A@ );' ':lf+ 6 0 $ S'DC:-R<4<!+&5-/+c2 +&5 Y!':),+ 7C*'O$
A@ +&5'L)&' '*lq+ 6 0 P +&' 0 + %-R<4< IF CN);'D-/+;' #g# 1) 0E +;5' 0 ?+ 7-=<C:-=<%<+ qp -/);' @ 1< r
-=+&1);-=+&'b-/)&'H$ 0E <%' Y!'*),+ 7C*'O$ A@ +&5' @ );-R3 ' @ ) 6 5 7C;5 -=<%< 0 C # ' 0 +g' # E 'O$
-=)&' EA # 2 FUi ) P $ 4+ 0 s aVB
5' 0 < Fg +;5'*) P '*).-/+ 0 +&5-=+ CN)&'O-/+&'O$J- 0
1 0 $,- @ 'g)&' ':l + 6 0 0 P 7$ lV+&' 0 # r L' '*l 0 -R3 'D< F -/+ (q, t) j ' 5-DY!'
-=) E 1' # 0 p -R$(' -/2 Y!'k +&5-/+X+&5' +;5'*
) = qt ' # E ' qq 0 C # ' 0 +X+ q
7$ P ':) @ 'OCN+ %3 4<7-/);< F! 6 'h5-DY!' $(5 /60 +&5-/+ @ t %$ -`$ 00E <%' Y!'*),+&':l A@ P
+;5' 0 +&5' +&5'*A
) = qt ' # E ' tt 0 C # ' 0 ++ t 7$ P '*) @ 'OCN+ fp <%'D-/);< F 0 +&5 7$
CD-R$('q-R<%$ qt %$ E_ # S +9);':3U- 0 0$S+ C 0 $ # '*)+&5' C:-$(' 6 5'*);' t - PP 'D-=)&' #
+ 6 7C*' 0 +&5' E ' # 'D$ 7C 0 l+&' 0 # r L' '*l ) t 7$o);'*Y %$ 4+&' # 2 F - E ' # 'D$ %C
# 1) 0E +;5' @ <%< /6 0E C:-R<4<+ -/+&1);-/+&' " <4+&5 1 E 5 qt %$ - 6 ' # E ' r ' # E ' 0
+;5 %$MC:-$(' +&5':)&' %$ EA # ' # E ' tt @K ) D 0 C # ' 0 +M+ t -=< A0E +;5' E ' # 'D$ 7C
+;5-/+[)&':Y 7$ 4+&' # t L'DCD-=<%<B+&5-/+[+&15' Y!':),+ 7C*'D$ A@ D -=< 0E +;5 %$ E ' # 'D$ %C -=)&'
) %' 0 +&' # - 6 - FH@ ) 3 +;5' )&' ':l CDC*1),);' 0 C:' A@ t 0 P - 0 # E 1-=) # ' # 2 F
+;5' $,- @ ' Y!'*),+;'*l r 6 4+&5 P $,$ 42< F r = t g
5':)&' @K );' t %$XC 0 Y!':l )X$&- @ '
0 +&5' )&'D$(1<?+ 0E @ );-R3 ' " $ 0 +&5 %$qCD1-R$(' +&5' 0 ':l+q).1- F s %$k$(5 + @ ) 3
0 # 4)&'OCN+ 0 −
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+&5 7$[);- F CD- 00 +o5 ?+
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- 0 # t -/)&' C 0 Y!'*l )$,- @ ' q
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0
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- );- F e $(5 + @ ) 3 $ 3g'q);' ':l Y!'*)&+&'*l a 0 +&5' 43g3g' # %-=+&':< F i P );'DC*' # 0E
D
Chapter 3. Hamiltonian Polygons
4+&':);-/+ 0 5 %$ 3 'O- 0 $ +&5-=+ 0 P ':);-/+ 0 6 -$k- PP < 4' # 0 +&5' P )&'*Y 1$
4+&':);-/+ 0 - 0 # 5' 0 C*' p %$ - C:- P +&5-=+ %$ 0 + P -=),+ A@ - 0 1 0 $,- @ 'M)&' '*l + 6 0
" <%$ s # # 0 +o5 ?+ - i EA # ' # E 'q2d'DCD-=1$(' +&5' 0 );- Er # E ' 6 1< # 5-:Y!'
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+;5'M' # E ' 0 C # ' 0 + + p +&5-=+ 6 -$J5 ?+ 2 F e 6 -R$C*':),+;- 0 < F E_ # 2d' @ )&' 4+
6 -R$S5 ?+ 5'*)&' -=)&'o+ 6 i P $&$ ?2<4' 43 P < %CD-/+ 0 $ 2 &+ 5 ' # E 'O$ 0 C # ' 0 +L+ a -/)&' E # " $ +&5');- F s @ ) 3 p 7$L$(5 +
0 # ?);'DCN+ 0 A@ +;5' ' # E '\+&5-/+ 6 -R$ 5 ?+ 2 F e ?+BCD- 00 + 5 4+ 2-RC;imo+ e 0 +['*Y!' 0 - @ +&'*)o- P $&$ ?2<4'q) +.-/+ 0 A@ s 0 +&' P # . S'DC:-R<4< +&5-/+
- @ +&'*) $(1C.5 - ) +;-=+ 0 ' 4+&5':) lV+&' 0 # r L' ':l %$ - PP < %' # )q- 0 ' # E '
0 C # ' 0 +J+ -o)&' '*lgY!':),+&'*l A@ D %$ 5 4+ X
5' EA # ' # E ' 0 C # ' 0 +J+ - 0 # 2 1 0 # 0E D 6 -$ 0 + 5 ?+ 2 F s ' ?+;5'*) " $X-/) E 1' # -/2 Y!'
a
+&5-=+ 6 1< # 5-DY!' 0 Y m!' # );- Er # E ' ' 0 C*' +;5'*)&' 7$ - EA # ' # E '
0 2 +&5H$ # 'D$ A@ s \0H0 ' $ # 'X+&5'o' # E ' 0 C # ' 0 +S+ p +&5-=+ 6 -R$
0 +5 4+S2 F e - 0 # A0 +&5' +&5'*)9$ # 'o+&5' E_ # ' # E ' 0 C # i ' 0 +S+ a 7$ - 0 - 0 + r CD- P ) P -/),+ A@ - 0 1 0 $&- @ '`)&' '*l + 6 0 @ 2 +&5 ' # E 'D$
a
0 C # ' 0 + + a -=)&' EA # +&5' 0 6 'UCD- 0 -/) E 1'U-R$ -/2 Y!' <7$(' a 6 -R$
E ' 0 ':);-/+;' # 2 F lV+&' 0 # r L' ':l +;5-/+ %$ a %$ +&5'o$,' E 3g' 0 +' 0 # P 0 + q
A@ +&5'f$(' E 3g' 0 + qt + k6 5 %C.5 6 'f':l+;' 0 # ' # 0 +&5' P );'*Y 1$ 4+&'*);-=+ A0
- 0 # t %$ - # 12<4' Y!'*)&+&'*l @ +&5' );'D$(1<4+ 0E @ );-R3g' ':+ q # ' 0 +;'
+&5' +&5':A
) = t 0 ' E 52 ) A@ a = q - 0 # # ' 0 +;' +&5'X+ 6 # 0%$,$,'DCN+ 0
P < FsEA0 $S+&5-=+-=)&' E ' 0 '*).-/+&' # 2 F +;5' $ P < 4+-R< 0E e 2 F D1 - 0 # D2
6 5':)&' q V(D ) - 0 # t V(D ) j '5-DY!'9-R<?);'D- # F -/) E 1' # -/2 Y!'
+&5-=+ 0 +&05 7$MCD-R$(' 1qq %$ E_ # @ 2) D - 0 # +&5'*)&' 7$-R< 6 - F $[- 0 ' # E '
0+ 7$
1
C
'
L
+
+
&
+
5
/
)
#
#
Bj ' 5-DY!'[+ $,5 /6 +&5-=+S- @ +&':)
0
0
E
_
@
tt1'
t
D
2
0 A@ +&5' D ) D 7$$ P < 4+M-=< 0E s +&5':)&' 7$$ + %<4< - E # ' # E ' 0
2 +&5h)&'D$,1<?+ 01E # 7$,$('O2CN+ 0 P < FsEA0 $ 1 PP $('J+&5-/+ D 7$ $ P < ?+ +&5-/+ 7$ p V(D ) " $ -/2 Y!' s CD- 00 +
5 4+ 2-C;m + e - 10 # ?+ # 'O$ 0 + 5 4+i qq ' 1?+&5'*) 2d'DCD-=1$,' qq 7$XE_ # ' # E ' - 0 # +;5' 0 );- Er # E ' 6 1< # 05-DY!'o2d':' 0 - PP < 4' # 0' 0 C:' +&5':)&' 7$k- E_ # ' # E ' 0 2 +&5 $ # 'D$ A@ s U0 0 ' $ # ' +&5' ' # E '
0 C # ' 0 + + p +&5-/+ 6 -$ 0 +X5 ?+X2 F e - 0 # 0 +&5' +&5'*)X$ # ' +&5'
E_ # ' # E ' 0 C i # ' 0 +S+ q = a +&5'*) 6 7$(' D2 %$9$ P < ?+M- 0 # pi V(D2) " E - 0 s # 'D$ 0 +5 ?+ e
0 ) tt 21+ 6 '[-=<7$ 5-DY!'9+ $(5 /6 +;5-/+ s # 'D$ 0 + 5 ?+J+&5' 6 ' # E ' r
1
' # E ' qt
' 4+&5'*) " $X2 +;5 q - 0 # t -=)&' $ +,) %C*+&< F C 0 Y!':l Y!'*),+ 7C*'O$ A@
0 ) +;-/+ 0HA@ s 0 +&' P # 6 %<4< 3U-=m!' s 5 4+ qt 9Q )&' Y!':) D2
7$ $(5 + 0 # 4)&'DC*+ A0 A@ −
- 0 # 0 # 4)&'OCN+ 0 A@ +&5'U' # E 'U5 ?+
e
s
tq
Either
Or
D
3.7. Algorithm Summary
2 F e ' 0 C*' @ e %$ 0 + ) +.-/+&' # 0 +&' P # +&5' 0 +;5'UC*<7- %3 %$
2Y 1$M-$ 6 ':<%< +)&':3U- 0 $M+ C 0 $ # '*)o- P $,$ 42<4' ) +;-/+ 0 A@ e +&' +&5-/+k$,1C.5 -b) +;-=+ A0 ) +.-/+&'O$ e + =6 -/) # $ t - 0 # - 6 - F @ ) 3
. " $o+&5'k) +;-=+ A0 $ + P $X-/+
7$ - Y!':),+&':l A@ D C 0 +&);-/) F
q0
pi pi
+ 1) -$,$(13 P + 0 p V(D ) B
5':)&' @K );' s # 'D$ 0 + 15 ?+ qt - 0 #
+&5':)&' -=)&' E_ # ' # E 'Di$ 0 2 +&25 $ # 'D$ A@ s 0 0 ' $ # ' +;5'U' # E '
0 C # ' 0 + + p +&5-/+ 6 -$ 0 +X5 ?+X2 F e - 0 # 0 +&5' +&5'*)X$ # ' +&5'
E_ # ' # E ' tt i
1
5V1$ 5 < # $+&5) 1 E 5 1+ " < E ) 4+&53 V !
5 < # $ C:<4'O-/)&< F 2d'DC:-R1$(' +&5':)&' %$ 0 1 0 $,- @ ' )&' ':l + 6 0 - 0 # 0
1 0 $,- @ 'X- 0 + r C:- P -/++&5' ' 0 # @ +;5 %$ ?+&':);-/+ 0J
51$ -=<7$ 5 < # $+&) ?Y r
7-=<%< F - 0 # $ %3 %<7-/)&< F 5 < # $ @ p - 0 # r -=)&' $(':+ 0` +&' P 2 L' E -/) # 0E - 0 # 2$(i'*),Y!'J+&5-=+ pi %$ $,- @ 'L- @ +&':) +&5 7$ 4+&'*).-/+ 0
- 0 # 6 5' 0 ':Y!'*)M- 0 ' # E ' 0 C # ' 0 ++ -k)&' '*l Y!'*),+&':l A@ D 7$5 4++&5' 0 p %$
$,'*+ + +;5 %$ );' ':l Y!'*),+;'*l Z F - 0 # 0 );- F CD- 0 5 4+ - 0 ' # i E '
+;5-/+ %$ 0 C # ' 0 ++ - 0 1 0 $,- @ '[)&' '*lU+ 6 0c\
5 %$ P ) Y!'D$ . & - 0 #
&\ +9-=<7$ %3 P < %'D$ 0 CD-R$('X+&5-=+
- 0 # r -/)&' $('*+ 0` +&' P @ pi
5 %$JC 0 C*<%1 # 'O$ +&5'MC:-$('[- 0 -R< F $ 7$J- 0 # P ) Y!'D$+&5-=+ +&5'C*<7- %3g' # 0 YA-/) r
- 0 +.$L5 < # -R<4<+&5) 1 E 5 " < E ) ?+;53 s !
+\)&':3U- 0 $+ - 0 -R< F e:'9+&5'L2d':5-:Y 1) A@cp 5 P r>j ' # E 'O$ 0 +&5'S<%-$ +J$ +&' P
A@ " < E ) ?+;53 s ! Z F 1)f<7-/2d':< 0E $&C;5':3g'q'*Y!':) F # 12<%'qY!':),+&'*l A@ +&5'
@ );-=3g' 7$M<7-/2d'D<4' # -R$f- 6 ' # E 'q'*l-RC*+&< F A0 C*' Mj ' 5-:Y!' + $(5 /6 +&5-=++;5'D$('
Y!':),+ 7C*'O$S-/)&' )&'D-R<4< F 6 ' # E 'D$9-R$ # 8' 0 ' # 0 E' 0 ?+ 0 s s
Corollary
3.41
4 5
D&@5 703=&
&'79 ?'
2 B (A 5'507 8 5 5
D K(
" 0 - 0 + r CD- P CN)&'O-/+&' # # 1) 0E +&5' 0 ?+ %-R<[CD-=<%<[+ -/+&1);-/+&' 7$ 0E
A@ +&5' )&'D$(1<?+ 0E @ ).-=3g'`2 F i ) P $ 4+ 0 s a +&5-=+ 7$ $&- @ '
- @ +&':)+;5' );$ + 4+&'*).-/+ 0 -$-/) E 1' # -/2 Y!' 0Hi ) P $ 4+ A0 s s " CDC ) # 0E
+ i ) P $ ?+ 0 s $&- @ ' Y!':),+ 7C*'O$c)&':3U- 0 $,- @ ' +&5) 1 E 5 1+ " < E ) 4+&53 V !
j ' P ) Y!' # i ) P $ 4+ 0 s @K ) " < EA ) 4+&53 s7Ds Z1+ ?+ %$ 'D-R$ 4< F $(':' 0
+ 5 < # @ ) " < E ) 4+&53 s ! -$ 6 ':<%<d2d'DCD-=1$('[+;5'2-R$ 7C P '*);-=+ 0 $- PP < %' #
0 2 +&5H-=< E ) ?+&53g$L-=)&'o+&5' $,-=3g' " 0 1 0 $,- @ 'f- 0 + r C:- P CN);'D-/+;' # 2 F l+&' 0 # r L' '*l %$ 0 + )&':Y 7$ 4+&' # # 1) 0E
+;5' @ <%< /6 0E C:-R<4<M+ -/+&1).-/+&' 2 F pJ ) <%<%-=) F s R 5'bY!'*),+ %C:'D$U-=< 0E
+;5' E ' # 'O$ %C 0 lV+&' 0 # r L' '*lb-/)&' ) %' 0 +&' # + /6 -/) # $+&5'q$ +,) %C*+&< F C A0 Y!'*l
Proof.
$ <%'`Y!'*),+&':l
D
Chapter 3. Hamiltonian Polygons
Y!':),+&':l p n-R< 0EUE ' (p , x, q) )M+ =6 -/) # $S+&5' $,- @ ' # 1'X+ Y!'*)&+&'*l
r " $9+&5i' ) 4' 0 +.-/+ 0 Ai@ +&5' @ );-R3g' # 'D$ 0 +fC 0 +;- 0 - 0_F -R<?+;'*) 0 -=+ A0 -=+
+;5-/+ P 0 + !6 'M3g- F C 0 C*<%1 # '92 FUi ) P $ 4+ A0 V +&5-/+ +&5' @ <4< /6 0E C:-=<%<
+ U -/+;1);-/+;' # 'D$ 0 +9CN);'D-/+;' - 0F 1 0 $,- @ ' - 0 + r CD- P 8 0 -=<%< F! C A0 $ # '*)- 0 1 0 $,- @ 'X- 0 + r CD- P CN)&'O-/+&' # 2 Fh ).- Er # E ' " C:C ) # r
0E + i ) P $ ?+ 0 V ! 4+SCD- 0 - PP 'O-/)9-=+ p 0 < F! Z FbpJ ) <4<7-/) F s +&5'
Y!':),+&':l p %$ 0 +9)&':Y 7$ 4+&' # # 1) 0E +&5' @ <%< =6 ` 0E CD-=<%< + -/+;1);-/+;' " $S+&5'
0 '*lV+ );- F `7$J$(5 + @ ) 3 p @ p %$ )&' ':l 0 D p 2^'OC 3g'D$ $,- @ ' 0 +&5' 0 '*lV+
4+&':);-/+ 0gj 'g5-DY!'U-/) E `1' # `0 +;5' P ) _A@ A@i `) P $ 4+ 0 s ! +&5-=+ p %$
0 + P -=),+ @ - 0_F - 0 + r C:- P @ 4+ %$ -=+ 0 D 5'*)&' @ )&' - 0 1 0 $,- @ ' - 0 + r `C:- P
C*)&'D-=+&' # 2 Fb );- Er # E ' )&'D3U- 0 $9- $ 0E <4'XY!'*),+;'*l @ +&5' @ );-=3g'o+;5) 1 E 5 1+
" < E ) 4+&53 V !
3.8 Induction
5 %$$('DC*+ A0 C 3 P <%'*+&'O$S+&5' 0 # 1CN+ 4Y!'
7$f).-/+&5':) $ +&);- E 5_+ @K ) 6 -/) # -R$ 3 $ + 6 +;5' # ':Y!':< P 3 ' 0 + A@ " < E ) ?+&53 s ! +;5' @ );-R3g' P - 0 # +;5' # 7$,$('OCN+ 0 C 3
P ) _A@ A@\
5' )&'D3 V 5' P ) _A@
),m 5-R$X-=<4)&'D- # F 2d':' 0 # 0 ' # 1) 0E
'*+ 1$J$(13g3U-/) %e:'S+;5' P ) P '*)&+ 4'O$ A@
P 1+&' # 2 F " < E ) ?+;53 s !
7 ( 59
D3( 3 3 5A( 5 & 85 ( K 4 B #
%< 5E7
Corollary
3.42
S
D 5
D&@5
C 0 Y
7
D ?=58705 3(A (
3
5 7E 5
{y,
z}
(∂S)
P
S
B #3F5 B I37 4 ( 5A '507 @=<=& B '5 # $K&' ( ( B ? ? ;
?=5 # 4 <& B D ( 785 ( ; ( 5
D ( 5
D&@5
{y, z}
P
7 5'5879$
50 ?=05 7
:D =
? 58705 ( s
s
P
D
S
74 ? ;
7
:D
s D
V(s) V(P)
s D =
D
(D)
(P)
D
5 45079$ # $&@ D DA587058 70
{y, z}
' 0 +&'
Z Fhi ) P
0 P - -0#
P < VF E0 $,-/+ 7$ @ F
Proof.
{y, z}+&'
? K(>
B 5
D&@54 ?
2 F (P, ) +;5' 1+ P 1+ A@ " < EA ) 4+&53 s ! @ ) 0
$ ?+ 0 s ! P 7$- @ ).-=3g'[2d' @ )&' p 5 P r j ' # E 'D$
2 F i ) P $ ?+ 0 V +&5'`)&'O$(1<4+ %$b- $ %3 P < F
0E I &
P
(
B ?
? ; 3(
P 1 + S - 0 #
7$S- PP < %' #
C 00 'OCN+&' #
3.8. Induction
" + +&5'b' 0 # A@ +;5' P )&'DC:' # 0E ?+;'*);-=+ 0 A@ " < E ) 4+&53 s ! -/+&1);-/+&'
5-R$L2d':' 0 C:-R<4<%' # 0` +;' P E %3 4<7-/);< F! 2d' @ )&'o+&5' );$ + 4+&':);-/+ 0 p -/);' @ 1< r
-=+&1);-=+&'5-R$ 2^'D' 0 C:-=<%<%' # -/+J+;5'Y!':) F 2d' E 00 0E " $ p 5 P r>j ' # E 'D$ # 'O$
0 +C;5- 0E ' V(P) -=<%<$(' E 3g' 0 +;$ @ ) 3 S -=)&'S$,-/+&1);-/+&' # 0 P - 0 # 5' 0 C*' 7$ @ 1< <%<%' # " C:C ) # 0E + U 1) 6 ' # E ' <%-=2^'D< 0E $&C;5':3g'X':Y!'*) F # 12<%'XY!'*),+&':l 0 P
7$ <7-/2d'D<4' # ':ls-CN+&< F 0 C:'-$B- 6 ' # E ' Z F pJ ) <%<7-/) F s -=<%<VY!'*),+ 7C*'O$ <7-/2d':<%' #
-$X- 6 ' # E 'UC ),)&'O$ P 0 # + CD- P $ +&5-=+ 7$ +;5' F -/)&' 6 ' # E 'O$X-$ # 8' 0 ' # 0
8' 0 4+ 0 s sRL
5'*)&' @ )&' -=<%< Y!'*)&+ %C:'D$9<%-=2^'D<4' # -$M- 6 ' # E ' -/)&' )&'D3 Y!' #
0 p 5 P r>j ' # E 'D$- 0 # P %$[-g$ %3 P <4' P < FVE0 " $9+&5'*);' ':l 7$ +;$ 0 # 12<4'
Y!':),+&':l 0 P 7$ +,) 4Y 7-=<%< F @ 1< <%<4' # -$ 6 ':<%<9- 0 # +&5V1$ P 7$ 0 # ':' # @ );-=3g' 2$(':),Y!'U+&5-=+ %$ 0 + 0 'OC*'O$,$,-/) 4< F - # 7$,$('OCN+ 0 A@ P 2^'OC:-R1$(' (P)
0 C*)&'D-$('D$ @ p 5 P r>j ' # E 'D$ C;5 P $ T - 0_F 6 ' # E ' Z1+ C*<%'D-=)&< F -R<4< P < FsE0 $
0 -/);' C A0 Y!'*l 2 F +&5' ' 0 # C 0 # 4+ 0 A@ " < E ) 4+&53 s ! - 0 # +;5' F -/)&'
C 0 +;- 0 ' # 0 (P) 2d'DC:-R1$('9+&5' Fq@K );3 ' # - # %$&$('DC*+ 0UA@ +&5' @ ).-=3g'L2d' @ )&'
p 5 P r j ' # E 'D$ 6 -$- PP < %' # %$- 0 ' # E ' @ P # 1'o+ i ) P $ ?+ 0 s 7$- 0 'O-R$ F C A0r
{y,
z}
$,' 1' 0 C:' A@ & - 0 # & 8 0 -R<4< F 7$ %3 P < 4' # 2 F ;
W $ 0E +;5'D$(' P ) P :' ),+ %'D$ 6 ' C:- 0 P ) Y!' 5' )&'D3 s 0 # 1CN+ 4Y!':< F -R$
K@ <%< /6 $ +&5' @K <%< /6 0E $(+;-/+&' r
j ' P ) Y!'S2 F 0 # 1CN+ 0 0
Proof.
of
Theorem
3.2
|S|
3g' 0 + 8 )q- $('*+ S A@ # 7$3 0 +q< 0 ' $(' E 3g' 0 +;$ 0 + -=<%< C <%< 0 'O-/) - 0 # @K )
- 0_F lV' # ' # E ' {y, z} A@ C 0 Y (∂S) +&5':)&' 7$S- -R3 %<4+ 0 %- 0 P < FsEA0 H @ )
$(1C;5h+;5-/+ {y, z} 7$S- 0 ' # E ' A@ H S
5' $ +;-=+&':3g' 0 +q5 < # $ @ ) |S| = 2 1 PP $(' 4+ 5 < # $ @ )k-=<%< S 6 ?+;5
1 < |S | < |S|
pJA0 $ # '*) +&5' $ %3 P <4' @ );-=3g' P - 0 # +&5' # %$,$,'DCN+ 0 # 'O$,CN) ?2d' # 0kpJ ) < r
<7-/) F s _ @ 2 +&5 ' 0 # P 0 +;$ A@ ':Y!'*) F $(' E 3g' 0 + -/)&' 0 V(P) +&5' 0 +&5'
$(+;-/+&'D3g' 0 + 5 < # $ U @ +&5'*);' 7$X-h$(' E 3 ' 0 + s 6 5 $,' 0 ' ?+;5'*) ' 0 # P 0 + %$ 0
+&5' 0 2 F i ) P '*),+ F s 7$ 0 +&5' 0 +;'*) ) A@ $ 3g' D Z F
V(P)
P ) P ':),+ F D 5-$ - C 3g3 A0 ' # E ' ab = yz 6 ?+&5 P Z F - 0 #
2d'OC:-=1$(' D 7$MC A0 Y!'*l 6 ' 5-DY!' C(D) := C 0 Y (∂S D ) D Q )&' Y!'*) 5-$ - 0 ' # E ' cd $(1C.5g+&5-=+\2 +&5 ac - 0 # bd -=)&' 0A0r C*) $,$ 0E Y 7$ ?2 4< r
C(D)
4+ F ' # E 'O$ @ c d , . . . , c d @ )9$ 3 ' m -/)&' +&5' $(' E 3g' 0 +;$ 0 D - 0 # +&5' F -=)&'g-=1<%<\1C <4< 0 'D-/m) m0 +&5 7$ ) # '*) +;5' 0 )&' P <7-RC*' +&5' ' # E ' ab @ P
Chapter 3. Hamiltonian Polygons
2 F +&5' P -=+&5 (a, c , d , . . . , c , d , b) +&5':) 6 7$(' +&5'*);' 7$ 2 F 0 # 1CN+ 0
m
- -=3 %<?+ 0 7- 0 P 1< FsE10 H(D)
@ m)+;5' $(' E 3g' 0 +;$ @ ) 3 S 6 5 7C.5 < %' 0 D $,1C.5 +&5-/+ cd %$o- 0 ' # E ' @ H(D) '*+ H (D) 2d'q+&5' P -/+;5 @ ) 3 c + d
+;5-/++,);-DY!'*).$('D$ H(D) '*lC*' P + @ )M+;5' ' # E ' cd L' P <7-RC*' +;5'q' # E ' ab A@ P
2 F +&5' P -/+;5 (a, c) H (D) (d, b) 9 0E $ f@K )'D-C;5 D +;5-/+\C 0 +.- 0 $
$,' E 3g' 0 +;$ @ ) 3 S )&'O$(1<4+;$ 0 - -R3 %<4+ 0 %- 0 P < FsEA0 @ ) S W P + 0/6 6 ' 5-DY!'H2d'D' 0 6 )&m 0E 1 0 # ':) +&5' -$,$(13 P + 0 +&5-/+ -=<%<
$,' E 3g' 0 +;$ 0 S -=)&' 00sr # ' E ' 0 ':);-/+;' Z1+ +&5' E ' 0 '*);-R< %eD-=+ A0 + P $,$ ?2< F
# ' E ' 0 ':);-/+;'`< 0 ' $(' E 3g' 0 +;$ %$ );-=+&5':)h$ +&);- E 5_+ @K ) 6 -/) # j ' CD- 0 P )&':Y!' 0 +
- 0_F );- F +&5-/+ %$ $(5 + # 1) 0E " < E ) 4+&53 s ! @ ) 3 5 4+,+ 0E - # ' E ' 0 ':);-/+&'
$,' E 3g' 0 +J2 F P ':),+&1)&2 0E ?+ 0 0 4+&'D$ 43U-=<%< F
5' 0 - # ' E ' 0 ':);-/+&'f$(' E 3 ' 0 + %$
' ?+;5'*)Y 7$ 4+&' # 2 F - E ' # 'D$ %C ) 4+ - PP 'D-=);$ 0 $ 3g'LC 0 Y!':lq51<%<s2 1 0 # -=) F
n0 2 +&5 CD-R$('O$ +;5' $,' E 3g' 0 + 7$ %3g3 ' # 7-/+&'D< F $,-=+&1);-=+&' # 0 P -=),+ 7C*1<7-/) 0
# ' E ' 0 ':);-/+;' $(' E 3g' 0 +f'*Y!'*)o- PP 'D-/).$o-R$f-g);' ':lHY!'*),+;'*l A@ +&5' @ ).-=3g' X
5 7$
P ) Y!'D$k+&5-=+ 5' )&'D3 s 5 < # $g-R$U$(+;-/+&' # @ )U- 0F $('*+ A@ # 7$3 0 +g< 0 '
$,' E 3g' 0 +;$ +&5-/+9-=)&' 0 +-R<4< C <%< 0 'D-=) 3.9 Runtime Analysis
n0 +&5 7$o$('DC*+ A0 6 'k$(5 /6 +&5-=+ 5' )&'D3 s P ) Y # 'D$f- P < Fs0 3 7-=<B+ %3 '
-R< E ) 4+&53 + C 0 $(+,)&1C*+ -=3 4<4+ 0 7- 0 P < FsE0 $ @K ) # 7$3 0 + < 0 '9$(' E g
3 ' 0 +;$ 7 $ ( 5 "<74 3(5 D 3( 3A%< 5 (5 & 05 ( K
Theorem
3.43
n
4 B #3%< 5
7 $&@ B 05 B (8 7 B 5ED < #
O(n2 )
< 5 :D
(+" B 5 O(n)
L'
)&'D$,' 0 +M+&5' @ ).-=3g'q- 0 # ?+.$ # %$,$,'DCN+ 0 C <4<%'DC*+ 4Y!':< F 0 - D $
op a! n >0 +&5 7$ 6 - F - 0 ' # E 'XC:- 0 2^' )&' P <%-C*' #
2 F - P < FVE0 -=< P -/+&5 0 + %3 ' P ) P ),+ 0 -R<+ +;5' $ %e:' A@ +&5' P -/+&5 " $ 'D-C.5
Y!':),+&':l A@ +&5 7$ P <%- 0 -/) $(12 # 4Y 7$ A0 %$ 0 C # ' 0 +o+ -/+o3 $ + @ 1)f' # E 'D$ 6 '
CD- 0 ':ls-R3 0 'f+;5' 0 ' E 52 )&5 _ # A@ Y!'*)&+ %C:'D$S- 0 # ' # E 'O$ 0 C 0 $ +;- 0 +S+ %3g' + )&' @K ) 'D-C;5 0 P 1+ $(' E 3g' 0 + +&5' 0 C # ' 0 +XY!'*),+ 7C*'O$ A@ +&5' @ );-=3g' @
- 0_"F . - 0 # Y 7C*' Y!'*);$,- @ ) 'O-RC.5 Y!':),+&':l A@ +&5' @ ).-=3g' $ + );' 4+;$ n1 0 1' 0 C # ' 0 + 0 P 1+k$(' E 3g' 0 + 8 ) 'D-RC.5 # %$,$,'DCN+ 0 P < FVE0 $ + )&'U+;5' < %$(+ A@
0 P 1+S$,' E 3g' 0 +;$+&5-=+9-/)&' 0 +&'*) )S+ +&5' P < FsEA0c
1) 0E " < E ) 4+&53 s ! 6 ' 3U- 0 +;- 0 - < 7$ + @ +;5 $('oY!':),+ 7C*'D$+&5-=+9-/)&'
1 0 $,-/+;1);-/+;' # - < 7$ + @ +&5 $(' Y!':),+ 7C*'D$ +&5-=+ -=)&'g)&' '*l 0 $ 3g' # 7$,$('DC*+ A0
P
Proof.
B 5 B 5
D 5 D&'5 %3(8
A
3.9. Runtime Analysis
P < FVE0 - 0 # -f< %$(+ A@d6 ' # E ' r ' # E 'D$ \p <%'D-=)&< F! +&5' Y!':);-=<%<$ P -RC:' 0 'D' # ' # 2 F
+;5'D$(' # -/+;- $ +,)&1CN+&1);'D$ 7$L< 0 'D-=) 5'C 0 Y!':l 5V1<%<C 3 P 1+;-=+ A0 $ C:- 0 2d' # 0 ' 0 O(n2 ) + %3g'-R<?+ AE ':+&5'*)
1$ 0E -=),Y 7$ j );- P A n " 0_F Y!'*),+&':l C:- 0 - PP 'O-/) 0 -/+ 3 $ + 0 ' A@
+;5' C A0 Y!'*l 51<4<2 1 0 # -=) 4'O$ 5' 2) # E ' ' # E 'D$ 1$(' # + C 00 'DC*+ +&5'
0 # 1CN+ 4Y!':< F C 0 $ +&)&1CN+;' # P -/),+ 7-=< $ <41+ 0 $ + +;5' E < 2-=< @ );-=3g' C:- 0 2^'
2+.- 0 ' # 2);1+&' r>@ );C:' 0 Y!'*);-R<4<< 0 'O-/)+ %3g' 5' E ' # 'O$ 7C:$qC:- 0 2^' C 3 P 1+&' # 0 - 2);1+&' r>@ );C:'g3g- 00 ':)q$ P ' 0 # 0E
< 0 'D-=) + 43g' P '*)hY!'*)&+&'*l " $ 'D-C;5 Y!':),+&'*l CD- 0 - PP 'O-/) 0 -/+b3 $(+ + 6 E ' # 'O$ %CD$ # 1) 0E " < E ) 4+&53 s ! +&5' Y!'*);-R<4<+ %3g' 0 ':' # ' # %$ - E - 0 O(n2 ) %3 4<7-/);< F! +&5' );- F $(5 _ + 0E - 0 # - P $,$ 42<%' ) +;-/+ 0 A@ +&5' );- F -/)&' # 0 '
2)&1+&' r @K ).C*' 0 < 0 'O-/)\+ %3g' P ':) );- F - 0 # 5' 0 C*' O(n2 ) + 43g' Y!'*).-=<%< S- F $
-=)&' $,5 + @ ) 3 )&' ':l Y!':),+ 7C*'D$ @ # %$&$('DC*+ 0 P < FVE0 $ 0 < F " @ +&':)X$(1C.5 ).- F 5-R$X2d':' 0 P ) C*'O$,$(' # +&5'gC ),)&'D$ P 0 # 0E Y!'*),+&':l %$ $,- @ ' - 0 # # 'D$ 0 +
- PP 'D-=)M-R$-q)&' ':l Y!':),+&'*l @ - # %$&$('DC*+ 0 P < FVE0 - 0F 3 )&' 5' Y!'*),+ 7C*'O$ +;5-/+U- PP 'O-/) -=< 0E +&5'bC A0 $ +,)&1C*+&' # E ' # 'D$ 7C:$ 5-DY!' + 2d'X3g-=),m!' # -R$ n1 0 $,-=+&1);-=+&' # - 0 # 5-DY!'o+ 2d' 0 $,'*),+&' # ))&'D3 Y!' # @ ) 3
+;5'XC )&)&'D$ P 0 # 0E < %$(+ M %3 %<7-/)&< F Y!':),+ 7C*'O$9-/);'X3U-/)&m!' # @ +;5' F -/)&' )&' '*l
0 $ 3g' # 7$,$('OCN+ 0 P < FsEA0 - 0 # ' # E 'O$U-/)&' 3U-/),m!' # -$ 6 ' # E ' r ' # E 'O$ @
- 0 0 C # ' 0 + Y!'*),+;'*l %$X<%-=2d':<%' # -R$ - 6 ' # E ' 8 0 -=<%< F! -=+ +&5' ' 0 # A@ " < EAr
) ?+&53 V ! -=<%< 6 ' # E 'D$[-=)&' )&':3 Y!' # " $-=<%< +&5'O$(' '*Y!' 0 +;$C:- 0 C:C:1) 0 < F
0 C:' P '*)SY!':),+&'*l )' # E ' +&5' F C:- 0 2^' P ) C:'D$,$,' # 0b Y!'*).-=<%< < 0 'O-/)M+ 43g' +X);':3U- 0 $ + C 0 $ # ':) +;5'k+ 43g'U$ P ' 0 +X+ 3g- 0 +;- 0 +&5' < 7$ +;$ A@ $(' Er
3g' 0 +;$ 0 +;'*) )+ +&5' # %$,$,'DCN+ 0 P < FsEA0 $ B Y!'*) F + %3 '- # %$&$('DC*+ 0 P < Fr
E0 %$ $ P < 4+ !6 'S5-DY!'L+ +&'D$ + @ ) 'D-C.5 @ +&5' 0 +&'*) ) $(' E 3g' 0 +;$ 0g6 5 %C.5 A@
+;5' + 6 )&'O$(1<4+ 0E P < FsEA0 $ 4+ %$fC 0 +;- 0 ' # 5 %$ # 'OC 7$ A0 CD- 0 2d'q3U- # '
0 < 0 'O-/) + 43g'U2 F C 3 P 1+ 0E +&5' 0 '*lV+ P < FsEA0 ' # E 'U2d':< /6 0 ' A@ +&5'
$,' E 3g' 0 +\' 0 # P 0 +;$ " $ +&5' 0 13q2d'*) A@ $ P < 4+;$ 7$ < 0 'O-/) -$ 6 'D<4< +&5' Y!'*);-R<4<
+ 43g'f2 1 0 # @ )S+&5'O$('f+&'O$ +;$ 7$ O(n2 ) *' Y!'*).-=< A@ +&5'h+;-R$ m$ # 7$,C*1$,$(' # -/2 Y!' CD- 0 2d' %3 P <4'D3 ' 0 +;' # 0 $,12 r
1- # );-/+ %C9+ %3 '1$ 0E 3 )&'MC 3 P < 7C:-=+&' # # -/+;- $ +&)&1CN+;1)&'D$ B
5'3U- 0 2 + r
+;<4' 0 'DC;m $,':':3U$+ 2d' +&5'XC 3 P 1+;-/+ 0bA@ E ' # 'D$ 7C:$ 5' C 0 $ +,);1CN+ 0bA@
3U- 0_F E ' # 'D$ %CD$L5-R$2d':' 0 - )&'DC:' 0 + @ C:1$ @ )&'O$('D-=);C.5 !_ n Z1+ 0h 1)
-R< E ) 4+&53 +&5' E ' # 'O$ %CD$L+ 2d'XC 0 $(+,)&1C*+&' # 0 - 0 ?+&':);-/+ 0bA@ +&' 0 # ' P ' 0 #
0 +&5')&'D$,1<?+ A@ +&5' P );'*Y 1$ ?+;'*);-=+ 0
5':)&' @K );' ?+ 7$ 0 +LC*<%'D-/)S5 =6 + 2+.- 0 - $(1 UC %' 0 +&< F <7-/) E 'X$,'*+ A@ E ' # 'D$ 7C:$ @K )S21<?m r P ) C*'D$&$ 0E
Chapter 3. Hamiltonian Polygons
3.10 Remarks
" $[3g' 0 + 0 ' # 0 p 5- P +&':) R 4+ %$ 6 ':<%<Bm 0/60 +;5-/+f+&5'*)&'k-=)&' +&) %- 0E 1<7- r
+ 0 $\+;5-/+ # X0 + C 0 +;- 0 - -R3 4<4+ 0 7- 0 C F C:<4' a > " $ 6 'M3g- F C 0 $ # ':)
+;5'X' # E 'D$ A@ - +,) 7- 0E 1<%-=+ 0 -R$[-g$('*+ A@ 0 +&'*);$,'DCN+ 0E $(' E 3g' 0 +;$ +&5 7$-=<7$ $,5 /6 $ +;5-/+ 0 'LC:- 00 + $ %3 P < F # ) P +&5' # 7$3 0 + 0 'D$,$ C 0 # 4+ 0q@ ) 3 5' r
$(5 /6 $ -bC 0 E 1);-/+ 0 A@ $('*Y!' 0 $,' E 3g' 0 +;$
)&'D3 s >0 @ -CN+ 8 E 1);'
@K ) 6 5 %C.5 0 +9'*Y!' 0 +&5'XY 7$ 42 %< 4+ FhE );- P 5 7$ -R3 4<4+ 0 7- 0 +&5)&'D' $,' E 3g' 0 +;$
@K );3 - +,) %- 0E <4' +&5-=+ C A0 +;- 0 $ +&5' @ 1),+;5 $(' E 3g' 0 + - 0 # +&5' )&':3U- 0 0E
+;5)&':' $(' E 3g' 0 +;$o-=)&' P <7-RC:' # -R< 0E +&5'kC 0 Y!'*l 5V1<%< $(1C;5 +&5-/+ 0 +&) %- 0E <4'
' # E ' 7$ - 0 ' # E ' A@ +&5'JC A0 Y!'*lX51<%< 21+ +&5' Y!':),+ 7C*'D$ @ +&5' +,) 7- 0E <%' )&'D3U- 0
C 0 Y!':l 51<4< Y!'*),+ 7C*'O$ p1
p2
p9
p3
p8
s
p4
p5
( 5 !( 5 & 50 (
Figure 43: 7
p7
p6
4 ?' B ?
?=5
(
0 $
&7E ?
(! K
?=58785 ( ( 5 ( 5 4 50 "<74 3 (5 '587 @=3=& %< 5
Theorem 3.44
S
(5 & 50( < ?=5 5 7 4 ?' B ? 1 7$
( K
(S)
' 0 +&'H+&5'hY!'*)&+ %C:'D$U-R< 0E +&5' C A0 Y!'*l 5V1<%<2 F p , . . . , p -R$
0 # 7C:-=+&' # 0 8 E 1)&' s L'D3 Y-=< @ p - 0 # p # 7$,C 00 'OCN+;$ 1 1 %$ (S)9 0 + + 6 C 3 P A0 ' 0 +;$ 0 ' A@6 5 7C;5 7$ {p , p 1} 5-/+ 47$ - 0F -R3 4<4+ 0 7- 0 C F C:<4'
0 1 %$ (S) 3k1$ +LY 7$ 4+ p - 0 # p 0 22^':+ 36 ':' 0 p - 0 # p " 0 -=< E 1$(< F! p
- 0 # p 3 1$ +92^'XY %$ 4+&' 2# 0 2d'*+ 36 'D' 0 p - 0 # p1 - 0 # p4 - 0 # p 3 1$ +92^8'
Y 7$ ?+;' # 9 0 2d'*+ 6 'D' 0 p - 0 # p Z1+q7+;5' 0 +;51'*)&' %$ 0 5 6 - F + 6 Y %$ 4+q+&5'
Y!':),+ 7C*'O$ A@ +&5' 0 +&':) 4)X$,' E 3g' 70 + s 5':)&' @K );' +&5':)&' 7$ 0b -=3 %<?+ 0 7- 0
C F C:<4' 0 1 %$ (S) " 0 +&5':) 0 +;'*)&'O$ + 0E P ) 2<4'D3 %$ +&5' E ' 0 '*);-R< %eD-=+ A0 A@f
5' )&'D3 s + P < FVE0 $ @ 6 ' C 0 $ # '*)< 0 ' $(' E 3g' 0 +;$-$ C 0 Y!':l P < FsE0 $ 0 + 6 Proof.
3.10. Remarks
Y!':),+ 7C*'O$ +&5' 0 '*lV+ 0 +;'*)&'O$ + 0E C:<%-$,$ A@ P < FVE0 $-/)&'L+,) %- 0E <4'O$ L'DCD-=<%<+&5-=+
- -R3 %<4+ 0 %- 0 P < FsEA0 @ )B-9$,'*+ A@ +,) 7- 0E <%'D$ 3U- FX0 + ' 0 +&':) +&5' 0 +&'*) ) A@
- 0_F +,) 7- 0E <%' B
5' 1'D$(+ 0 -=2 1+ +&5' '*l %$ +;' 0 C:' A@ -R3 4<4+ 0 7- 0 P < FsE0 $
@K ) # 7$3 0 +S+,) 7- 0E <%'D$ 7$S$ + %<4< P ' 0
7 $
5 ( 59A "37 4 ( 5D ( < 79 =&# 59(
Conjecture
3.45
4 B #3%< 5
7 ?=58785 5 (8( 6 $&@ #
K
5 ' $ +;-=+&':3g' 0 + %$ 0 + +,)&1' 0 E ' 0 ':);-=< @ ) 1- # ) 4<7-/+;'*);-R<%$ -$X+&5'U'*l r
-R3 P <4' 0 8 E 1)&' $(5 /6 $ 5'X'*l-=3 P <%' C 0 $ %$ +.$ A@ +&5)&'D'o<7-/) E 'X$ 1-=)&'D$
- 0 # s +;5)&':' 3g' # %13 $ 1-/);'D$ s s - 0 # s - 0 # $('*Y!' 0 $(3U-=<%<
s$ 1 1-=s)&3'D$
5
2$ 14-=)&'D$ -=)&' 6 );8
1
&
)
'
&
+
5
'
(
$
U
3
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<
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# 60 3 1C.5q<7-/) E '*)
>
0
E
A,
.
.
.
,
G
+;5- 0 +;5' F -/);' # 'D-R<4< F 0 ) # ':)+ 0 CN)&'O-R$(' +&5' ?)MY 7$ 42 %< 4+ F! 2$,'*),Y!'X+&5-=+
@K )M+&5'X<%-=) E ' $ 1-=)&'D$9+&5);':'XY!'*),+ 7C*'O$9-=)&' A0 +&5' C 0 Y!'*lh51<%< 6 5'*)&'O-R$ @ )
+;5'3g' # %13 $ 1-/)&'O$ - 0 # @ ) A B - 0 # C + 6 Y!'*)&+ %C:'D$ -=)&' 0 +&5'9C 0 Y!'*l
5V1<%<S2 1 0 # -=) F " + +&5'bC*' 0 +;'*) +&5' +&5)&':'b<7-/) E 'H$ 1-=)&'D$g-/);'bY!'*) F C:< $('
+ AE ':+&5'*)$(1C.5H+&5-/+ @ )M':ls-R3 P <4' s - 0 # F C:- 00 +$,':' +&5' C*' 0 +&':)9Y!':),+&'*l
6 C:&
+
5
U
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:
C
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1
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00 + $,':' - 0F +&5 0E '*lC*' P + @K ) +&5'
c+;51 )&':'XsC*'1 +&':)SY!'*),+ 7C*'O$
G
0
- 0 # c @ +;5'o<7-/) E ' $ 1-=)&'D$ c c
1
3
5
z1
C
s6
z5
F
G
s1
s5
c1 c5
c3
x1
A
D
s2
x3
E
s3
y3
Figure 44:
( 5 A
D ( 3 ( K 6 7859(
7
s4
y5
B
4 ?' B ?!?=5
3(
9% $>&7E
?> (
Da
Chapter 3. Hamiltonian Polygons
?=58785 ( ( 5 '? <7 5858!I374 ( 5 D ( < ( 7859( 3 Theorem
3.46
?=5 5 ( B ?? ; ?=5 (
9 $ & 78 ? ?=5837 4507 B 05 ( 4 ?7850("5 B 6 ?=5 ( 7859( ( K '*+
2d' - -R3 %<4+ 0 %- 0 C F C*<%' 0 +&5' Y 7$ 42 4< 4+ F E );- P 5 1 %$ (Q)
@K
A@ $ 1-=)&'D$$(5 =60 0 8 E 1)&' \pJ0 $ # ':) +;5'+&5)&'D'C A0 Y!'*l
Q
5V1<%< Y!':),+ 7C*'D$ @ +;5' <7-/) E ' $ 1-=)&' s 0 ' A@ +&5'O$(' Y!':),+ 7C*'O$ 5-R$ # ' E )&':'
+ 6 0 1 7$ (Q) - 0 # 5' 0 C*' ?+ 7$ - P1P 'D-/).$ 0 2d'*+ 6 ':' 0 +&5' +;5'*)q+ 6 A0
" 0 -=< AE 1$(< F! +&5' C 0 Y!':l 51<4< Y!'*),+ 7C*'O$ A@ s - 0 # s );'D$ P 'DC*+ 4Y!':< F -/)&'
H
Y 7$ ?+;' # # 4)&'DC*+&< F - @ +;'*) 'D-C.5 +;5'*) 43 %<%-=)&< F 3+;5'hY!'*),5+ %C:'D$ A@ +&5'H$(3U-=<%<
$ 1-=)&'D$ A B - 0 # C 3 1$(+ 2d'gY %$ ?+&' # 0 2d':+ 6 ':' 0 +;5'g+ 6 C A0 Y!'*l 5V1<%<
Y!':),+ 7C*'O$ @ s s - 0 # s )&'D$ P 'OCN+ 4Y!':< F
2
4
6
5'HY!':),+ 7C*'D$ 0 X := V(A) V(D) V(s ) -=)&' C 00 'OCN+&' # + +&5'
);':3U- 0 0E Y!':),+ 7C*'O$ 0 +&5)&'D' 6 - F $ 0 < F Y %- +&5' 2C A0 Y!'*l 51<4< Y!'*),+&':l x A@
Y %- +;5' C 0 Y!'*l 51<4< Y!':),+&':l x A@ s - 0 # Y 7- +&5' C:' 0 +&'*) Y!':),+&'*l 1c s12$('*)&Y!'S+&5-=+ 4+ 7$ 43
P $,$ 42<4'+ Y 37$ ?+ -R<43<Y!'*),+ %C:'D$ A@ X 0 - P -/+;5g2^':+ 6 ':' 10
- 0 # x 6 4+&5 1+ Y 7$ 4+ 0E c `
5V1$ c 7$ C 00 'DCN+;' # + -=+ <4'O-R$ + 0 '
x1
Y!':),+&':l A@ 3X 0 H 8s1)&+&5'*);3 )&' 1 x 3 1$ + 2d1'MC 00 'DCN+;' # + ' 4+&5':) -fY!':),+&'*l
@ ) 3 X ) c 0 H - 0 # @ x 7$SC 100 'OCN+&' # + - Y!':),+&'*l @ ) 3 X 0 H +&5' 0
3 1$ +2d'k1C 00 'DCN+;' # + 01 ' A@ c ) c 0 H " <4+ E ':+&5':) 0 H +&5'*)&'
x
-=)&3'q-/+f<%'D-R$(+M+ 6 ' # E 'D$92d'*+ 6 ':' 0 +;5'1 +&5)&'D'3 C:' 0 +&'*)Y!'*)&+ %C:'D$ c c - 0 # c
1
3
5
- 0 # +&5'oY!'*),+ 7C*'O$ @ ) 3 X {x , x } 1 3
" $ %3 4<7-/)-=) E 13g' 0 + @ )J+;5'f$('*+ Y := V(B) V(s ) V(E) $(5 =6 $ +&5-/+
0 H +;5'*)&'q-=)&'q-/+[<4'O-R$ +[+ 6 ' # E 'O$92d':+ 6 ':' 0 +&5'X+&5);':4'qC*' 0 +&':)MY!':),+ 7C*'O$ c 1+
;
+
5
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Y
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,
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+
7C
*
O
'
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3
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1
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$
<
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5
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'
=
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o
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#
#
"
0
0
@
0
A
E
F
c<%'D3-$ + + c5'
6 # E 'O$\2d'*+ 6 ':' 0 +&5'9+&5)&'D'9YC*' 0 {y+;'*3) ,Y!y':5),}+ 7C*'D$ c c - 0 # c - 0 # +&5'
Y!':),+ 7C*'O$ @ ) 3 V(C) V(s ) V(F) {z , z } Z\1+1+&5' 0 3 +&5':)&' 7$5 0g6 - F
+ 0 C*<%1 # 'q+;5' Y!'*),+ %C:'D$ A@ 6+&5' C*' 0 +;'*)o$ 11-/)&5' G 0 + H 0 C A0 +,);- # 7CN+ 0
+ H 2d' 0E - -=3 4<4+ 0 7- 0 C F C*<%' 5'*)&' @ )&' +&5'*);' %$ 0g -=3 %<?+ 0 7- 0 C F C:<4' 0 1 %$ (Q) Proof.
) +&5'f$(':+
H
Chapter 4
Alternating Paths
5 %$LC.5- P &+ '*) %$SC 0 C*':) 0 ' # 6 4+&5
$,' E 3g' 0 +;$ 0 +;5' P <7- 0 ' 465879:; 3=& ";?@(
+&5) 1 E 5 # 7$3 0 +< 0 '
(E D 507 ( 5 D 5 &@50 5878;5 D ( < 3 5( 5 &
Definition
4.1
S
n
50( 3 ?=5 5 (
3
5 I; ?
3 1 7$
P
=
(v
,
.
.
.
,
v
)
k
(S)
1
k
( B 4 5ED :D %$ B (
3(0( (5 & 50 5
D&@50(
alternating path
D (
0 $ 5ED&'59( 3 45079:; 3=& 7ED 507 ? ; (
7
v
v
S
2i−1
2i
5 '5879$
7
7 5'5879$
1 i
k/2
v2i v2i+1 S
1 i
(k − 1)/2
L +&'o+&5-/+ 0 P -=),+ 7C*1<7-/)f- 0F P /- +&5 A@ $ %e:'q-/+3 $ +S+ 6 0 1 %$ (S) %$M-=<4+&'*) r
0 -/+ 0E 8 E 1 )&' $(5 /6 $q- 0 '*l -=3 P <%' @ )q- 0 =- <4+&'*) 0 -/+ 0 E P -/+;5 +;5) 1 E 5
$ 3g' # %$3 0 S+ < 0 'X$,' E 3g' 0 +;$ Figure 45:
465879:; 3=& I; ? ?'78
&?!D ( 3 ( 5 & 58 (
D
Chapter 4. Alternating Paths
> 0 P -=),+ 7C*1<7-/) 6 ' -=)&' 0 +&'*);'D$ +&' # 0 +&5' @ <4< /6 0E 1'O$ + 0[j 5-=+ %$
+;5'o3U-/l 43k13 0 13q2d'*) A@ Y!'*),+ %C:'D$+&5-=+C:- 0 2^'oY 7$ 4+&' # 2 F - 0 -=<4+&'*) 0 -/+ 0E
P -/+&5 5' @ <%< /6 0E +&5' )&'D3 $ +;-=+&'D$+;5-/+L+&5':)&' 7$-R< 6 - F $S- 0 -=<4+&'*) 0 -/+ 0E
P -/+&5 A@ $ %e:' < E -/) 4+&53 7C 0 +&5' 0 13q2d'*) A@ $(' E 3g' 0 +;$ 7$ (59
D ( 3A 3 5 (5 & 50( < ?=5 5
Theorem
4.2
S
n
?=58705 3( 465879:;<=& ";? < 1 %$
? ; 3 (
(
< E (n + 2) − 3
(S)
2
2
'507 B 59( " P /- )&+ @ ) 3 - C 0 $ +;- 0 + @ -CN+ ) +&5 7$ %$L2d'O$ + P $,$ 42<4'
7! $
?=58785!5 (8( :DC (59
Theorem 4.3
n0 n > n0
S
D3( 3 3 5 ( 5+& 50( 3 ?=5A 5 ( B ? ? ; 1 %$
D 50( K
n
< (S)
E n − 17 <
B +3 45070:;<& ";? 3 (
3=& 87 5? 12
2
3
2
<
'507 B 59( E
7.57
2n
5 ' P ) _A@A@ 5' )&'D3 1$('O$ +&5'M':l 7$ +&' 0 C*' A@ - -R3 %<4+ 0 %- 0 P < Fr
E0 @ ) S +&5-=+ 6 -R$ P ) Y!' 0 0 p 5- P +&':) s Q );' Y!':) 4+ %$g-=< E ) ?+;53 %C
- 0 # CD- 0 5' 0 C*'o2^'X1$(' # + C 0 $ +,);1CN+9- 0 -=<4+&'*) 0 -/+ 0E P -/+;5 A@ $ 4eD' -=+<4'O-R$ +
< E
SS=6 '*Y!':) 0 +&' +;5-/++;5' C 0 $ +,)&1CN+&' # P -/+;5 7$ 0 + 0 'DC r
2'O$,$,-=) 24< (n +;+
2)
−
3
F 5' <%-=) E 'D$ +o-=<4+&':) 0 -/+ 0E P -/+;5 @ )[+&5' E 4Y!' 0 $('*+ A@ < 0 ' $(' E 3g' 0 +;$ 5' P ) 2<4'D3 A@ C 3 P 1+ 0E +&5' <%-=) E 'D$ +o-R<?+&':) 0 -=+ 0E P -/+;5 @ )f- E 4Y!' 0 $(':+
' n3 1C.5 3 )&' # UC*1<?+ A@ $(' E 3 ' 0 +.$ 3U- F 2d
5' P ) _A@Â@^
5' )&':3 $ %3 P < F $ P 'OC 'D$\- @ -R3 4< FkA@ $,'*+;$ A@ # %$3 0 +
$,' E 3g' 0 +;$ 6 5 $('gY 7$ 42 %< 4+ F E );- P 5 # 'D$ 0 +qC 0 +;- 0 - 0 -=<4+&':) 0 -/+ 0E P -/+;5
0 3 )&'[+&5- 0 7.57 < E n Y!'*),+ 7C*'O$ 2
4.1 Lower Bound
" $o-=<4)&'O- # F 3g' 0 + A0 ' # +&5'k2 1 0 # $ +;-=+&' # 0 5' )&'D3 );':< 4'O$ 0 +&5'
@ -RCN+X+&5-=+X'*Y!':) F $(' E 3g' 0 +X' 0 # P 0 +oY %$ ?2 %< ?+ F E );- P 5 C 0 +;- 0 $ - -=3 4<4+ r
0 %- 0 P < FsE0 +;5-/+ %$ ^
5' )&'D3 V j 'qC 0 $ # '*)f- P <7- 0 -=)[$(12 E );- P 5 A@
+;5' $(' E 3 ' 0 +o' 0 # P 0 +oY %$ ?2 %< ?+ F E ).- P 5 6 5 7C.5 7$ - 1 0 0 A@ - -=3 4<4+ r
0 %- 0 C F C*<%' - 0 # -kC 3 P <%'*+;'o3U-/+;C.5 0E C ),)&'O$ P A0 # 0E + +&5'X$(' E 3g' 0 +;$ A@
0 +&5'9$(' E 3g' 0 + ' 0 # P 0 +;$ " $ +&5 7$ E ).- P 5 7$ P <7- 0 -=) -=<%< P -=+&5$ 0 ?+J-=)&'
S
$ 43 P <%' L'DCD-=<%< +&5-=+ 5' )&':3 s $ +;-/+;'D$9+&5-/+ 1 7$ (S) C 0 +.- 0 $M- -=3 4<4+ 0 7- 0
0 @f0 +U-R<4<$(' E 3g' 0 +;$ 0 S -/);' C <%< 0 'D-=) p <4'O-/)&< F @ -=<%<M$,' Er
P < FVE
D
4.1. Lower Bound
3g' 0 +;$c-/);' C <4< 0 'D-=) +&5' 0 +&5'*)&' 7$ - 0 -=<4+&':) 0 -/+ 0E P -/+;5 +&5) 1 E 5 -R<4<V$(' E 3g' 0 +
' 0 # P 0 +;$ +&5':) 6 %$,' 6 ' CD- 0 )&':< F 0 +&5'U'*l 7$ +&' 0 C*' A@ - -=3 %<?+ A0 %- 0
P < FVE0 2$('*)&Y!' +&5-=+ 7$ 0 + 0 'OC*'O$,$,-/) 4< F -=<4+&'*) 0 -/+ 0E 2d'DCD-=1$(' 4+
3U- F C 0 +;- 0 $,'*Y!'*).-=< Y %$ ?2 %< ?+ F ' # E 'O$ 0 - ) /6 n$(':' 8 E 1)&' !a 2 M@ ) - 0
':ls-R3 P <4' . C:- 0 P $,$ ?2< F C 0 $ 7$ + A@ Y 7$ 42 %< 4+ F ' # E 'O$ A0 < F!
i
e
$%
(S)
( 5+& 05 50:D <
Figure 46: 6 "#%$&'(
Proposition 4.4
7 4507 65
v
26
;
?=50785
v
5ED&'5 # ' E
(v)
5 =<&':4 :D 2$(':),Y!'+&5-/+
'5879$ '587
5 ?=58 ( 5 &
785
3(A3 B D
v ?=50
=
3
v
4 3(
9% $
7$L-
.
(
9 $ & 7E ? :D 5 " $(' E 3 ' 0 + s S 6 5 7C.5 7$ 0 +
' P AE 0 -R< A@ >0 8 E 1);' !a >2 . @ )
# %- EA0 -R< \6 5 4<%'h$(' E 3g' 0 + e %$ - $(' E 3
+;5' -R3 4<4+ 0 7- 0 P < FsEA0 + E '*+;5'*)
' P EA0 -R<%$ - 0 #b# ' 0 +&' V := V( ) - 0 #
( 7$ 0 'OC*'D$&$,-/) %< F - # %- EA0 -R< )
'*l-=3 P <%' $(' E 3g' 0 + i 7$ - $,' E 3g' 0 +
' 0 + ' P EA0 -R< @ ' 0 +;' 2 F 6 4+&5 -=<%< ?+.$ $(' E 3g' 0 + # 7- E0 -=<7$ - 0 #
0 +&5' @K <%< /6 0E
E := E( )
K( D 5 &785E5 4 7 ?'705E5 # ' E
(v) =
0
?
50D&' 4 7 5 = &@:4: ( 3 B D 05 50 6 (
0 $!5
D&'5 :D ( 5 & 50
( < B D 58 ( 5 & 50 D &@:4 7
5ED&'59( 4 & ' 0 C:'o- 0 F P -/+;5 0
%$L-q$ %3 P %< ' P -/+&5 0
%$ (S) 3 - +;5-/+ A 0 ' CD- 0 21 4< # - 0 -R<?+;'*) 0 -=+ 0 E P -/+;5
j 'f$(5 / 6 0 +&5' 0 :' l+<4'D3 U
' + - 0F !Y '*)&+&'*l 0 @ ) 3 - 0F ,$ ' E 3g' 0 +L' # E f
1
75 '5879$ ( 5 & 58 :D 5'5879$ '587 5 Lemma
4.5
e
E
f V
0 e1 S
?=5 &7E ? 5079:; 3=& ";?
B + 3 ( 4
= (V,
(e0 , e1 , . . . , ek = f)
7 ( 5
E)
k Chapter 4. Alternating Paths
5 ')&'D3U- 0 # '*) A@ +&5 7$J$,'DCN+ 0 7$ # '*Y +&' # + +;5' P ) _A@ A@ ':3g3U- _
5' P ) @ A@B
5' );':3 +&5' 0h@ <%< /6 $2 F 'D<4'D3g' 0 +;-/) F -/) E 13g' 0 +;$ 8' 0 'H- # 7$ +;- 0 C*' @ 1 0 CN+ 0
0 +&5' Y!'*)&+&'*l $(':+ V -$ @ <4< /6 $ 8 )
d
- 0_F v V v = e <4':+ d(v) 2d' +&5' $ %e:' A@ +;5' $,3g-R<4<%'D$(+ 0 + 0 'OC*'D$&$,-/) %< F
-R<?+;'*) 0 -=+ 04E P -/+&5q0C 00 'DCN+ 0E v - 0 # f -R< 0E +;5-/+ # 'D$ 0 + P -R$,$c+;5) 1 E 5
1C;5 - P -=+&5 -R< 6 - F $ ':l 7$ +;$ $ 0 C*' 7$ - C F C*<%' @ e = f <4':+
e0
'D<%$,' d(e ) := ∞ b ':l+ 6 ' ) 4' 0 + -=<%<\Y 7$ 42 4< 4+ F ' 0# E 'D$ 0 d(e
)
:=
0
0
$,1C.5h
+&5-/+S+;5' F -=)&' # 0?)&'OCN+&' # + /6 -/) # $+;5'fY!'*)&+&'*l 6 4+&5 $,3g-R<4<%'*)9Y-=<%1' d( ) 6 '*l-=3 P <%'D$L-=)&' # ' P %C*+&' # 0 8 E 1)&' +&'+&5-=+ 6 ' # 0 +SC 0 $ # ':)
-R$L- # 4)&'OCN+&' # E );- P 5 +;5' ) 4' 0 +;-=+ A0 0 # 1C:' # 2 F d( ) %$3g'*);':< F - 0 - #
+ C 0 $(+,)&1C*+ P -/+&5$ 6
7
e0
3
5 e1
∞
4
3
5
2
4
∞ e0
1
5 e1
3
2
f 0
2
1
4
1
1
0
f
3
2
4 254 59( 4 ? 7950; ( D D (8+ B 50( ?' (
Figure 47: D 7E4 3 =& 4 5 58 ? K(
5 ?=5 6 (87 B 705 ?=58785 785 B 78(E(
3=&(32 ?=5 97 &3:4 . ? 7850("5 B 6 ?=5 70 50+; EB B 78D<=& 5
Proposition
4.6
d(
)
5079$ '587 5 D 5+& 87 585 ?'785E5 3 ( 3 B D 50 ; 5
K(8 5 &@<=&
D ; (8 5 <&? f
05 3 B D 50 4 3 B <& (
9 $ 5
D&'59(
j 4+&5 5 'D< P A@ ;+ 5 %$ ) %' 0 +;-/+ 0J6 'HCD- 0 +,) F + 21 %< # - 0 -=<4+&'*) 0 -/+ 0E
P -/+&5 P 0 $(+;-/),+ 0 E @ ) 3 (e0 , e1) - 0 #H# 4)&'DC*+&' # + /6 -=) # $ f R- $ @ <%< =6 $ Algorithm 4.7
- 0 '# E
'
(v0 , v1 )
@
- 0 # -k$(':+
X
V
@ Y!':),+ 7C*'D$ s
4.1. Lower Bound
4 ; &'79 ?'
P ← (v0 , v1 )
j 5 4<%' P 7$- P -=+&5H- 0 # 5-R$ 0 +L)&'O-RC.5' # - 0F Y!':),+&'*l @ ) 3 X # - (u, v) ← <7-R$(+9' # E ' A@ P 2 @ # ' E (v) = 2 +;5' 0 - PP ' 0 # +&5' +&5'*) = u B0 ' E 5_2 ) A@ v 0 + P
C <7$(' @ {u, v} 7$ -LY 7$ ?2 4< ?+ F ' # E ' +&5' 0 - PP ' 0 # +&5' 1 0 1'J$(' E 3 ' 0 +
' # E ' 0 C # ' 0 +S+ v 0 + P # <7$('- PP ' 0 # -oY 7$ ?2 4< ?+ F ' # E ' 1+ E 0E n-RCDC ) # 0E + d( ) @ ) 3
+ P
v
P
C;5'DC;m +&5-/+ " < E ) ?+;53 %$ 6 ':<%< # 'E 0 ' # )&' @ '*)+ i ) P $ ?+ 0 $ - 0 # asBj 5' 0 C:-=<%<%' # 6 4+&5 (e , e ) - 0 # {f} +&5' -R< E ) 4+&53 ' 4+&5'*)S+;'*)&3 r
0 -/+&'O$2 F )&'D-C;5 0E f d )M2 F +&5' 0 P -/1+&5 P )&'O-RC.5 0E - Y!'*)&+&'*l @ )9+;5'q$('DC A0 #
+ 43g' - 0 # +;51$X2d'DC 3 0E - 6 -=<4m 5'g-R< E ) 4+&53 +&'*)&3 0 -/+;'D$ 2d'DC:-R1$('
+;5' 0 13 2^':) A@ ' # E 'D$ 0 P $ +,) %C*+&< F 0 C*)&'D-$('D$ 0 '*Y!'*) F ?+&':);-/+ 0 @ +&5'
P -/+&5 # 'O$ 0 + )&'O-RC.5 f 6 'b-/)&' <%' @ + 6 ?+&5 - P -/+&5 C 00 'OCN+&' # + -`C F C*<%' 6 5 %C.5 < _ m$[< 4m!' - 2-=<%< 0 6 ?+;5 - C ) # -/+,+;-C.5' # + 4+ '*+o1$ # '*) 4Y!' 3 )&' @K );3g-R< - 0 # $(< E 5+&< F 3 )&' E ' 0 '*);-R< # 'D$,C*) P + A0 @ ) +&5 7$\+ F P ' A@
C 0 E 1);-=+ 0
( +&7E ?
1 %$
( B 4 5
D
70 4507 65
Definition 4.8
G
(S)
walkable
:D! $ 75 '5070$ '587 5 ?=50785 3(
v V(G)
u V(G) \ {v}
45070 ; 3=& I; ? 4 ?'3
78
6
4 ?=
( 5 5
D&@5 3 B D 50 (
G
v
u
u
( 5+& 50 5ED&'5 - $ 0E <%'9Y!':),+&'*l )J+ 6 +&'+;5-/+ 0 P =- ),+ 7C*1<7-/) - E );- P 5 C 0 $ %$ + 0EqA@ X
Y!':),+ 7C*'O$ C 00 'OCN+&' # 2 F -o$,' E 3g' 0 +\' # E '9=- < 6 - F $ @ )&3 - 6 -=<4m-=2<4'[$(12 E .) - P 5 ?=5 4 ( +&7E ?@( 1 %$
4 ?
Definition 4.9
B=G P
(S)
:D
( B 4 5
D :D %$ V(G)
V(P)
=
{v}
E(G)
E(P)
=
balloon
3( 4 4; 5 78
:D
( G
v
P
=
(v
=
v
,
v
,
.
.
.
,
v
=
u)
k
0
1
k
45070 ; 3=& ";? < 1 %$
( B ? ? ;
5 B 4 V$ );C
?=5
(S)
vv
S
(B)
:=
u
1
5),+ (B) := v ?=5 heart 2 # F (B) := G ?=5 body :DbC ) (B) := P
source
?=5
cord
B
8 E 1)&' $(5 /6 $9- 0 '*l-=3 P %< ' A@ -q2-=<%< A0c
Chapter 4. Alternating Paths
v = (B)
u = (B)
843F
Figure 48:
B
?=5 8 D$
(E? @D 5
D ( 4 4 # 5
78
v
(
D 507 ?=5 ";?
&'79 ?' 7
B 5ED 9$
Proposition
4.10
P
?=5 < :D
4 =
? 58705
?=58 85 ?=507 07 5
B ?=59( '587 5 X
(v
,
v
)
X
f
P
0
1
78
7
70 ( B $ B F 5 7
70 ( E4
F X
P
P
Proof.
4+&5H-
6
@ &+ 5'-R< E ) 4+&53 # 'O$ 0 +);'D-RC.5 - 0_F Y!':),+&':l @ ) 3
6 -=<4m
X
4+\+&'*);3 0 -=+&'D$
P = (v0 , v1 , . . . , vs , vs+1 , . . . , vk = vs ),
@K )o$ 3g' 0 s < k $,1C.5 +&5-/+ (v , . . . , v ) %$o- P -/+&5 X>@ s = 0 +;5' 0
0 43
k−1
&
)
U
3
$
q
C
*
C
%
<
'
;
+
5
*
'
)
7$
(
'
o
'
:
C
%
<
&
+
5
=
+ 2 +&5 v v - 0 # v v -/)&'
@
F
6
s
6
P
k−1 k
s s+1
Y 7$ ?2 4< ?+ F ' # E 'D$ >0 # 'D' # -g$(' E 3 ' 0 +[' # E ' 7$ 0 C:<41 # ' # 0 + +;5' P -/+&5 # 4)&'DC*+&< F - @ +&':) 0 '
A@ 4+;$ 0 C # ' 0 + Y!'*),+ %C:'D$ 5-R$q2d'D' 0 )&'O-RC.5' # ' 0 C*' v v C:- 00 + 2d' $,' E 3g' 0 + ' # E ' +&5':) 6 %$,' +&5' -=< EA ) 4+&53 6 1< # 5-DY!'J$('Dk−1
<4'OCN+&' k# v 0 $(+&'D- #
' # 0E ' # E '
A@ v -R$ - $(1C:C:'D$,$ ) A@ v @ v v S +&5' 0 +&5' P )&'DC:k−1
s+1
s
s s+1
%$X- Y 7$ ?2 4< ?+ F ' # E ' 6 ?+;5 d(v ) < d(v ) 2 F C A0 $ +,)&1C*+ A0c 8 )
v+;5s−1
v
s
' $&-=3g' )&'D-$ 0 +&5' ' # E ' v v %$s # 4)&'DC*+&' s−1
# + /6 -/) # $ v = v 5' 0
k−1
k
i ) P $ 4+ 0 a +&':<%<7$ 1$ +&5-/+ v = f - 0 # +&5' -R< E ) 4+&53 k 6 1s< # 5-DY!'
s
$(+ PP ' # +&5':)&' i ) P $ 4+ A0 %3 P < 4'O$k+;5-/+g'*Y!':) F $('DC A0 # ' # E ' 0 P 7$ - $(' E 3g' 0 +
' # E ' 0 P -/)&+ %C:1<7-/) @ v v 7$ -SY %$ 42 %< 4+ F ' # E ' +&5' 0 v v %$ -M$(' E 3g' 0 +
' # E ' 51$ '*Y!':) F Y!'*)&+&s'*l s+10 +&5' P -=+&5 (v , . . . , v s−1
CD- s0 2d'q)&'O-RC.5' #
)
$ 6 ?+&5 - $,k−1
' E 3g' 0 +[' # E ' ' 4+&5':)
@ ) 3 v 0 - 0 -=<4+&'*) 0 -/+ 0E P -/+&5`+&5-/+f' 0 # s+1
s
Y 7- v ) Y 7- v -=< 0E P " <?+ AE ':+&5'*) L6 ' 5-DY!' $(5 /60 +&5-=+ +&5'
C 0 $ +,s+1
)&1CN+&' # P -/+&5 Pk−1
@ )&3U$S- 2-=<%< A0H6 ?+&5H$ 1);C*' v - 0 # 5'O-/),+ v 0
Proposition 4.11
7 $
'587
5
v
3
?=5 8 D$
s
E4 B
?=58785
4.1. Lower Bound
(
4
5079:; 3 =&>";? 3
78 5),+
4 ?' B ?
B
(B)
v
(
9 $ 5
D&'5 :D 50:DK(54 ? ( 5 & 58 5
D&@5 (8+7 ( 4 ?
0 *C 'M+&5'$(' E 3g' 0 +J' # E 'O$ -/);' P - ?) 6 %$(' # 7$3 0 + +;5'*)&'C:- 0 0 < F 2d'
0 E 3 ' 0 + ' # E ' 0 C # ' 0 + + - 0F Y!'*),+&':l h
5' $(' E 3g' 0 +X' # E ' 0 C # ' 0 +
+ 5),+ (B) %$ P -=),+ A@ C ) (B) 2 F # '8 0 4+ A0 _5' 0 C:' +&5':)&'f-/)&' 0 < F Y 7$ 42 %< 4+ F
' # E 'D$ 0 C # ' 0 + + 5)&+ (B) 0 2 # F (B) 5'hC*<7- %3 @ <4< /6 $ @ ) 3 +&5' @ -RC*+
+;5-/+S2 # F (B) 7$ 6 -R<?m-=2<%' Proof.
'U$('
E(Bi )
50 K65
`
i=1
2 #
i 1
7 $
V(Bi )
i ` B ( 843F 3 1 %$ (S) :D $
i 1 i < `
j i < j `
7 $
( 5 58 B 5
3 ( B 4 5ED 0
=
(B
1 , B2 , . . . , B ` ) ` :D> $ A ( ; ( 59
( ?=5 #3F 4 <=& B :D (
Definition 4.12
3
balloon-path
i
$ );C (B ) V(Bj ) =


E(Bj ) =
.
F| (B| :=) ` V( ) :=
i
V(
j
2 #F
(Bi ))

`
i=1
V(Bi )
$();C (
) :=
$ );C (B
1)
j=i+1
K ?=5874 3(5
D2
#F
( ) :=
j ' 2 $('*)&Y!'k- @ ' 6 %3 3g' # %-=+&'kC A0 $(' 1' 0 *C 'D$ @ +;5 %$ # E' 0 ?+ 0 pJ0sr
$ # :' )M- 2R- <4< _0sr P -=+&5 = (B , B , . . . , B ) 0 1
2
`
i `
7 $
Proposition
4.13
i 2
3( (
0 $ 5 D&@5 Bi
?=55
D&@5 < B D 58 6
$ );C (B )
<
i
0 C:' $ ).C (B ) V(2 # F (B )) 2 F # 'E 0 4+ 0 +&5'*);' 7$X- 0 -=<4+&'*) r
i−1
0 0E P -/+&5 @ ) 3 $ );C i(B ) + $ );C (B
0 B +&5-/+ ' 0 # $ 6 ?+&5 - $(' E 3g' 0 +
i−1
i)
' # E ' 5'*)&' 7$k':ls-CN+&< F A0 ' $(' E 3 ' 0 + ' # E ' i−10 C # ' 0 +k+ ':Y!'*) F Y!':),+&'*l 0
1 %$
- 0 # E(B ) E(B ) = s
5V1$ - 0F ' # E ' 0 C # ' 0 ++ $ );C (B ) 0
(S)
i−1
i'
i
k
3
1
$
L
+
d
2
X
'
Y
%$
?2
%<
?+
'
#
F
E
B
Proof.
-/+
i
2
+&'b+&5-/+ i ) P $ ?+ 0 7 %3 P < 4'O$U+&5-=+
i `
|V(
C ) (B ))|
i
3
@ ) - 0 F
Chapter 4. Alternating Paths
Proposition 4.14
7 5'5879$ '587 5 B <( 465879:; 3=&!I; ?
u
V( )
70 $();C
6
(B1 )
u
2 # F ( ) B <( 465879:; 3=& I; ?
3 7 5 '5879$ '587 5 u
70 $();C
6
? ; 50:DK5
( 4 ? ( 5 & 58 5
D&@5 (B1 )
u
5 'X$ +.-/+&'D3 ' 0 + 7$ 2VY 1$ @ ) u V(C ) (B )) +&5'*) 6 7$(' +&5'*)&'
0 4+ 0 - 0 -R<?+;'*) 0 -=+ 0E P -/+&5 0 B @ ) 3 $();C 1(B ) + 5),+ (B ) +&5-/+
' 0 # $ 6 4+&5H- $(' E 3g' 0 +L' # E ' Z F i ) P $ ?+ 10 7 - 0F 1Y!'*)&+&'*l 0 2 1# F (B )
CD- 0 2d'f)&'O-RC.5' # @ ) 3 5),+ (B ) 6 4+&5 0 B 0 - 0 -R<?+;'*) 0 -=+ 0E P -/+&5b$ +;-=),+ 01E
6 4+&5 - Y 7$ 42 4< 4+ F ' # E ' - 0 # 1 ' 0 # 0E 6 4+1&5 - $(' E 3 ' 0 + ' # E ' 0 C*' 2 +&5
P -/+&5$ C:- 0 2d'HC A0 C:-/+;' 0 -=+&' # + - $ 0E <4'b-R<?+&':) 0 -=+ 0E P -=+&5 \6 'H-/)&' # 0 '
@K ) +&5'bCD-R$('b+;5-/+ u V(2 # F (B )) +&5':) 6 7$(' J6 'HCD- 0 1$('h+;5'H$,-=3g'
-=) E 13g' 0 + @ )q$ );C (B ) +&5-=+ < %'D$ 10 2 # F (B ) 2 F # '8 0 4+ 0 + C 0 $(+,)&1C*+
- 0 -R<?+&':) 0 -=+ 0E P -/+;5 2@ ) 3 u + $ ).C (B ) 6 5 %C1.5 ' 0 # $ 6 ?+;5 - $(' E 3g' 0 + ' # E ' 5' 0 (B , . . . , B ) @ )&3U$- E - 0 -f2-R2<4< _0sr P -=+&5 !6 5 7C;5 %$ $(+,) 7CN+&< F $(3U-R<4<%'*)
+;5- 0 2 2F i ) P `$ ?+ 0 7 sB
5' C*<7- %3 @ <4< /6 $2 F 0 # 1C*+ 0h0 | | Proof.
7$S2 F # 'E
7 $ 4507 5
Proposition
4.15
 1 6 ' 5-DY!' v V(2 # F (Bi )) @ v = 5),+ (B ) 4+ 7$ v V(2 # F i(B )) n0 +&5'
);':3U- 0 0E CD-R$(' # ' E i−1 (v) = 2 0 C:'XiC ) (B ) 7$S- 0 -R<?+&':) 0 -=i+ 0E P -=+&5 0 ' @ +&5'o' # E 'O$ 0 C # ' (B0 +Si )+ v 0 C ) (B ) 3 1$(+i2d' - $(' E 3g' 0 +S' # E ' i
Proof.
<4':+
@
j ' 5-DY!' 0/6 C <%<%'DCN+;' # -=<%<+ _ <7$c+ # 'D$,C*) ?2d'- 0 -R< E ) 4+&53 + C A0 $ +,)&1C*+
-`2-=<%< A0r P -/+&5 @ ) 3 (e , e ) 5'D- # ' # + =6 -/) # $ f +&5-=+ 6 %<%< P ) Y # ' +&5'
P ) _A@ A@ 'D3 3U- _ 0 1
Algorithm 4.16
- 0 '# E
'
4 ; (κ, µ)
←
←
(e0 , e1 )
A@
()
(e0 , e1 )
- 0 # -qY!':),+&'*l
f V
61%:% 11
#
?5
! (
4.1. Lower Bound
&'79 ?'
L' P 'D-=+S+&5' @K <%< /6 0E 1 0 + %< f V( ) 0 +;' P 'f2d':< /6X
- '*+ P := (v = κ, v = µ, . . . , v ) 2d'`+&5' 1+ P 1+
) 4+&53 - PP < %' # + -0#
1
2
2
(κ, µ)
V @ 9) $ 3g'
V(B )
(B1 , . . . , B` ) ←
C @
vk k
i
←
1
V( )
{f}
i
`
A@ " < Er
&+ 5' 0
B1 , . . . , Bi−1 , P
`
Bj .
j=i
# <7$(' ← (B , . . . , B , P) 1
`
' @ f V( ) +&5' 0 '*l ?+ -qY!':),+&'*l @ ) 3 2 # F (B ) +&5-/+3 0 43 %e:'D$ d( ) @ κ←
| |
E µ ← -kY!'*),+;'*l 6 ?+&5 {κ, µ} E( ) - 0 # d(µ) < d(κ) n C @i ) P r
#
$ 4+ 0 ;a L +&'+&5-/++&5'C.5 7C*' A@ κ 0 +;' P @ 7$ 0 + 0 'DC:'D$,$,-=) %< F 1 0 1'L2d'OC:-=1$('S+ 6 0 # 'O$ 3 E 5+ 5-DY!'+&5'$,-=3g' d( ) Y -R<41' 0 &+ 5-=+JC:-$(' 6 M' CD- 0 2)&'D-=m +;5'+ %'
-=),2 4+,);-=) 4< F!
Proposition
4.17
&@79 ?' ?=5 85 & < < =& $ 507E; 79( E4 3I; ? 4 ? ( 7 B 5
?=5 23
e0
5' $ +;-=+&':3g' 0 + 7$ +,) ?Y %-R< @ ) &+ 5' );$ + 4+&':);-/+ 0 $ 0 C*'L-/+ +&5-/+ P 0 +
< E ) ?+;53 %$\C:-R<4<%' # 6 4+&5 P -=);-=3g':+&'*);$
P $ 4+ A0 7Ds @ +&5' P -/+;5 P # 'O$ 0 +
F C:<4' )U-`2-R<4< _0 5' @ 1 0 C*+ 0 d( )
# 0 #
0
F ' # E ' %$ # ?);'DCN+;' # + /6 -/) # $ e 1 0 <%'D$,$
0
5
H
'
(
$
'
g
3
'
+
'
'
C
'
+
+
7$
=
4
<
&
)
D
'
#
#
#
E
0
E
0
0
F
0
@) 3
e2d0' = f 5' C*' 4+ 7$
e
e
e
P
0
1
0
E 0 0
0
0 + P $&$ ?2<4'U+ )&'*Y 7$ ?+ e -R< 0E -b$(' E 3g' 0 +q' # E ' ' ?+;5'*) L
5'*)&' @ )&' +&5' P -=+&5 P )&'*+&1) 0 ' # 2 F " < E0) ?+&53 0 +&5' $('DC A0 #
4+&':);-/+ 0 C:- 00 +S2d' -kC F C:<4' ?+3 1$ + @ )&3 - 2-=<%< 0
" $,$(13 ' = (B , . . . , B ) 7$M- 2-=<%< A0r P -/+&5 -/+2d' E 0À@ $ 3g' ?+;'*);- r
+ 0 " < EA ) 4+&53 1)&':+&1) 0 $\`- P -/+&5 P = (v = κ, v = µ, . . . , v ) 6 5 7C.5 1
2
k
Proof.
B>0 +&5'$('OC 0 # 4+&'*);-=+ A0c "
= () 0 # {f} " CDC ) # 0E + i )
(e
,
e
)
0
1
);'D-RC.5 f +;5' 0 ?+ @ )&3U$ ' 4+&5':)U- C
7$ '8 ' $,1C.5 +&5-/+
Y %$ ?2 %<?+
a
Chapter 4. Alternating Paths
-C:C ) # 0E + i ) P $ ?+ 0 DV ' ?+&5'*)[)&'D-C.5'D$- Y!'*),+;'*l @ ) 3 {f} V( ) ) @K );3g$ - 2-R<4< _0 + 7C*'9+&5-=+ @ P @ )&3U$ - C F C:<4' ?+ 0 'DC*'O$,$,-=) 4< F )&'O-RC.5'D$
-qY!'*),+&':l A@ V( ) $ 0 C:' κ = v V(2 # F ( )) k
' :+ 1$ ).$ + C 0 $ # ':) +&5'`C:-$('`+&5-/+ P )&'D-C.5'D$ - Y!'*)&+&'*l v V( ) j ' C:<%- 43 +&5-/+ v v %$U- Y 7$ ?2 4< ?+ F ' # E ' ` @ v = e );'DC:-R<4<M+&5-=+U2 F
+&5'U$(k' E 3 ' 0 + ' # E ' vw 0 C # ' 0 + + v <0%'D$ 0 -R$ 6 ':<%< i ) P $ 4+ 0 7 k−1
5V1$ " < E ) 4+&53 $ + P $ @ P )&'D-C;5'D$ w %3 %<7-/)&< F +;5'q$(' E 3g' 0 +9' # E '
0 C # ' 0 + + e 7$ P -=),+ A@ - 0 # +&5'g-=< E ) ?+;53 $ + P $ @ P )&'O-RC.5'D$
e0 e1
0
&
+
5
:
'
&
)
'
/
;
)
o
'
+ 6 $,12 C:-R$,'D$S+ C A0 $ # '*) J
/
6
e
1
2 # F (B )) 8 E 1)&' ! - .d@ ) $ 3g' 1 i ` 5'*)&' %$ 2 F
# 0 A 0c - 0 -Ri <?+;'*) 0 -=+ 0E P -/+&5 @ ) 3 $ ).C (B ) + v +;5-/+9' 0 # $ 6 4+&5 $(' E 3 ' 0 + ' # E ' ' 0 C*' - 0F Y!':),+&':l v , . . . , vi CD- k0 2d'S);'D-RC.5' # @ ) 3
$ ).C (B ) A0 - 0 -=<4+&'*) 0 -/+ 0E P -=+&5 ' 0 # 2 0E 6 ?+&5 k−1
- $(' E 3g' 0 +9' # E ' ' 4+&5'*)
i
Y %- v - 0 # P )qY 7-b+&5' 2-R<4< _0sr P -=+&5 (B , . . . , B ) + κ - 0 # +&5' 0
` 3U$k- C C:<4' +&5'
n$(':k' i ) P $ ?+ 0 . L +&' +;5-/+ i @
&
)
@
F
0
P
P
- 0 # +;5' -=) E 13g' 0 + $ + %<4< E 'O$ +&5) 1 E 5 >0 @ -RC*+ \0 ' C 1< #
v$(5 k = +&κ5-/+
/6
0 '*Y!':) )&'D-C;5'D$ κ - E - 0 0 +&5'MC 1).$(' A@ " < E ) ?+;53 :as
P
8s1),+&5':)&3 );' +&5' $,-R3 ' -=) E 13g' 0 +hCD- 0 2d' - PP < %' # + +&5' Y!':),+ 7C*'O$
0 ` V(C ) (B )) U
51$ P ` B %$ - 2-=<%< 0 6 4+&5 $ 1);C*'
j=i j
$ ).C (Bj=i+1
-) 0 # 5'D-=),+5j ),+ (B ) i
i
v V(C ) (B )) 8 E 1)&' ! 2 @ )S$ 3g' 1 i ` '*+
k
i
C ) (B ) = (u = $ );C (B ), . . . , u = v , . . . , u ) ,
i
1
i
s
k
r
@ )X$ 3g' 1 < s r L +&' +&5-/+ @K ) s = 1 6 ' 5-DY!' u = $ );C (B ) 2
:
C
&
+
V
5
1
$
/
)
1
'
R
$
(
$
'
sL'OC:-R<4<[+&5-/i+
#
#
F
0
0
E
0
p
V('D$ (B+Ui−1
# 0 )&':Y ))7$ 4+ e 0 C:' -=<%< P -/+;5$ 0 -/)&'`C 0 $ +,);1CN+&' # 1$ 0PE
" < E ) 4+&53 -=<%<c0Y 7$ 42 %< 4+ F ' # E 'O$ 0 +&5' C ) # $ @ +&5'f2-R<4< _0 $ 0
-=)&' ) 4' 0 +&' # @ ) 3 +&5'[$ 1);C*'M+ +&5'[5'O-/),+ A@ +&5'2-R<4< _0\ ' 0 C:' 6 '
CD- 0 -=) E 1' -R$ 0 i ) P $ 4+ A0 D +&5-/+ u u S Z F +&5' $,-=3g'
)&'O-R$ 0 0E -R$ -=2 Y!' P ` B 7$ -[2-R<4< _s−1
0g6 4+&s5 $ 1);C:'$ );C (B ) - 0 #
i
j=i j
5'O-/),+ u = v vk'8 4+V(
s
k
+o)&':3U- 0 $f+ C 0 $ # '*)X+&5' 'D<%$,' r 2);- 0 C.5 +&5-=+ %$ +&5' CD-R$(' +&5-/+f+&5' P -/+;5
+&5-=+ 7$hC 0 $ +&)&1CN+;' # );'DC*1);$ 4Y!':< F 2 F " < E ) 4+&53 5 ?+.$ 4+;$('D< @ 2d' @ )&'
P
);'D-RC.5 0E - 0F Y!'*),+&':l @ ) 3 V( ) 8 E 1)&' ! nC 5' 0 2 FXi ) P $ ?+ 0 D
' 4+&5'*)X)&'D-C.5'D$ ) 4+ @ )&3U$o- 2-=<%< _0 6 4+&5 $();C (P) = κ U @ P );'D-RC.5'O$
P +&5' -R< ) 4+&53 f+&':)&3
E
0 -/+&'O$ +&5'*) 6 %$(' (B , . . . , B , P) @K );3g$ - E - 0 f2-=<%<
1
`
A0r P -/+&5h2d'DCD-=1$,' κ 2 # F (B ) `
4.1. Lower Bound
B2
κ
µ
e0
B1
vk
P
vk−1
P
$ #&'&%(
vk = u s
u1
κ
µ
B2
e0
B1
vk−1
P
$% )
P
#&%&'
B2
κ
µ
e0
B1
P
*) Figure 49:
P
$% $%& 3 (0 7E; (
7
&@70 ?'
Chapter 4. Alternating Paths
" < EA ) 4+&53 7Da %3g3 ' # 7-/+&'D< F P ) Y # 'D$+&5' P ) _A@ @ ) ':3g3U- - 0 # f S @6 '
j 'q- PP < F " < E ) 4+&53 7Da + Proof.
of
Lemma
4.5
(e
,
e
)
0
1
CD- 0 $,5 /6 +&5-=+S+&5' -R< E ) 4+&53 -=< 6 - F $L+&'*);3 0 -=+&'D$ +&5'XC*<7- %3 @K <%< /6 $ @ ) 3
i ) P $ 4+ 0 $ - 0 # 7 \ +,) %C*+&< F $ P 'D-=m 0E - @ +&':) +&':)&3 0 -=+ A0 =
%$ 0 + -R< 6 - F $ -f2-=<%< 0sr P -/+&5 - 0F 3 )&'21+ (B , . . . , B )
(B
,
.
.
.
,
B
)
1
`
7$ - 2-=<%< 0sr P -/+&5 - 0 # B 7$ - 0 -R<?+&':) 0 -=+ 0E P -/+&5 6 5 $,' 10 'H' 0 # P `−1
0+
`
< %'D$ 0 2 # F (B ) - 0 # %$ 0 C # ' 0 +9+ - Y %$ ?2 %< ?+ F ' # E ' 0 B ' 0 C:' +&5'
`
-=) E 13g' 0 + @ ) `−1
3 i ) P $ ?+ 0 7 EA 'D$+&5) 1 E 5 $(5 /6 +;'*)&3 0 -/+ 00 +&' );$(+ +&5-=+ 0 ' # E ' 7$'*Y!'*) # 7$,C:-=) # ' # @ ) 3 +;5-/+ 7$ |E( )| 0 C*)&'D-$('D$3 0 + 0 ':< FX Y!'*) +&5' ':lV'OC*1+ 0kA@ +&5'-R< E ) 4+&53 Q );' Y!':) +&5 7$ 0 CN)&'O-R$(' %$f$ +&) %C*+[2^'OC:-R1$(' 0 '*Y!':) F 4+&'*);-=+ A0 -=+[<%'D-$ +f+&5'
' # E ' {κ, µ} 7$- ## ' # + E( ) + 3 E _5 +X2^' 6 )&+&5 6 5 4<%' + H0 +&' +&5-=+ 6 ' # # 0 +X1$(' - 0F6 5'*)&' +&5'
@ -RCN+X+&5-=+ 7$ P <%- 0 -/) 5' P ) @A@
5' );':3 7$ C 3 P <4':+&' # 2 F +&5'
@K <%< /6 0E 'D<4'D3 ' 0 .+ -/) F -=) E 13g' 0 + pJA0 $ # '*) - 0 -/),2 4+,);-=) F # 4)&'DC*+&' # $(' E 3g' 0 +
Proof. of Theorem 4.2
e 0 e1 S
8 ) i # ' 0 +&'f2 F V +&5' $(':+ A@ Y!'*)&+ %C:'D$ 0 V +&5-/+9C:- 0 2d'o)&'O-RC.5' # 0
- 0 -R<?+&':) 0 -=+ 0E P -/+&5 0 i '*l-RC*+&< F i + 1 Y!'*)&+ %C:'D$ $ +;-=),+ 0E 6 ?+;5 (e , e ) 8 )f':ls-R3 P <4' V = {e } " C:C ) # 0E + i ) P $ 4+ 0 ^6 ' 5-DY!' 0|V |1
1
1
iD< $
)
*
'
!
Y
'
)
:
'
,
+
+
4'
#
#
#
#
@
0
0
@
0
E
F
2|V
|
i
|V
|
|V
|
i
n
:=
|V
|
i−1
i
i−1
i
i
+;51$
i ':Y!' 0
2
ni i ## 2
i−1
2
i
2
' 0 C:' Pk |V | 2 − 3 @ ) k ## - 0 # Pk |V | 3 2 − 3 @K )
i
i
i=1
i=1
:
'
!
Y
'
0
k
Z F 'D3g3g- +&5':)&'k':l 7$ +;$X- 0 ` 1 ` n $(1C.5 +&5-/+ V \ {e } =
0
q
3
2
&
+
5
7$
?+
;
5
&
+
5
'
'
1
=
<
?+
)
3
=
2
!
Y
'
4'
D
<
$
#
J
p
0
0
E
6
0
F
@
F
`
n−1 =
i=1 Vi
P`
&
+
5
=
+
7$
<
< E 3 E
|V \ {e0 }| |V
|
3
2
−
3
`
2
(n
+
2)
−
2
i
5' C*<7- %3 @ <%< i=1
/6 $ $ 0 C*' +&5' E ).- P 5 C 0 +;- 0 $ - 0 -=<4+&'*2) 0 -/+ 0E P -=+&5 0 2 -=+
<%'D-$ + ` + 1 Y!'*),+ %C:'D$ k+3
2
k
2
`
2
5'H-/2 Y!' P ) _A@ $(1 EE 'D$ +;$U- $ 43 P <%' -R< E ) 4+&53 + C 0 $(+,)&1C*+U- < 0E
-R<?+;'*) 0 -=+ 0E P -/+&5 7 $ ( 5 D ( < 3 5 ( 5 & 58 ( 32 ?=5 5
Corollary
4.18
S
n
D $ 6 "#%$&'
7
4
5070:;<&";? 3 1 7$
H
S
(S)
R
4.2. Upper bound
? ;
O(n)
3(
( ; F5
K(8
2
3 5 < E
2 (n
+ 2) − 3
4507
B 50( B 05
B 5
D
3
pJ0 $ +,)&1CN+k+;5' E );- P 5 +;5-/+UC 0 $ 7$ +;$ A@ H + E ':+&5':) 6 4+&5 -=<%<
E 0 + # 7- E0 -=<7$h- 0 # ' P E0 -=<7$ +;-=),+ - 0 -R<?+;'*) 0 -=+ 0E 2)&'O- # +;5 ).$ +
$,'D-/).C;5 @ ) 3 - 0 -=),2 4+,);-=) F $(' E 3g' 0 +M' # E ' (e , e ) >@ +&5' # %$ +.- 0 C:' @ +&5'
C:1),)&' 0 + Y!':),+&':l 0 +&5'2 @ $ r +,)&'D'9+ e 7$ ## 0C 0 1$ # ':) $(' E 3g' 0 + ' # E 'O$ 0 < F
+&5'*) 6 7$(' C 0 $ # ':) Y 7$ ?2 4< ?+ F ' # E 'D0$ A0 < F! " $ -/) E 1' # -=2 Y!' +&5' <7-R$ +
Y!':),+&':l Y %$ 4+&' # # 1) 0E +&5 %$X$('D-=);C.5 7$f)&'O-RC.5' # 0 - 0 -R<?+;'*) 0 -=+ 0E P -/+;5 0
-=+ <4'O-R$ + 2 < E (n + 2) − 3 Y!'*),+ 7C*'O$ 0 +&5'L2 @ $ r +&)&':' " $ - P <7- 0 -=) E );- P 5 C 0 +;- 0 $S- < 0 2'O-/) 0 13 2^':) @ ' # E 'D$ 0h@ -CN+ +&5':)&'X-=)&'X-=+3 $ + 3n ' # E 'D$ - 0 # +&5V1$+&5'o$,'D-/).C;5 C:- 0 2^' # 0 ' 0 < 0 'D-=)+ %3g' Proof.
$,' 3g'
4.2 Upper bound
pJ 3 P <%':3g' 0 + 0E +;5' );'D$(1<4+;$ @ ) 3 +;5' P )&'*Y 1$ $('DC*+ 0S6 ' $(5 /6 5'*)&'
- 0 -R$ F 3 P + + 7C:-R<4< F 3U-=+;C.5 0E 1 PP ':) 2 1 0 # +&5-/+ 7$ 6 '`C A0 $ +,)&1C*+ $('*+.$
A@ # 7$3 0 +M< 0 'q$(' E 3g' 0 +;$L+;5-/+ # U0 +95-DY!' <%-=) E ' -R<?+;'*) 0 -=+ 0E
,
k
k
P -/+&5$ j 'b)&':3U-=),m +&5-=+g- 0 O(< E n) 2 1 0 # 6 -R$U-R<?)&'O- # F m 0=60 2 F C 0 $ +,)&1CN+ 0 # 1' + WL)&)&1+ 7- ! n 21+ 6 ?+&5 - <%-=) E '*)fC 0 $ +;- 0 +[C ' UC %' 0 +
+;5- 0b 1);$ j ' C 0 $ +,);1CN+ +&5' $(':+;$ A@ $(' E 3 ' 0 +.$
k Proof.
of
Theorem
4.3
k
);'DC*1);$ 4Y!':< F -$ @ <%< =6 $ " <%< < 0 ' $(' E 3 ' 0 +.$ -/)&'hC.5 ) # $ A@ -`C 4);C:<4' c C 0 $ 7$ +;$ A@ +&5);':' $,' E 3g' 0 +;$q-/),).- 0E ' # 0 -H+,) 7- 0E 1<7-/) @ -R$(5 0 ' 4 $,1C.5 1
+;5-/+ 1 7$ ( ∼ K 5'k' 0 # P 0 +;$ A@ +&5' C;5 ) # $ P -/)&+ ?+ 0 c 0 + -/).C:$ 7$ 2+;- ) 0 1' # =@ ) 36
2 F 0 $('*),+ 0E -M$(' 1' 0 C*' A@ +&5)&':'$(' E 3g' 0 +;$ ' 4
k
k−1
-qC P FUA@ \0 ':Y!'*) F -=);C A@ c +&5-/+ %$ 2 1 0 # ' # 2 F 0 < F 0 'f$(' E 3 ' 0 + A@
8 E 11)&' $(5 /6 $ - 0 # k−1
1
2
5'+ 6 ' 0 # P 0 +;$ A@ - 0_F $(' E 3 ' 0 + 0 +&5 7$C 0 $ +,);1CN+ 0 -=)&'f- # (-RC:' 0 +
+ +&5' $&-=3g' $(':+ @ $(' E 3 ' 0 + ' 0 # P 0 +;$ 0 +&5'gY 7$ 42 4< 4+ F E ).- P 5 1 7$ ( ) ' 0 C:' !6 '9CD- 0 0 +&'*) P )&'*+ 1 7$ ( ) -R$ C 3 P <4':+&'S+&':) 0 -/) F +,)&'D' A@ # ' P +&5 k −k 1
6 5'*)&' 'O-RC.5 Y!'*),+;'*l 7$ @ )&3g' # 2 F k-9C*< 1' A@ +&5)&'D' $(' E 3g' 0 +;$ 8 E 1)&' nC $ 43'*+ P λ<%k' P2d-/'[+&+&5 5'o0 $ %e- :' +,)&'D@ ' - @<7-/# ) E ' P'O$ +&+95 -=<4+&'*) 0 -/5+ -0$ME $ P4eD-/' +&5 0 k - 0 # 0 C*$ 'o0 +&C:5' 'fY <77$-/) 4+E 'D0$(E +
- =r C*< 1' A@ $(' E 3 ' 0 +.$J3g'D- 0 $JY 7$k4+−01E a Y!'*),+ %C:'D2$ k6 −'[1C 0 C*<%1 # '+&5-=+ λ =
k
Chapter 4. Alternating Paths
s4
s12
s5
s1
s1
s6
s3
s3
s2
s2
1
s2
s1
s4
s5
s3
...
s6
2
s12
*) ?=5 B (8 7 B Figure 50:
12 k − 6
k
C A0 +;- 0 $L':ls-CN+&< F
λk
< E
= 12
=
< E
< AE
12
2
3
3
nk
2
nk := 3k+1 − 3
< E
−1+
nk
3
k
Y!'*),+ %C:'D$ ' 0 C:' 3k
−6
3k − 1 < E
− 18 + 12
3
3k
3k − 1
- 0 # +&5'C:<%- 43g' # )&'D$,1<?+ @ <4< /6 $ 2d'OC:-=1$('+&5'9<7-R$(+J+&':)&3 %$J<4'O$,$ +&5- 0 0 ' @ )
k 3
+&'S+&5-/+ +&5'C.5 %C:'S+ C 0 $ +,);1CN+ 1$ 0E E ) 1 P $ A@ +&5);':'M$,' E 3g' 0 +;$
7$ 0 +J-=),2 4+,);-/) =F 1 PP $(' 7$ C 0 $(+,)&1kC*+&' # 1$ 0E E ) 1 P $ @ x $(' E 3g' 0 +;$ @K )$ 3 ' P $ ?+ 4Y!' )&'D-R< 0 13q2dk':) x 5' 0 +;5' CD-=<7C*1<7-/+ 0k@ ) 3 -=2 Y!' F 4'D< # $
λk
= 4x
< E
x nk +
< E
x
x − 1 < E
+
2x
x
xk
− 2x .
xk − 1 3 0 %3 4eD' λ @ ) ($ 1 UC %' 0 +&< F <%-=) E ' k =6 \' 5-DY!'+ 3 0 %3 %e:' x/ < 0 x 5 %$
+;'*)&3 %$S3 0 %3 k%e:' # @ ) x = e 2.718 - 0 # ;+ 5)&':' %$L+;5' C*< $('D$(+ 0 +&' E ':) Chapter 5
Chordless Paths
5 %$ C.5- P ;+ '*) # %$&C*1$,$,'D$ +&5' P =- );-=3g':+,) 7CUC 3 P 4< ':l 4+ FÀ@ 0 # 0E C.5 ) # <%'D$&$
P -/+&5$ 0bE ;) - P 5$ L'OC:-=<%<+&5' # 8' 0 4+ A0bA@ +&5' P ) 2<%':3 G /IHKJ 450 :D3785 B 5
D & 78 ?
"(E '5
G /IHKJ
Problem
5.1
(
)
G
=
(V,
E)
365 &'507
:D ?'785E5 D (8 3 B 4507 B 50(
( ?=50785 B ?=87 D= 59(E(
k
s,
t,
v
V
3I; ?> (
K5; (8
3 (s, v, t)
k
G
8 ?).$ + 6 ' $(5 /6 0 'DCN+ 0 7 +&5-=+ G /IHKJ 7$ -H3g'D3q2d'*) A@ +&5' C 3 r
P <4':l 4+ F C:<%-$,$ W[1] 1$ 0E -k);' # 1C*+ A0 + 5 ),+ 0 # '*+&':)&3 0 7$ + 7C 1) 0E
Q -RC.5 0 ' pJ 3 P 1+;-=+ 0 5' C 3 P <4'D3 ' 0 + 0E W[1]r 5-/) # 0 'D$&$ )&'O$(1<4+ );' # 1C*+ A0 @ ) 3 >0 # ' P ' 0 # ' 0 + '*+ %$f+&5' 0 $(12 'DCN+ A@L 'OCN+ 0 _ pJ 3 r
2 0 0E 2 +;5 );' # 1C*+ A0 $ F %':< # $L+&5'o3U- 0 +&5' )&'D3 A@ +&5 7$SC.5- P +&'*) Theorem 5.2
G /IHKJ
3(
W[1]
B F5965 4 ? 7050(+ 5 B k
'OCN+ 0 - 0 # O' CN+ 0 _ # %$,C:1$,$ ':l+;' 0 $ 0 $ A @ +&5'D$,')&'O$(1<4+;$+ ,$ '*Y!'*).-=<
);':<7-/+&' # P ) 2%< ':3U$ &) ' E -/) # 0 E +&5'f'*l %$ +;' 0 C:' A@ C;5 ) # <4'O$,$ P -=+&5$S- 0 # C F C*<%'D$ 5.1 Membership in W[1]
n0 +;5 %$f$,'DCN+ 0`6 'k- 0 -=< F e:'q+&5' P -/).-=3g'*+&':) %e:' # C 3 P <4':l 4+ FHA@ G /IHKJ - 0 #
P ) Y!'+&5' P ) 2<%':3 + 2d' 0 W[1] 6 4+&5 )&'D$ P 'OCN+S+ 4+;$ 0 -=+&1);-R< P -/).-=3g'*+&':) +;5' P -/+;5 $h$ %e:' 8 ?);$(+ 0 +&' +&5-=+b- C.5 ) # <4'O$,$ (s, t)r P -/+;5 P %$b-R<?);'D- # F
!
Chapter 5. Chordless Paths
# '*+&':)&3 0 ' # 2 F ?+.$M$(':+ A@ Y!'*),+ %C:'D$ 8 )['*l-=3 P
CD- 0 2d' 0 V(P) 2^'OC:-R1$('k- 0_F - PP 'D-/).- 0 C:' @
s 1<
6 # 0 +,) # 1 C*' - C;5 ) # %3 4<7-/);< F! '*l-RC*+&< F
+;5- 0 s C:- 0 2^' 0 V(P) n0 +;5 %$ 3U- 00 '*) P CD- 0
@ ) 3 V(P) <%' ^0 < FbA0 '
- 0 +;5'*) 0 '
0 ' 0 ' E 52
2d'1 0 1'D< F
0 ' E 52 ) x A@
E 5_2 ) 0 V(P)
) A@ x +&5':)
)&'OC 0 $ +&)&1CN+;' #
( +&7E ?
3( B ?=7ED=F50(8(
3";?C :D
Proposition
5.3
P
G
(s,
t)
$
( B 5 B 65
D
:D
? '5 D 5 &87 5E5 5 <
:D 4
P
s
t
G[V(P)]
'507 B 59( K ?=587 ? :D ? 45 D 5 & 705E5 4 <
s
t
G[V(P)]
j ' # g0 +m 0/6 5 =6 + # ?);'DCN+;< F )&' # 1C*' G /IHKJ + j ' E 5+&' # qr G r
-=+ 7$ -/2 4< ?+ F! >0 $ +&'O- # 6 'h);' # 1C:'h+ - # T ':)&' 0 + P ) 2<%':3 CD-=<%<4' # 5 ),+
L0 # '*+;'*)&3 0 %$(+ %C 1) 0E Q -RC.5 0 ' pJ 3 P 1+;-=+ A0 ) @ ) $(5 ),+ +;5-/+ 7$ # E' 0 ' # 2d':< /6X 7$[m 0/60 + 2d' W[1]r C 3 P <4':+&' _ >
- 0 # )&' # 1CN+ 0 + - 0 # 4+;$)&'D<%-=+ ?Y!'O$\5-R$ P ) Y!' 0 + 2d'-f1$(' @ 1<+ _ <
+ 'D$(+;-/2< 7$(5 3g':3 2^':);$(5 P )&'D$,1<?+.$ 6 ?+;5 0 +;5' W r 5 4':);-/);C.5 F :V n
Problem 5.4 (Short Nondeterministic Turing Machine Computation)
450 (E<=&# 5 ++ 5 4 4 $C< 65 (
3=&# 5 ?=5
@D :D 5 50703
(8 B 793=&2 B ?'< 5
4 7ED
?=5 4 ? 859 :D M
x
M
(
45 365 &'507
( ?=50785 B ; < ? ;
k
M
x
705
B ?=59( :4 BEB 5 3=& (8+;65 3 ; (8
(85 (
k
Theorem 5.5
G /IHKJ
3( <
W[1]
4 ?
7850("5 B k
pJ0 $ # ':)X- 0 0 $(+;- 0 C:' (G, s, v, t, k) A@ G /IHKJ 6 5'*)&' G = (V, E)
0 0 # 4)&'DC*+&' # E ).- P 5 s, v, t V - 0 # k 7$M- P $ ?+ 4Y!' 0 +&' E ':) j ' 6 %<%<
C 0 $ +,)&1CN+- 0 0 $ +;- 0 C*' (M, k ) A@ H$(1C.5h+&5-/+L+&5':)&' 7$S-kC 3 P 1+.- r
+ 0h@ ) M +&5-/+S)&'O-RC.5'D$M- 0 -=< -RC:C:' P + 0E $ +.-/+&' 0 -=+93 $(+ k = k2 + 4k
$(+&' P $ @ - 0 # A0 < F @ +&5':)&'o'*l 7$ +;$S- C.5 ) # <%'D$&$ (s, v, t)r P -/+;5 A@ $ 4eD'X-=+3 $ +
0 G + %$ %3 P ),+.- 0 ++;5-/+ k # ' P ' 0 # $ A0 k 0 < F - 0 # 0 + 0 n "
k
$&C;5':3U-/+ 7C[Y 4' 6 A@ +;5' C 0 $(+,)&1C*+ 0 7$S$(5 /60 0 8 E 1)&' s
6 5'*);' +&5'X-=< P 5-=2d'*+ Σ %$ # E' 0 ' # -R$ Σ :=
':+ M = (Σ, Q,V∆,+&5g' 1$(,+;{A})
-/+&'X$('*+ 7$
{}
{σ | u V}
Proof.
7$9- 1
u
Q := {A, R}
{gi |1
i
k}
{a, b, c, d, l, r} {pu |u V}
{qu |u V} ;
+;5'+,);- 0 $ ?+ 0 )&':<7-/+ 0 ∆ : Q Σ Q Σ {+, −, 0} 7$ # 'E 0 ' # 2^'D< =6 +&5'
0 4+ 7-=< $ +;-/+;' 7$ g +&5' 0 -R<c-C:C*' P + 0E $ +;-=+&' 7$ A - 0 # &+ 5' 0 -R< );'+'OCN+ 0E
$(+;-/+&' %$ R j 5' 10 +;5' 1 ) 0E 3g-C.5 0 'k$ +;-=),+;$ -=<%<B+;- P 'k*C 'D<4<7$oC 0 +;- 0 +&5'
5.1. Membership in W[1]
2<%- 0 m $ F 3q2 <: .
5'fC 3 P 1+;-=+ A0 C 0 $ %$ +.$ A@ +&5)&'D' P 5-R$,'D$ ).$ + +&5'
-=+3 $ + k Y!'*),+ %C:'D$ @ - C.5 ) # <4'O$,$ (s, v, t)r P -/+&5 P 0 G -=)&' E 1'O$,$(' # 2 F
6 ) ?+ 0E +&5' $(' 1' 0 C*' A@ C ),)&'O$ P 0 # 0E $ F 3 2 <7$ σ u V(P) 0 + +&5'
+.- P ' 5' 0 '*lV+ + 6 P 5-$('D$-/)&'[C 3 P <%'*+&'D< F # ':+&'*)&3 u0 7$ + 7CM- 0 # C.5'DC;m +&5-=+
Y 7$ 4+;$ s v - 0 # t 0 +&5' ) # ':) E ?Y!' 0 - 0 #
P
%$S- C;5 ) # <4'O$,$ P -=+&5 0 G P
n 0 +&5' @ <4< /6 0E P -/).- E );- P 5$ 6 ' # 'O$,CN) ?2d'+&5'9+&5)&'D' P 5-$('D$ A@ +&5'C 3 P 1 r
+.-/+ 0 8 )S'O-RC.5 $(+&' P s6 ' );$ + E ?Y!'f-XY!'*)&2-=< # 'D$,C*) P + 0 - 0 # +&5' 0 # 'E 0 '
+;5' C ),)&'O$ P A0 # 0ES@ )&3U-=<+,).- 0 $ 4+ 0XA@ +&5' 1) 0E 3U-RC.5 0 ' $(12$(' 1' 0 +&< F
5 ' 1) 0E 3U-RC.5 0 'U3U- F 6 ) 4+&'U1 P + k -/),2 ?+,).-/) F Y!':),+&':l
F 0 +&5'k+;- P ' (g , , g , σ , +) ∆ @ )X-=<%< u V - 0 # -=<%<
i
i+1
u ) -R<4<
- 0 # -=<%< 1 i k #
0
@
1 5'\i+,);<- k$ 4+
(g
,
,
a,
σ
,
0)
u
∆
V
i
u
0 0 $ P 'DC 'D$ 0 ) # '*) C*1),)&' 0 + $ +;-=+&' $ F 3q2 <!1 0 # ':) +&5' 5'O- # 0 ' 6 $ +.-/+&' - @ +;'*)9+&);- 0 $ ?+ 0 $ F 3 2 <+ 6 ) 4+&' + +;5' +.- P ' - 0 # 3 Y!':3g' 0 +
A@ +&5'L5'D- # + @K )) E 5_+ − @ ) <%' @ + - 0 # 0 @ )$ +.- F! " @ +&':) +;5' );$ + P 5-R$,' +;5' 1) 0E 3U-C;5 0 ' 7$ 0 $ +;-=+&' a - 0 # +&5' $(' 1' 0 C*' A@ 2d'*+ 6 ':' 0 0 ' - 0 #
Y!':),+&'*l $ F 3 2 <7$S$ +;-/)&+;$S-/++;5'oC*1)&)&' 0 +S+;- P ' C:':<%< '*lV+&' 0 # 0E + +&5' <4' @ + k
First
$ 3q2 Phase.
<7$ +
p 5 'OC&m 6 5'*+;5'*)f+&5' E 1'D$,$(' # $(' 1' 0 C*'qY 7$ ?+.$ t v - 0 # s 0 # ) E 5+&3 $ + $ F 3q2 <$(5 1< # 2d' σ (a, σ , b, σ , −) ∆ 5' 0 $ 3g' 6 5':)&' σ (b, σ , b, σ , −) ∆ @ t ) -R<4< ut V t\ {v} - 0 #
v
u+&3
u
5
k
'
4
<
'
$
+o$ F 3 2 < 7$ σ (c, σ , c, σ , −) ∆ @
(b,)kσ-Rv<4< , c, σv , −) ∆
s
u
u
#
@K
0
8)
u V \ {s} (c, σs , d, σs , −) ∆
(d, , l, , +) ∆
-R<4<$ +;-=+&' R$ F 3q2 <9C 3q2 0 -=+ 0 $ +&5-/+ -/)&' 0 +g':l P < 7C 4+&< F 3g' 0 + 0 ' # @ )
':ls-R3 P <4' (b, ) ) (a, σ ) +&5'*)&' %$k-H+,).- 0 $ 4+ 0 + +&5' 0 -=<)&''OCN+ 0E
$(+;-/+&' R " @ +&':) +&5'o$('DC A0 # v P 5-R$(' +&5'3U-C;5 0 ' 7$ 0 $(+;-/+&' l - 0 # +&5'f5'D- #
P 0 +;$f+ /6 -=) # $[+;5' <%' @ +&3 $(+ @ +&5'g$ F 3 2 <7$[+&5-=+ 5-DY!' 2d'D' 0 E 1'O$,$(' # 0
i 5-$(' 5' C 0 +&' 0 + A@ +&5'o+;- P 'f)&'D3U- 0 $S1 0 C.5- 0E ' #h# 1) 0EUi 5-R$,' Second
) '*Phase.
) 5'
C:- 0 - 0 # )&'D3 Y!'k+;5' );$ + Y!':),+&'*l (l, σ , p , , +) ∆ Third
Phase.
@K ) -=<%< u V @L0 3 )&'kY!':),+ 7C*'O$X-=)&'g<4' @ + -=+ +&5 %$ P u0 + cu6 'U-=)&' # 0 ' ^@ )f-=<%< u V <7$('q+&5' 0 '*lV+Y!':),+&':lH5-R$+ 2d'k- # r
(p
,
,
A,
,
0)
∆
u
(-RC:' 0 + @ )[-R<4< u, w V @ ) 6 5 %C.5 {u, w} E (p
u , σw , qu , σw , +) ∆
j 5-=+&'*Y!':) @ <4< /6 $ 3 1$(+ 0 + 2d' - # -C*' 0 + (q , σ , q , σ , +) ∆ @K )
-R<4< u, w V 6 4+&5 u = w - 0 # {u, w} / E u @ w-=<%<Y!u'*)&+ %Cw:'D$ 5-DY!'h2d':' 0
Chapter 5. Chordless Paths
C.5'OC;m!' # )&':+&1) 0 + +&5'f<%' @ +;3 $ + (q , , r, , −) ∆ V@ )-=<%< u V - 0 #
u8
)
=
%
<
<
K
@
0 -R<4< F )&' r ?+&':);-/+;' (r, , l, , +) (r, σw ,- r, σw-=,<%< −)
∆
w
V
" E 0c $ +;-/+;' R$ F 3q2 <\C 3q2 0 -=+ A0 $f+&5-=+X-/);' 0 +X'*l P < 7C 4+&< F 3 ' 0sr
∆
+ 0 ' # <4'O- # + +;5' 0 -=<M)&+' 'DCN+ 0E $ +.-/+&' R L +&'b+&5-=+ - @ +&'*) +;5'H+&5 ?) #
P 5-R$(' -R<4<+;- P ' C*':<%<7$C 0 +;- 0 +&5'f2<%- 0 m $ F 3q2 < - E - 0
i 5 -$(' ' 0 $,1)&'D$q+&5-/+k-R<4< Y!'*)&+ %C:'D$ E 1'O$,$(' # 0 i 5-$(' -/)&'
# 0 +&5'*) 6 7$('o+;5'f) E 5_+$,CD- 0 0 $ +;-=+&' q @ )9$ 3g' u V V@ - %<7$ Q );' Y!':) 2d'DCD-=1$(' A@ +&5'f+,).- 0 $ 4+ 0h@ ) 3 p + uq +;5'fY!'*)&+ %C:'D$SC.5 $(' 0
@K );3 - P -=+&5 P 0 G 5') E 5+$,CD- 0 0 $ +;-/+;' qu -R<%$ u' 0 $(1);'D$ +&5-/+ 0 + 6 LA@
+;5'9Y!':),+ 7C*'O$ -=)&'fC 00 'OCN+&' # ':lsC:' P +L-=< A0E P 8 u 0 -=<%< F! 0hi 5-$(' o6 'oC.5'OC&m
+;5-/+ +&5'U' 0 # P 0 +;$ @ P -/);' s - 0 # t - 0 # +&5-=+ P Y %$ ?+;$ v " <?+ E '*+;5'*) +;5' 3U-RC.5 0 ' )&'D-C.5'D$ - 0 -RC:C:' P + 0E $(+;-/+&' @ - 0 # 0 < F @ 4+ E 1'O$,$('O$U+&5'
Y!':),+ 7C*'O$ @ -HC.5 ) # <4'O$,$ (s, v, t)r P -=+&5 A@ $ %e:' -=+q3 $ + k 0 i 5-$(' " 0
'O-R$ F C:-=<7C*1<%-=+ 0 )&':Y!'D-R<%$M+&5-=+ @ k $ F 3q2 <7$M-=)&' 6 ) 4+,+&' 0`0 + +&5' +.- P ' 0
i 5-$(' +&5'\C 3 P 1+;-/+ 0 C 0 $ 7$ +;$ A@ ':ls-CN+&< F k2 +4k +,).- 0 $ 4+ 0 $ p <%'D-=)&< F!
C:- 0 2d'fC 0 $(+,)&1C*+&' # 0 + %3 ' P < Fs0 3 %-R< 1- # );-=+ 7C 0 2 +&5 n - 0 # k M
Correctness.
%$ + C*+ -$
5.2
n0 +&5
@) 3
<%':3U$
Hardness for W[1]
7$ $('OCN+ 06 ' P ) !Y ' +&5-/+ G /IHKJ %$ W[1]r 5-/) # 1$ 0E - );' # 1 C*+ A0
>0 # ' P ' 0 # ' 0 + '*+ ^6 5 %C.5 7$ 0 ' A@ +&5 ' &C*<7-R$,$ 7C:-R< W[1]r 5-/) # P ) 2 r
n
'
50 (
45 < 65 @& 587 :D D
Problem
5.6
(Independent
Set)
k
705 B 65
D &7E ?
( ?=58705 2< D 56 85 D 85 ( 5 E( 5; E5 K(8
G = (V, E)
3
k
G
pJ0 $ # ':) - 0 0 $ +;- 0 C*' A@Ln0 # ' P ' 0 # ' 0 + '*+ +&5-=+ 7$ - E );- P 5 G = (V, E)
- 0 # - 0 0 +&' E '*) k 1 k |V| - 0 # <%'*+ V = {v , . . . , v } j ' C A0 $ +,)&1C*+
- E );- P 5 G @ ) 3 G $(1C.5 +&5-/+o+&5' - 0 $ 6 ':) + +&15' G /IHKnJ P ) 2<4'D3 A0 G
P ) Y # 'O$ +&5' $ <%1+ 0 + +&5' 0 # ' P ' 0 # ' 0 + r $('*+ P ) 2<4'D3 0 G 5' 3U- 0 0E )&' # 4' 0 + @K ) 1) C 0 $ +&)&1CN+ 0 7$ CD-=<%<4' # '587 5 B ?= B 5
D :D + C 0 $ 7$ +;$ @
Y!':),+ 7C*'D$ vi , . . . , vi P <%1$\+ 6 '*lV+,);-oY!'*),+ %C:'D$ si
n
- 0 # ti C 00 'OCN+&' # + 'O-RC.5 A@ +&5' n Y!1'*)&+ %C:'D$ vni 1 j n -R$o$(5 /60 0
8 E 1)&' p <%'D-=)&< F +&5':)&'k-/);' '*l-RC*+&< F n C.5 ) # j <%'D$,$ (si , ti)r P -/+;5$ 0 $(1C;5
!
5.2. Hardness for W[1]
/σu /0
/σu /0
...
g1
gk
a
σt /σt /−
/σu /+
/σu /+
/σu /0
σu /σu /−
u=s
σu /σu /−
u=v
σs /σs /−
σv /σv /−
c
d
A
b
//+
//−
//0
σy /σy /+
σy /σy /+
{y, u} / E
y=u
r
σy /σy /−
qu
{y, u}
E
σu //+
pu
l
σy /σy /+
{y, w} / E
y=w
qw
σy /σy /+
{y, w}
pw
σw //+
E
//0
//−
//+
( B ?=58 ; B
Figure 51: 3 ?=5 =78 85 F5
D 9$ 3
:D?=5
@D 5 B 3=&(8+;65
D 50( B 97 ?=5
?=5E7850 7ED 507 (E$K 8# :D
450 58 < B 785EK(
:D 4 7E (
(
79<=& B ?'< 5 D 5 5
D
? 5 7E (
7078 4 ( 785
=
507 ?=5
@D (
$K 8# 4 70 5
5 785
@D@ 9% $ ?=5 4785
? 45 85E50 65
D a
Chapter 5. Chordless Paths
- # 7-=3 0 # " $+&5' 0 -R3 0EhA@ +&5' Y!'*),+ 7C*'O$o$(1 EE 'D$(+;$ 6 ' -$,$ C 7-/+&'g'D-C.5
A@ +&5'D$,' P -/+&5$ 6 4+&5 - Y!':),+&'*l @ ) 3 G 0 - 2 'DC*+ ?Y!' 3U- 00 '*) ) 1+ 0E P -/+&5U+;5) 1 E 5 vij _@ ) $ 3g' 1 j n %$ 0 +&':) P )&':+&' # -$J$('D<4'OCN+ 0E vj + 2^'
P -/),+ A@ +&5' 0 # ' P ' 0 # ' 0 +$(':+ I + 2d' C 0 $(+,)&1C*+&' # 5'XC 0 $ +,);1CN+ 0 1$('D$
$(1C.5 Y!'*)&+&'*lkC.5 7C*' # 7-=3 0 # $ A6 5 %C.5 -/);'C 00 'OCN+&' # 2 F # ' 0 + @ F 0E si+1
k
- 0 # ti @ ) -=<%< 1 i < k '*+o1$oCD-=<%<B+;5' E );- P 5 # 'D$&CN) 42d' # $ @ -/) G 6 5'*)&' 1Mp $ +;- 0 # $ @ )SY!':),+&'*l C;5 7C*' Proposition
5.7
(":DK( (E 5
<
$ B =
? 7ED=F50(E(
1 k 3I; ?
(s
,
t
)
`
( 59 '587 B 59( 3 `−k−1
G
vi1
i
ti s
vin
*) Figure 52:
4507
vin
vi1
(si , ti )
vi2
i
ti s
vin
5 B ?= B 5
3";?@(
ti
vi2
vi2
vi1
si
B 79785
vi1
vi2
si
G
D :D
:D
?'78585
ti
vin
(
n
B ?=7ED 59(E(
5' 0 '*lV+B$ +&' P %$ + ' 0 $(1)&' +;5-/+ +&5' Y!'*),+ 7C*'O$ C.5 $(' 0 2 F +,);-DY!'*).$ 0E G
0 - C;5 ) # <4'O$,$ P -=+&5 C ),);'D$ P 0 # + - 0 0 # ' P ' 0 # ' 0 +k$,'*+ 0 +&5' ) E 0 -=<
E );- P 5 G -C:C 3 P < %$,5 +&5 7$ S6 ' C 0 $ +,)&1CN+ G @ ) 3 + 6 $ F 3 3g':+,) 7C
C P 4'O$ A@ G \ ' 0 +&'M+&5'9Y!'*),+ %C:'D$ 0 +&5' );$ + C P F C A@ G 2 F si vi - 0 # ti 6 5'*)&'O-R$ +&5'kY!'*),+ 7C*'O$ 0 +&5' $('DC A0 # C P F Γ @ G -=)&' )&' @ ':),)&' j#
!
5.2. Hardness for W[1]
+ R- $ σi ϕi - 0 # τi ^@ ) 1 i k - 0 # 1 j n o
5' E );- P 5$ C - 0 #
-/)&' C 00 'OjCN+&' # 2 F # ' 0 + @ F 0E tk - 0 # τk 5' C 0 $ +,);1CN+ 0 A@ G %$
ΓC 3 <4':+&' 2 ' +&5' - # -C*' 0 C F A@
P # F ## 0E - 0 13q2d':) @ ' # E 'D$f+&5-/+X' 0 C # k
" 0 '*l-=3 P <%' %$9$(5 /60 0 8 E 1)&' s
G
)
)
)
5':)&' %$- 0 ' # E ' 0 G 2d':+ 6 ':' 0 vi - 0 # ϕi @ )[-R<4< 1 i k - 0 #
-=<%< 1 j, ` n 6 ?+&5 j = ` 1C;5 -j 0 ' # E ' `%$[CD-=<%<4' # - B (
(850 B $
5
D&@5 8 )S'*Y!':) F ' # E ' {v , v } 0 G C 00 'DCN++&5'Y!':),+&'*l $,'*+;$ {vi , ϕi } - 0 #
p q
p
p
4+
&
5
2
C
3
4
<
:
'
&
+
'
2
-=),+ 4+&'
6
F
j
j @ )X-R<4<
P
P
{v
1
i,
j
k
i
=
j
q , ϕq }
$(12 E ).- P 5 0 G \
5'O$('o' # E 'D$-=)&' C:-R<4<%' # <:D 5 58:D 50 B 55
D&@59( pJ00 'OCN+o+&5' Y!'*)&+&'*l $('*+;$ {vi , ϕi } - 0 # {vj , ϕj } @K ) -=<%< 1 i, j k
6 4+&5 i = j - 0 # -=<%< 1 ` n` 2 F` -qC 3 P `<4':+&'[`2 P -/)&+ ?+;'o$(12 E );- P 5 0
\
5'D$('o' # E 'D$S-=)&'XCD-=<%<4' # ( 5 5
D&@59( G
B ?=7ED=F50(8(
3I; ? k <
Lemma 5.8
(s1 , σ1 )
t
G
5ED&'5 7 3:D 5 "50:D 50 B 55
D &@5 7 ( 5 5 D&@5 Proof.
4+ 7-=<
( 50( B (E3(8658 B $
P -/)&+'*+ @ P 2d6 'g5 - %C.C.5q5 +& );) -D# Y!<%'*'D);$&$($ 'O$(s P ,-=σ),+ )Ar @P -/+&+&55 ' Y 7-);$ + t Y!'*)&0 +&'*Gl C;g5 pJ7C*' 0 $ # 7-# =':3 )o+&0 5# '
0
A @ C Z F C 0 $ P+,)&1CN+ 0 A@ G ':ls-CN+&< F 0 ' A@ +;5' Y!'*),+ %C:'D$ v1 1 j n
7$ A0 P " <%< A@ +&5'D$(' Y!':),+ 7C*'D$-/);' 0 ' E 52 );$ A@ s1 H 43 %<%-=j)&< F! +&5' 0 -=<
P -/),+ A@ P C 0 +.- 0 $['*l-RC*+&< F A0 ' @ +;5' Y!'*),+ %C:'D$ ϕ1` 1 ` n @ ` = j +;5' 0 +&5'MY!'*),+ %C:'D$ v1 - 0 # ϕ1 -=)&'fC 00 'OCN+&' # 2 F - C 0 $ 7$ +&' 0 C F ' # E ' 0 G ' 0 C:' +&5' F + E '*+;5j'*) 6 4+&5 `s1 - 0 # σ1 0 # 1C*'[- 0 (s1 , σ1)r P -/+&5 0 G +&5-=+
# 'D$ 0 +Y %$ 4+ tk 9 1C.5 - P -/+;5 C:- 00 +92d' '*lV+&' 0 # ' # + - C.5 ) # <%'D$,$ P -/+;5
+;5-/+ Y %$ 4+;$ tk >0 @ -CN+ -HC.5 ) # <%'D$&$ (s, t)r P -/+&5 C:- 00 + 2d' ':l+;' 0 # ' # 0
$ 6 - F - 0 # -/+X+&5' $&-=3g'k+ 43g' );':3U- 0 - C.5 ) # <%'D$,$
r P -/+&5 5V1$ (s,
t)
6 ' C 0 C:<41 # 'o+&5-/+ ` = j 8s1),+&5'*)&3 )&' v1 - 0 # ϕ1 5-DY!' +&5'q$,-=3g' 0 ' E 52 );$-=< 0E 2 +&5 0 # ' r
P ' 0 # ' 0 C*'S- 0 # $('*+\' j# E 'O$ 2 F Cj 0 $ +,);1CN+ 0 1 PP $('+&5-/+ P C 0 + 0 1'O$ @ ) 3
Y %- - 0 0 # ' P ' 0 # ' 0 C:' )q$('*+ ' # E 'g+ -hY!'*),+;'*l w p <4'O-/)&< F w = tk
v1j
2d'OC:-=1$(' tk 7$ 0 + 0 C # ' 0 +q+ - 0F 0 # ' P ' 0 # ' 0 C*' )k$(':+ ' # E ' " $ w %$
-R<%$ - 0 ' E 52 ) @ ϕ1 +&5'oY!':),+ 7C*'O$ {s1, v1 , w, ϕ1, σ1} V(P) 0 # 1C*' - 0
j
j
j
# 'D$ 0 +\Y 7$ 4+ tk ' ?+&5'*) 1
1 r P -/+&5 6 5 7C.5 # 'O$ 0 + Y %$ ?+
k B ' 0 C*' (s
,
σ
)
t
P
C 0 +,).-/) F + X 1)-R$,$(13 P + 0 5':)&' @ )&' P C:- 00 +J1$('[- 0F 0 # ' P ' 0 # ' 0 C*'
)9$(':+' # E ' 0 C # ' 0 +S+ v1 ) ϕ1 1
j
j
1
k
Chapter 5. Chordless Paths
v2
v1
1
s1
G
1
2
3
2
3
3
2
3
1
Figure 53:
v3
σ1
2
1
G
5 #
F5 (8 7E;<& ?=5 B (8 7 B 7
G
k
=
2
(
3(050 B $ 5
D&@50( 705 (E?= 4 9$ ( #% D %< 59( 3:D 5 "50:D 50 B 5
5
D&@59( 9$ D@K(E?=5
D %< 50( D(59 5
D&@50( 0$ D K 5ED%< 50( ?=5
'587 5 85 %( 3
< D B ;65 ?=5 B 797859(+" D 58 B 56 ?=5 '587
G
B 59( 78
7 5 4 5 ?=5 '587 B 59( 85 5ED
B 79785
G
1
(+ :D6
?=5 4507 65 k k 3((E? @D 5
D D@7 v1
t =τ
5.2. Hardness for W[1]
1
s1
1
2
2
3
*$%
G
1
?=554
5
D&@50(
G
σ1
2
σ1
1
3
3
2
3
2
" ) &
2
3
Figure 54: 1
1
2
3
2
3
s1
3
1
1
$%! $% " ) 5 70
<&
785
&78
5
D <6 DA587058 $"59(
!
Chapter 5. Chordless Paths
+S)&':3U- 0 $+ C A0 $ # '*)S+&5'XC 0 $ 7$ +&' 0 C F ' # E 'O$ 0 C # ' 0 +S+ v1 - 0 # ϕ1 1 PP $,' P C A0 + 0 1'D$ @ ) 3 v1 0 -C 0 $ 7$ +&' 0 C F ' # E 'J+ /6 -/) # $ -SY!j':),+&'*l ϕj1 @K )f$ 3g' 1 ` n 6 4+&5 ` j= j " <4< 0 ' E 5_2 );$ @ v1 -R< 0E C 0 $ 7$ +&' 0 C `F
' # E 'D$ -=)&' A@ +&5 7$+ F P ' 5' 0 +&5'SY!':),+ 7C*'D$ {s1 , v1, ϕ1j, σ1 } V(P) 0 # 1C*'
- C.5 ) # <%'D$&$ (s1, σ1 )r P -/+&5 +&5-/+ # 'O$ 0 +JY 7$ 4+ tk j ' `0 C:' P # 'O$ 0 + Y 7$ 4+
' 4+&5':) C A0 +,);-=) F + 1) -R$&$(13 P + 0 5'*)&' @ )&' P C:- 00 +k1$,' - 0_F
tC k $ 7$ +&' C '
0
0 F # E ' 0 C # ' 0 +S+ v1 ) ϕ1 j
j
>0 $(13 3U-=) F +&5' 0 ?+ %-R< P -/)&+ A@ P E 'D$ @ ) 3 s1 Y 7- v1 S@ )h$ 3 '
+
- 0 # +&5' 0 -R< P -/),+ A@ P 7$9C 3 P <4':+&':< F j$ F 3g3g'*+,) 7C 1) 3 j nY 7- t1 + = s2 Z
@ τ1 ϕ1 σ1 F 0 # 1C*+ 0 0 k 6 ' CD- 0 P ) Y!' +&5' @ <4< /6 0E
$(+;-/+&'D3g' 0 + H
5j' 0 4+ 7-=< P -/),+ A@ P %$U- 0 (s1, tk)r P -/+&5 Q +&5-=+gY 7$ 4+;$U-=<%<
0 0 CN);'D-R$ 0E ) # ':) @ ) 1 i n 6 4+&5 1+q1$ 0E - 0F C 0 $ 7$ +&' 0 C F!
si ' '
0 # P 0 # ' 0 C*' ) $('*+ ' # E ' - 0 # +;5' 0 -=< P -/),+ A@ P 7$\- (τk , σ1)r P -=+&5 +&5-/+
7$C 3 P <%'*+;':< F $ F 3 3g':+,) 7C+ Q 5'o- 0 -R< F $ 7$ @ ) 3 -=2 Y!' P ) Y # 'O$ +&5'2-R$,' C:-$(' A@ +;5' 0 # 1CN+ 0 @ )
" +f+&5'g$,-R3 ' + %3g' 4+X-=<7$ P ) Y # 'O$[+&5' 0 # 1CN+ 0 $ +;' P " $[+&5'
k 4+=7-=1< 0
0 # 0 -=<c$('DC*+ 0HA@ +&5' P -/+;5 -/)&' # '*+;'*)&3 0 ' # s6 'XCD- 0 );':3 Y!'[+&5'
);$ +Y!'*),+&':l # 7-=3 0 # - 0 # ?+;$L$ F 3g3g'*+,) 7C[C P FU@ ) 3 +&5' E );- P 5 - 0 # - PP < F
+;5' 0 # 1C*+ 0 5 F P +&5'O$ %$+ +&5'[)&':3U- 0 0E k − 1 # 7-=3 0 # $ Theorem 5.9
G /IHKJ
3(
W[1]
? 7ED
4 ?
7859(+"5 B k
] 4Y!' 0 - 0 0 $ +;- 0 C*' (G, k) A@\>0 # ' P ' 0 # ' 0 + ':+ 6 'qC A0 $ +,)&1C*+9+&5'
E P G -R$ # 'O$,CN) 42d' # -=2 Y!' 5' E ).- P 5 G C 0 +;- 0 $ 2k(n + 1) + 1
Y!':),+ 7C*'O$ C 3 P 1+;'U+&5' 0 13 2^':) A@ ' # E 'O$ 0 G 0 +&'U+;5-/+ +;5'*)&' -/)&'
' # E 'D$ 0 +&5' 2k Y!':),+&':lkC;5 7C*' # %-R3 A0 # $ P <41$ kn(n − 1) C 0 $ 7$ +&' 0 C F
4kn
' # E 'D$ 4mk(k − 1) 0 # ' P ' 0 # ' 0 C:'9' # E 'D$ - 0 # 2nk(k − 1) $('*+ ' # E 'D$ !6 5':)&'
- 0 # m := |E(G)| 9 ' 0 C:' G C:- 0 2d' C 0 $ +&)&1CN+;' # @ ) 3 G 0
n
:=
|V(G)|
+ 43g' - 0 # $ P -C*' P < FV0 3 7-=< 0 2 +&5 n - 0 # k Z':+ F P 2d':3g'g3U- - C.5 )s# <%'D$&$ 5-(s$q1-h, σY!1'*)) r F P -=$ +&P 5 'DC Y 7-%-=< t@ k )&3@ $ 4eD0 'UP -/-=+X),+3 7C* 1$ <7+ -/) 4k 4+;+$q1$ %e:0 '
G
7$['*l-RC*+&< F 4k + 1 - 0 # P ?+[Y %$ ?+;$[':ls-CN+&< F 0 ' Y!':),+&':l vi @ ) 3 'D-C;5 A@ +&5'
Y!':),+&':l C.5 7C*' # 7-=3 0 # $ 6 4+&5 1 i k '*+ I := j{v V(G) | vi @ )
M 1 $,'X+;5-/+ v = v @ )[$ 3 ' j1 i, ` k 6 j 4+&5
V(P) $ 1 -i k} -=)&' PP C
j - $('*j+S'
D
'
N
C
;
+
'
2
"
#
#
# E ' 0 G +&5-/+ %$ 0 + 0 P
0
0
0
F
i
`
i2 =
v
v
`
F 'D3 3U- j s +&5 7$ j $(':+ ' # E ' @ )&3U$ -XC.5 ) # @ P 0 C 0 +,);- # %C*+ 0 + 1)
-$,$(13 P + 0 +&5-/+ P %$ C.5 ) # <%'D$&$ b
5':)&' @ )&' +&5' Y!':),+ 7C*'O$ vi Y 7$ ?+;' # 2 F
j
Proof.
);- 5
i
i
i
i
i
`
`
i
5.2. Hardness for W[1]
C ),);'D$ P 0 # + 3 1+&1-=<%< F # 7$ + 0 CN+Y!'*)&+ %C:'D$ 0 G +&5-=+ 7$ + E '*+;5'*) 6 ?+&5
i P $ 4+ 0` 4+ @ <4< /6 $ |I| = 2k + 1 − k − 1 = k 8s1),+&5'*)&3 )&' 6 'kC*<7- %3 +;5-/+ I %$[- 0 0 # ' P ' 0 # ' 0 +o$(':+ 0 G X 1 PP $('
+;5-/+ @ )L+ 6 Y!'*),+ 7C*'O$ vi - 0 # v` 0 P 1 i, ` k - 0 # i = ` +&5' C )&)&' r
$ P 0 # 0E Y!':),+ 7C*'D$ v - j0 # v -/j)&' 0 ' E 52 );$ 0 G 5' 0 2 F C 0 $ +,);1CN+ 0
- 0 # v` -=)&' C 00 j 'DCN+;' # 2 jF - 0 0 # ' P ' 0 # ' 0 C*'X' # E ' 0 G \
5 %$L' # E ' %$
vij +
0 0 P 2 j F ':3g3U- s +&5-=+ %$ 4+ @ )&3U$S- C.5 ) # A@ P 0 C 0 +,).- # 7CN+ 0
+ 1) -R$&$(13 P + 0 +;5-/+ P 7$oC.5 ) # <%'D$,$ 5'*)&' @ )&' 0 + 6 Y!'*),+ %C:'D$ 0 I
-=)&' - # -C*' 0 + 0 G pJA0 Y!'*).$(':< F@ ) - 0F 0 # ' P ' 0 # ' 0 +J$('*+ I = {v , v , . . . , v , . . .} A@ $ 4eD'M-/+
<%'D-$ + k 0 G +&5'*);' %$9-kC.5 ) # <%'D$,$ (s1, σ1 )r P -=+&5 1P Y 27- tk Ak@ $ 4eD' 4k + 1 0
0 +&5' r +;5qY!'*),+;'*lqC;5 7C*' # 7-=3 0 # $ Y 7$ ?+.$ i - 0 #
R@ ) 1 i k G
i
P
vi
ϕii
5'*)&' @ )&' s6 'o5-DY!'o- P -/);-R3 ':+&'*) 4eD' # )&' # 1CN+ 0 @ ) 3 - 0 0 # ' P ' 0 # ' 0 +
$,'*+ 0 $(+;- 0 C:' (G, k) + - G /IHKJ 0 $(+;- 0 C:' (G , 4k + 1) 'O$ +;-/2< 7$(5 0E W[1]r
5-/) # 0 'D$,$ A@ G /IHKJ >0 E ' 0 ':);-=< W[1]r 5-/) # 0 'D$&$ # 'O$ 0 + n-/+ <4'O-R$ + 7$ 0 + m 0/60 + 4 %3 P < F
/ r 5-=) # 0 'O$,$ A@ +;5'HC ),)&'O$ P 0 # 0E 1 0 P -/);-R3g'*+&':) 4eD' # P ) 2<%':3 Z1+ 0
+;5 %$ P -/),+ 7C*1<%-=)MC:-$(' +&5'f);' # 1C*+ A0 CD- 0 2d'o':l+&' 0 # ' # 'D-R$ 4< F
P)
i
`
i
i
`
`
? 5 ?=507 ?=58785!5 (8( $
3( / B F5965 6 2D 5 B D 5 4 =
Corollary
5.10
3";? 7C?'785E5 & '58 4507 B 50(
:D
2
B ?=7ED 59(E(
(s,
v,
t)
s
v
t
:D3785 B 5
D & 7E ? 5' P ) 2<%':3 %$qC:<4'O-/)&< F 0 / S' E -/) # 0E +&5'g5-=) # 0 'O$,$ P ) @
2$,'*),Y!' +&5-=+ +&5' );' # 1C*+ A0 # 'O$,CN) 42d' # 0 5' )&':3 C 0 $ +&)&1CN+.$U+&5'
E );- P 5 G 0 + %3g' - 0 # $ P -RC*' P < FV0 3 7-=< 0 2 +;5 n - 0 # k - 0 # k n Q );' Y!':) +&5'[$ 4eD' A@ - 0_F C.5 ) # <%'D$,$ (s1, σ1 )r P -/+&5 Y 7- tk 0 G 7$ '*l-RCN+;< F
5-/+ 7$ 0H ) # '*)+ $ <4Y!'f+&5' 0 # ' P ' 0 # ' 0 +9$('*+ P ) 2<4'D3 0 G ?+
4k
+
1
7$ $(1 UC %' 0 + + # 'DC # ' 0 +&5'':l 7$ +&' 0 C*' @ - 0_F C;5 ) # <4'O$,$ (s1 , σ1)r P -=+&5gY %0 G
tk
" 0 -R< E 1$(< F +&5' P ) 2<%':3 )&'D3g- 0 $ W[1]r C 3 P <%'*+;' @B6 ' %3 P $,' - # r
# ?+ 0 -=< C 0 $(+,);- 0 +;$ 0 +&5' C.5 ) # <4'O$,$ P -=+&5H1 0 # '*)9C 0 $ # ':);-/+ 0 $(1C.5H-R$
- 0 '*l-RC*+9$ 4eD' ) P -=) ?+ F!
Proof.
5 5 4 7 F D 5 B D 5 4 ?=59 ?=587 ?=58785
(
B Corollary
5.11
W[1]
k
5 3(8 ( B ?=78D= 59(E(
3I; ?> (
55 # B %$
7 ?'785E5 & '58 '507
(s,
v,
t)
k
B 50(
:D
:
D3785 B 5
D &7E ? :D "(E '5 <65 &@587
s v
t
k
Chapter 5. Chordless Paths
5.3 Chordless Cycles
n0 +;5 %$f$,'DCN+ 0`6 ' # %$,C:1$,$[+;5' )&':<7-/+ 0
A@ # 'DC # 0E A0 +&5'b':l 7$ +&' 0 C*' A@ - C;5 Y!':),+ 7C*'O$- 0 # +&5' P ) 2<%':3 I/J2 Q
Y!':),+ 7C*'O$ # A@ # 'DC # 0Ek0 +&5'['*l %$(+&' 0 C:'
r -/+&5$ @ )S$ 3g' ` (s, t) P
$(5 P 2d'*+ 6 'D' 0 +&5' P ) 2<%':3 G /IHKJ
) # <4O' $,$ P -/+;5 +&5) 1 E 5 +&5);':' E ?Y!' 0
- 0_F C.5 ) # <%'D$,$ P -/+&5$+&5) 1 E 5h+ 6 A@ - # 7$3 0 +1 0 0 A@ ` C.5 ) # %< 'D$,$
/ J
Problem
5.12
(Many
Chordless
(s,
t)-Paths
(
))
"(E '5>35+&'5079(
:D
450 :D3785 B 5ED & 7E ?
G
=
(V,
E)
k
`
D 4 D (8< B '507 B 59(
; (8
( ?=50785 ( 59
U
V3( D3 ( 3
'587 B 59( ( B ? ? ;A?=5 ( s,
+t&7E ?V2<:D B 5
D 9$
<
k U
G
3";?@(
B ?=7ED 59(E(
`
(s, t)
" $(12 E ).- P 5 H A@ G 7$X-!D ( < A@ C;5 ) # <4'O$,$ (s, t)r P -/+&5$ @ 0
'O-RC.5HC 3 P A0 ' 0 + 7$
H \ {s, t}
) ' 4+&5':)M- 0 7$ <7-/+&' # Y!'*),+;'*l - # -C*' 0 +S+ 2 +&5 s - 0 # t 0 G ) )9- P -=+&5 (p , . . . , p ) r 2 s@K ) 6 5 7C.5
1
r
%$9- # (-RC*' 0 +S+ s 21+ 0 +S+ t 0 G – p1
%$- # -C*' 0 +S+ t 21+ 0 +L+ s 0 G – pr
'*Y!':) F p 2 i < r %$- # -C*' 0 +S+ 0 ' 4+&5':) s 0 ) t 0 G –
i
' $(5 /6 +&5-=+ G /IHKJ - 0 # I/J -/)&'X' 1 ?YA-=<%' 0 +L1 0 # '*)S2 &+ 5 P -=);-=3 r
j X
':+&'*) 4eD' # )&' # 1CN+ 0 $[- 0 # C:%< -$,$ 7C:-R< -/) P );' # 1*C + A0 $ f
5 7$ 43 P < %'D$M+&5-=+
2 +&5 P ) 2<%':3U$S-=)&'XC 3 P <%'*;+ ' @ )S2 +&5 W[1] - 0 # / $
Theorem
5.13
6
7 $
k
` 2
?=7ED=F50(E(
(s, t)
;?@( (
W[1]
? 7ED
4 ?
7850("5 B 8 ) ` = 2 6 ' @ -C*'+&5' P ) 2<%':3 I/J $ +&5'*);'- .C 5 ) # <%'D$,$C F C*<%'
-/+3 $ + k +&5) 1 E 5 s - 0 # t A@
5' P ) A@ 7$ 2 F )&' # 1CN+ 0 @ ) 3 G /IHKJ pJ0 $ # '*)k- G /IHKJ r 0 $ +;- 0 C*'
XpJ0 $ +,);1CN+ - E ).- P 5 G @ ) 3 G 2 F - ## 0E - 0 ' 6 Y!'*),+;'*l c
(G,
s,
v,
t,
k)
+;5-/+ 7$ - # (-RC:' 0 + + s - 0 # t 0 < F! " 0_F C.5 ) # <4'O$,$ C F C*<%' A@ $ %e:' -=+ 3 $(+
+&5) 1 E 5 c - 0 # v 0 G C ),);'D$ P 0 # $+ -kC.5 ) # <4'O$,$ (s, v, t)r P -=+&5 A@
k
+
1
$ 4eD'X-/+3 $ + k 0 G - 0 # Y 7C*' Y!'*);$&- Proof.
$ %e:'
5.3. Chordless Cycles
8 ) ` > 2 6 ' - ## ` − 2 - ## ?+ 0 -=<oY!'*),+ 7C*'O$U+ G 'D-C.5 A@ +;5':3
- # -C*' 0 ++ c - 0 # v 0 < FJ
5' 0 - 0_F $('*+ @ -=+93 $ + k + ` − 1 Y!':),+ 7C*'D$ 0
+;5-/+ @ )&3 - # 7$3 0 +S1 0 0bA@ ` C.5 ) # <4'O$,$ (c, v)r P -/+&5$ C )&)&'D$ P 0 # $S+ G
-kC;5 ) # <4'O$,$ (s, v, t)r P -/+&5 @ $ 4eD'X-/+3 $ + k 0 G - 0 # Y 7C*' Y!'*);$&- 5' -=2 Y!'q)&' # 1CN+ 0 + E '*+&5'*) 6 ?+;5 pJ ) <4<7-/) F 7D %$o'D-R$ 4< F $(':' 0 + P ) Y!' / r C 3 P <%'*+;' 0 'O$,$ A@ I/J Z\1+c+&5 7$ );'D$(1<4+ %$ -=<4)&'O- # F m 0/60 ! >
n0 @ -CN+ ?+ P ) Y # 'D$- 0 -=<4+&':) 0 -/+ ?Y!' 6 - F + P ) Y!' pJ ) <%<%-=) F 7Ds -R$ 1+ r
< 0 ' # 2d':< /6X
5' P ) 2<4'D3 7$C:<4'O-/)&< F 0 / 5'
Proof.
(of
Corollary
5.10
(alternative))
5-/) # 0 'D$,$ %$ P ) Y!' 0 2 F )&' # 1CN+ 0 @ ) 3 I/J pJ0 $ # ':)h- 0 0 $ +;- 0 C*'
@ I/J ':+ N (s) = {x , . . . , x } 6 ?+&5 d = # ' E (s) (G, s, t, k)
G
1
d
@ t N (s) +&5' 0 $ 43 P < F );':3 Y!'[+&5' ' # E ' {s, t} " C.5 ) # G<4'O$,$ (s, t)r
P -/+&5 0 +&5'XG)&'D$(1<?+ 0E E );- P 5 C ),)&'D$ P 0 # $S+ - C.5 ) # <%'D$,$C F C*<%' +&5) 1 E 5 s
- 0 # t 0 G - 0 # Y 7C*'fY!':);$,- @ t / N (s) +&5' 0 C 0 $ +,)&1CN+X- E );- P 5 G @ ) 3 G 2 F - ## 0E - 0 ' 6
Y!':),+&':l s - 0 G# 0 ' 6 Y!':),+ 7C*'O$ y , . . . , y - 0 # z , . . . , z L':3 Y!'f-R<4<c' # E 'O$
0 C # ' 0 +9+ s - 0 # C A00 'DC*+ s + 2 -=<%< @ dy , . . . ,2y 0 $ +&d'D- # pJ00 'DC*+ y + 2
d
i
k
)
=
%
<
<
4+
&
5
8
R
4
<
<
C
O
'
N
C
+
#
@
0
6
0
!
F
0
0
x
+ j -R<4< A@ z 2, . . . ,iz - d0 # C 010 'O CN+ j z + d x @ i) = -=<%< j 2 i j d " s0
':ls-R3 P <4' 7$ 2 # ' P %C*+&'d# 0 8 E 1)&' 9ip <4'O-/)&< jF! G C:- 0 2d' C 0 $ +,);1CN+&' # @ ) 3
0 O(|V(G)|2 ) + 43g' G
y2
x1
s
x1
y4
s
t
x4
x4
s
Figure 55:
t
z2
G
z4
54
G
5 (8 78; 3=&?=5 B (8 7
j ' *C <7- %3 +;5-/+9- 0_F C;5 ) # <4'O$,$ (s, t, s )r P -=+&5 A@ $ %e:'
$ P 0 # $ + -;C 5 ) # <4'O$,$BC F C:<4' A@ $ %e:' k +&5) 1 E 5 s - 0 # t 0
B k +- 3
0#
G
0
Y
G
G
7C*'
C ),);' r
Y!'*);$,- Chapter 5. Chordless Paths
n 0 # ':' # - 0_F C.5 ) # <%'D$,$ C F C*<%' C +;5) 1 E 5 s - 0 # t 0 G P -$,$('D$ +&5) 1 E 5
':ls-CN+&< F + 6 0 ' E 52 ).$ x - 0 # x @ s 6 5':)&' 1 i < j d 1C.5
- C F C*<%' C )&)&'D$ P 0 # $U+ - i C.5 ) # <%j'D$,$ P -/+&5 (s, y , x , . . . , t, . . . , x , z , s )r
j j
r
P -/+&5 0 G 6 5'*)&' +&5' # +&+&' # P -=),+ 7$ C \ s pJj 0 iY!':);$(':< F! - 0_F (s,
t, s )
P -/+&5 0 G @ <4< /6 $ +&5' P -/+,+;'*) 0 (s, yj, xi , . . . , t, . . . , xj, zj, s ) 9@ ) 1 2 F C 0 $ +,)&1CN+ 0À@ G - 0 # C ),)&'D$ P 0 # $+ - C.5 ) # <%'D$,$oC F C:<4'
i<j d
0 5 7$ P ) Y!'O$ +&5' C:<%- 43 - 0 # +&5'f+&5' )&':3 (s, xi , . . . , t, . . . , xj , s) G
+&'[+&5-=+S+&5'f);' # 1C*+ A0 # 'D$,C*) ?2d' # -/2 Y!' 7$ P -=);-=3g':+,) 7C " $S-kC 0 $,' r
1' 0 C*' I/J 7$ 0 W[1] - 0 # 5' 0 C:' 2 F`
5' )&':3 _ W[1]r C 3 P <%'*+&' n0 ) # '*) + ':l+;' 0 # +&5 7$q$ +;-=+&':3g' 0 + + +&5' Q - 0F p 5 ) # <%'D$&$ (s, t)r i -/+&5$
i ) 2<%':3 V6 'XCD- 0b0 C*'X- E - 0 1$(' - )&' # 1CN+ 0 + $ -?=7ED 59(E(
Theorem
5.14
(s, t)
("5 B 7 $
k
` 2
;?@( (
W[1]
B F5965 4 ? 785
5' 5-/) # 0 'D$&$ P -=),+ 6 -$ $(5 /60 0 5' );':3 7 ?+ )&':3U- 0 $o+ P P V6 5 7C.5 %$ # 0 'f2 F )&' # 1CN+ 0 + pJA0 $ # '*) - 0 0 $ +;- 0 C*' (G, s, t, k, `) A@ I/J 6 5'*);' G = (V, E) 7$
- 0 1 0 # 4)&'DC*+&' # E );- P 5 s, t V - 0 # k - 0 # ` -=)&' P $ ?+ 4Y!' 0 +&' E '*);$ 8 )
+;5' $&-/m!' A@ 2);'*Y ?+ F6 ' # U0 +':l P < 7C 4+&< F # '8 0 'q- 1) 0E 3U-RC.5 0 ' -R$ 0
5' )&':3 5'*);' >0 $ +&'O- # 6 ' 0 < F $ m!'*+;C.5 5 /6 $(1C;5 - 3U-RC.5 0 'UC 1< #
2d'oC A0 $ +,)&1C*+&' # -R< 0E +&5'f< 0 'O$ A@ 5' )&':3 _ $(1C;5 +&5-/+L+&5' 0 13q2d'*) A@
$(+&' P $ 0 4+;$BC 3 P 1+;-/+ 0 C:- 0 2d'2 1 0 # ' # 0 +&':)&3U$ A@ k n0 +&5' @ <%< =6 0E
6 ' # ' 0 + @ F +&5'LY!':),+ 7C*'O$ @ G 6 4+&5 +&5' 4)JC ),);'D$ P 0 # 0E $ F 3q2 <7$ @ ) 3 +&5'
-R< P 5-/2d'*+ @ +&5' 1) 0E 3g-C.5 0 ' " $M+&5'k$ <%1+ 0 $('*+ U V(G) 0 # 1C:'D$o- # %$ 0 +[1 0 A0
First Phase.
A@ (s, t)r P -=+&5$ A6 '9C:- 0 ' 0 13g'*);-=+&'S+&5'LY!'*),+ 7C*'O$ A@ U -$ -o$ 0E <%' P -/+;5 @ ) 3
+ ' 4+&5':) s ) t # ' P ' 0 # 0E 0U6 5'*+&5':) ` 7$ ':Y!' 0 ) ## )&'O$ P 'DCN+ ?Y!'D< "F .
s
6 5'*)&' 0 < F s - 0 # t - PP 'D-=)3 1<4+ P <4'+ %3g'D$ 5'f3U-C;5 0 ' );$ + E 1'O$,$('O$ +;5'X-=+M3 $ + k Y!':),+ 7C*'D$ @ U 0 +&5 7$ P -/),+ 7C*1<%-=) ) # ':) +&5'*);'*2 F 6 ) 4+ 0E -/+
3 $ + k + ` − 1 Y!'*),+ %C:'D$ 0 + +&5'[+.- P ' 5' 3U-RC.5 0 ' C.5'OC&m$ 0 O(k) +&);- 0 $ ?+ 0 $ 6 5'*+;5'*)+&5'
Second
Phase.
$,' 1' 0 C:' A@ Y!'*),+ 7C*'O$ 0 +;5' +.- P 'b$ +;-/)&+;$ 6 4+&5 s ' 0 # $ 6 ?+&5 s ) t # ' r
P ' 0 # 0E 0h6 5':+&5':) ` %$9'*Y!' 0b ) ## );'D$ P 'DC*+ 4Y!':< IF . - 0 # 6 5'*+&5'*) s - 0 #
- 'O-/)M-R<?+;'*) 0 -=+&':< F 0 +&5 %$S$(' 1' 0 C:' - 0 # 2 +&5 + E '*+;5'*) ` + 1 + %3 'O$ t PP
5'X3U-RC.5 0 'XC.5'OC&m$ 0 O(k) +,);- 0 $ ?+ 0 $ 6 5'*+&5'*)9':Y!'*) F
Third
Phase.
Y!':),+&':l 0 +&5' +;- P ' +&5-=+ 7$X- # -C*' 0 + + s 0 G 7$ -=<7$ - # -RC:' 0 +X+ s 0
+;5' +.- P ' [ 43 %<%-=)&< F! 4+[C.5'OC;ms$ 6 5'*+&5'*)['*Y!'*) F Y!':),+&':l 0 +;5' +.- P ' +&5-/+ %$
Proof.
) Y!'f3g':3q2d':);$(5
5.4. Directed Graphs
- # - C*' 0 +S+ %$S-R<%$ - # -C*' 0 +S+ t 0 +&5'f+;- P ' 5'b3g-C.5 0 'b'*);-$('D$ +&5' Y!':),+ 7C*'O$ @ ) 3 +&5'h+;- P ' A0 '
Fourth
Phase.
- @ +&':)M+&5' +;5'*) 8 )['D-C;5`)&'D3 Y!' # Y!'*),+&':l v +&5' Y!'*),+;'*l v %3g3 ' # 7-/+&'D< F
@K <%< /6 0E 4+ A0 +&5' +;- P ' 5-R$c+ 2d'J- # (-RC:' 0 +c+ v @ v {s, t} +&5' 0 +;5'*)&' 7$
0 +;5 0E 3 )&'L+ # +;5'*) 6 7$(' @ )\'*Y!':) F )&':3U- 0 0E Y!'*),+&':l w / {s, t, v }
0 +&5'[+.- P ' 4+L5-R$+ 2d'fY!':) ' # +&5-=+ w = v - 0 # +&5-/+ w 7$ 0 +S- # (-RC*' 0 +
+ v 0 G 5 7$ P 5-$('XCD- 0 2d' %3 P <4'D3g' 0 +&' # 1$ 0E O(k2 ) +,).- 0 $ 4+ 0 $ >0 $(13 3U-=) F6 ' 5-:Y!' # 'O$,CN) 42d' # - 1) 0E 3U-RC.5 0 ' +&5-/+[C:- 0 2d'qC 0sr
$(+,)&1C*+&' # @ )- 0_F $ P 'OC C 0 $ +;- 0 C*' (G, s, t, k, `) A@ I/J 0 + %3g' P < F_r
0 3 7-=< 0 -=<%< A@ |V(G)| k - 0 # ` - 0 # 6 5 $(' C 3 P 1+.-/+ 0 )&'O-RC.5'D$S- 0 -=<
-C:C:' P + 0E $ +.-/+&' 0 O(k2 ) +,);- 0 $ ?+ 0 $ @ - 0 # 0 < F @ (G, s, t, k, `) %$$ <?Y r
-=2<%' t
0
G
5.4 Directed Graphs
5' 0 + 0À@ C.5 ) # <4'O$,$ P -/+&5$
+ # 4)&'DC*+&' # E );- P 5$ " P -/+&5 P
+;5' # 4)&'DC*+&' # $,12 E );- P 5 0 # 1C*' #
E ' 0 *' );-R< %e:'O$ 0 - $ +,);- E 5+ @ ) 6 -=) # 3g- 00 ':)
0 - # 4)&'DC*+&' # E );- P 5 G 7$ B ?=78D= 59(E( @ P %$
2 F V(P) 0 G I/
7 D3785 B 5
DA&7E ?
Problem 5.15 (Directed
Chordless
(s,
t)-Path
(
))
"(
'5 365 &'507
:D 4 D3(8< B 4507 B 50(
G
=
(V,
E)
k
s, t V
( ?=58705 B ?=7ED=F50(E( D<785 B 65
D
3";? A(
5; (8
<
(s, t)
k
G
8':<%< /6 $ ':+ -=< a $(5 =6 ' # +&5-=+ I/ 7$ / r C 3 P <4':+&'U'*Y!' 0 @ )&'D$(+,) 7CN+&' #
+ P <%- 0 -/) # E );- P 5$ 1) C 0 $(+,)&1C*+ 0 $ # 'D$,*C ) 42^' # -=2 Y!' CD- 0 O' -R$ %< F 2^'
- # - P +&' # + # 'O-=< 6 4+&5b+;5' # 4)&'DC*+&' # $,'*+,+ 0E -R$ 6 ':<%< Theorem 5.16
I/!3(
W[1]
B 5 5 4 ? 7859(+ 5 B 6
k
>0 +;5' 1) 0E 3g-C.5 0 ' @
5' )&':3 )&' P <7-RC:'J-=<%<sC 0 # 4+ A0 $ +;5-/+
+&5'X':l 7$ +&' 0 C*' A@ - 0 ' # E ' 2 F C ),)&'O$ P A0 # 0E C 0 # 4+ A0 $S)&' 1 4) 0E
+;5' P )&'D$,' 0 C:' A@ - # ?)&'OCN+&' # ' # E ' 43 %<%-=)&< F -=<%< C 0 # ?+ 0 $M)&' 1 4) 0E +&5'
-=2$(' 0 C*' A@ - 0 ' # E 'L-/);' );' P <7-RC:' # 2 F C ),)&'O$ P 0 # 0E C 0 # 4+ 0 $ # 7$,-R<4< /6 0E
2 +&5 # 4)&'DC*+&' # ' # E 'D$ 5'UC 0 $ +,);1CN+ 0 # 'O$,CN) ?2d' # 0 5' )&'D3 _ 7$X3 # ' # -$ @ <4< /6 $ n0 +&5' Y!':),+&':l C.5 %C:' # 7-=3 0 # $ # 4)&'OCN+X-R<4<' # E 'O$ @ ) 3 si + vi - 0 # @ ) 3
j
R
4
<
<
&
+
5
'
$
g
3
g
3
'*+,) %C C P F!
#
0
>
0
F
i + i @ )X-=<%<
v
t
1 i k
1 j n
Proof.
);' 1 ?);'
j
!a
Chapter 5. Chordless Paths
# ?)&'OCN+M-R<4< ' # E 'D$ @ ) 3 τi + ϕi - 0 # @ ) 3 ϕi + σi @K )-=<%< 1 i k - 0 #
-R<4< 1 j n 5'O$(' ) %' 0 +;-/j+ 0 $ 0 # 1C:' j- < 0 'O-/) ) # ':) (s1 , . . . , tk =
A0 V(G ) τk , . . . , σ 1 )
5' )&':3U- 0 0E ' # E 'D$ +&5-=+ 7$ +&5'bC 0 $ %$ +;' 0 C F 0 # ' P ' 0 # ' 0 C:' - 0 #
$,'*+X' # E 'D$ -=<%< -/)&' ) %' 0 +&' # @ ) 3 +&5' Y!'*),+;'*l`+&5-/+ %$ E )&'D-=+&'*) 6 ?+;5 )&'O$ P 'DCN+
+ +&5 7$ < 0 'D-/) ) # ':)X+ +;5' $,3g-R<4<%'*) Y!':),+&'*l U + 7$ 'O-R$ F + Y!':) @ F +&5-/+ 0
C.5 ) # <4'O$,$ # 4)&'DC*+&' # (s1 , σ1)r P -=+&5 C:- 0 1$(' - C 0 $ 7$ +&' 0 C F! 0 # ' P ' 0 # ' 0 C:' )X$,'*+ ' # E ' >0 @ -CN+ +&5' ) 4' 0 +.-/+ 0 %$XC;5 $,' 0 $(1C;5 +&5-/+ - 0F C.5 ) # <%'D$&$
# ?)&'OCN+&' # (s1, σ1 )r P -=+&5 0 G P -R$,$('O$ +&5) 1 E 5 tk = τk -=<4+&5 1 E 5 +&5 7$ %$
0 +L)&' 1 ?)&' # 2 F # E' 0 ?+ 0 0 C 0 +,);-$ +L+ G /IHKJ " $U- C A0 $(' 1' 0 C*' -=<7$ +&5' K@ %< < /6 0E P ) 2<4'D3 7$ W[1]r C 3 P %< '*+;' 6 ?+&5
);'D$ P 'DC*++ k 1$ + - ## -S$ 0E <4' # 4)&'DC*+&' # ' # E ' (σ1 , s1) + +&5'\C 0 $ +,)&1CN+ 0
# 'D$,C*) ?2d' # 0H
5' )&':3 7Das
7A D 3785 B 5
D & 78 ?
Problem
5.17
(Directed
Chordless
Cycle)
G = (V, E)
(
45 365 &'507
:D 4507 65
3( =
? 50785 B =
? 7ED=F50(8( D3785 B 5
D
k
s V
?'78 &?
3
B $ B 5 A(E 5; (0
k
s
G
L +&'o+&5-/+92 +&5 P ) 2<%':3U$-/);' P < FV0 3 7-=< @ +&5' P -/+;5 )C F C:<4' 7$ 0 +9);' r
1 4)&' # + 2d'HC.5 ) # <%'D$,$ -R$k+&5'h3U-/l %3 13 0 13q2d'*) k A@ Y!'*),+;'*l r # %$3 0 +
# ?)&'OCN+&' # (s, t)r P -/+&5$9C:- 0 2^' C 3 P 1+&' # 0 O(k|E|) + 43g' 1$ 0E /6 +&'OC.5 r
0 1'D$ n S/6 '*Y!'*) # 'DC # 0E 6 5':+&5'*)f+;5'*)&'k'*l %$ +o- # ?);'DCN+;' # (s , t )r
P -/+&5 - 0 # - # 4)&'OCN+&' # (s2, t2)r P -/+;5o+&5-=+c-=)&'BY!':),+&':l r # 7$3 0 + %$ / r C 3 1P <%'*1+;' ':Y!' 0h@ ) t = s - 0 # t = s >
1
2
2
1
" <7$ @ +&5' # E' 0 ?+ 0UA@ C.5 ) # <%'D$,$ %$\);':<7-/ls' # + -R<4< /6 (2-C&m r C*1+&+ 0E -=);C:$ 6 ?+&5 0 'O-RC.5 P -=+&5 I/ )&'D$(+,) 7CN+&' # + P <%- 0 -/) E );- P 5$ 7$ P < Fs0 3 %-R< ':Y!' 0 @ ) - 0 -/),2 ?+,).-/) F 21+ lV' # 0 13q2d':) A@ C;5 ) # <4'O$,$ (s, t)r P -=+&5$ a n
5'9'*l 7$ +&' 0 C*' A@ $(1C;5 -/);CD$ 7$ +&5'MCN)&1C %-R< # T '*);' 0 C:'92d'*+ 6 'D' 0 +;5' # 4)&'OCN+&' #
- 0 # +;5'X1 0 # 4)&'DC*+&' # P ) 2<%':3 0 - 0 1 0 # 4)&'DC*+&' # (s, t)r P -/+;5 P '*Y!':) F ' # E '
0 0E + 6 Y!':),+ 7C*'D$ +;5-/+ -=)&' 00sr - # -C*' 0 + -=< 0E
C:- 0 2^' 1$(' # -R$ P
$,5 ),+;C*1+ L
5-/+ %$ +&5' P )&'D$,' 0 C:' A@ - 0F (s, t)r P -/+&5 %3 P < %'D$9+&5' '*l 7$ +&' 0 C*'
A@ - C.5 ) # <%'D$,$ (s, t)r P -/+&5 JS/6 '*Y!'*) s6 ' 6 4<%< $(5 =6 2^'D< =6 +&5-/+- # 3 4+,+ 0E
2-RC;m r C:1+,+ 0E -/).C:$ # 'O$ 0 +hC;5- 0E '`+&5' P -=);-=3g'*+&) %C C 3 P <%'*l 4+ F A@ +&5'
P ) 2<4'D3 @ ) E ' 0 '*);-R< E );- P 5$ $%)& " $%
$% $' " $ #& ) 0&'$ $% ./
$') v *$%1$'& & ϕ $%$% #1
./
) $%) $ #1 $% $%
i
j
i
j
5.4. Directed Graphs
3I; ?
3 & 78 ?
( B 4 5
D
Definition 5.18
(s,
t)
P
G
=
(V,
E)
D $ 3( E( ?=7 650(0
3";?<
weakly
chordless
P
(s, t)
G[V(P)]
2$('*)&Y!'J+&5-/+B+;5'*)&' 7$ 0 # T *' )&' 0 C*'2d'*+ 6 D' ' 0 ;C 5 ) # <4'O$,$- 0 # 6 D' -=mV< F .C 5 ) # r
%< 'D$&$ 0 1 0 # ?)&'OCN+&' # E ).- P 5$ Z1+ 0 C 0 +&);-R$ +\+ I/ +&5' P )&'D$(' 0 C*' @ - # r
);'DCN+;' # 6 'D-=mV< F C.5 ) # <%'D$,$ (s, t)r P -=+&5 )\C;5 ) # <4'O$,$\C F C*<%'+&5) 1 E 5 s CD- 0 2d'
# 'DC # ' # 0 < 0 'D-=)[+ 43g' 2 F - 2);'D- # +&5 r );$ +f$('O-/);C.5 f/6 ':Y!'*) +&5' 2VY 1$
E ' 0 '*);-R< %eD-=+ A0 + $('*Y!':);-=< P -=+&5$ # 'E 0 ' # 2d':< /6 7$k- E - 0 W[1]r C 3 P <%'*+&' -R<?);'D- # F @ )L+ 6 P -/+&5$ 7 Problem 5.19"(Many
Weakly
Chordless
(s,
t)-Paths)
(E '5 35 &@5079(
:D
:D 4 D (8 3 B
G = ( (V,
E)
k
`
?=58785 !( 5 ; (8
4507 B 50(( B ?
V
U
V
k
D ( < 4 5
K $ B ?=7ED 59(E(
3";?@(
`
(s, t)
$
5
K $
?=7ED=F50(E(
; ?@( (
Theorem
5.20
(s,
t)
4 ? 7859(+ 5 B 6
7 $
k
` 2
D3785 B 5ED2&7E ?
'507 B 59(
s, (
t ? ;
G[U]
W[1]
B F5965
+ %$ C*<%'D-/) 5 /6 + - # - P + +&5' 1) 0E 3g-C.5 0 ' C A0 $ +,)&1C*+ A0 A@
+ 'O$ +;-=2< 7$(5H3g'D3q2d'*);$(5 P 0 W[1] 8 )o+&5' 5-/) # 0 'D$,$ P ) A@ C 0 $ # '*)X+&5' C:-R$,' ` = 2 5' C 0 $ +,)&1CN+ 0
# 'D$,C*) ?2d' # 0 5' )&':3 7$[3 # ' # -R$ @K <%< /6 $ 8 ?).$ + - ## - # 4)&'OCN+&' #
' # E ' @ ) 3 σ1 + s1 - 0 # <4':+ s = tk = τk - 0 # t = s1 >0 +;5' Y!':),+&'*l
C.5 7C*' # 7-=3 0 # $ # 4)&'OCN+-=<%< ' # E 'D$ @ ) 3 ti + vi - 0 # @ ) 3 vi + si V@K )S-=<%<
- 0 # -=<%< 1 j n ?m!' 6 %$(' 0 +&j5' $ F 3 3g':+,) 7CjkC P F # ?)&'OCN+
1-R<4< ' i 'O$ k ) 3
# E @ τi + ϕi - 0 # @ ) 3 ϕi + σi @ ) -=<%< 1 i k - 0 # -=<%<
j<4< 0 # ' ' 0 # ' 0 jC*' - 0 # $(':+ ' # E 'D$ 6 ?+&5 0 +&5' $,-=3g'
L
D
'
3
!
Y
g
'
R
P
1 j n
# %-R3 0 # C.5- 0 $(1C.5 +&5-/+o-=<%<B);':3U- 0 0E 0 # ' P ' 0 # ' 0 C*' ) $,'*+f' # E 'O$f-/)&'
2d':+ 6 ':' 0 vi - 0 # ϕp @ ) $ 3 ' 1 i, p k - 0 # 1 j, q n 4)&'DC*+
+;5 $('o' # E 'Oj$ @ ) 3 vqi + /6 -/) # $ ϕp J ':' 8 E 1)&' Ra @ )- 0 ':ls-R3 P <4' q
j
pJA0 $ # '*) (tk, s1)r P -/+&5$ P - 0 # Q 0 G $(1C.5 +&5-/+ G[V(P) V(Q)]
7$b- # 7$3 0 + 1 0 0 A@ 6 'D-=mV< F C.5 ) # <%'D$&$ (tk, s1)r P -/+&5$ 5' 6 - F +&5'
' # E 'D$S-=)&' # 4)&'DC*+&' # s0 ' A@ +&5' P -=+&5$ $&- F! P C 3g'O$JY 7- +&5'[Y!':),+&'*l C;5 7C*'
# %-R3 0 # $ - 0 # Y %$ ?+;$ +&5'`Y!'*)&+ %C:'D$ tk , sk, tk−1, . . . t1 , s1 0 ) # ':) 0
+;5' +&5':)L5- 0 # Q +,);-DY!':);$('D$+;5' $ F 3g3 ':+,) 7CfC P F - 0 # Y 7$ 4+;$+&5'fY!':),+ 7C*'O$
0g ) # '*) 2d' @ )&' 0 -R<4< F )&'O-RC.5 0E s1 Y %-+&5'L- ## ' #
τk , σk , τk−1 , . . . τ1 , σ1
' # E ' Z\'DCD-=1$(' +&5'*);' 3 1$ + 0 + 2d'U- 0F ' # E ' 2d'*+ 6 ':' 0 P - 0 # Q 6 'UC:- 0
-=) E 1' -R$ 0 5' );':3 +&5-=+[+&5' Y!'*)&+ %C:'D$ vi - 0 # ϕp Y 7$ 4+&' # 2 F P - 0 #
j0 +o$(':+ Aq
&
)
O
'
$
D
'
*
C
+
?Y
!
D
'
<
C
,
)
&
)
D
'
$
+
'
'
'
#
#
#
!
F
0
0
0
0
@ $ %e:' -/+f3 $(+ k 0
P
P
P
Q
G
Proof.
5' )&':3
!
Chapter 5. Chordless Paths
8 ) ` > 2 6 'U- ## ` − 2 - # # ?+ 0 -R<\Y!'*),+ 7C*'O$f+ G O' -RC.5 @ 6 5 7C;5 %$
C 00 'DC*+&' # + s1 # 4)&'DC*+&' # + /6 =- ) # $ s1 - 0 # tk # 4)&'OCN+&' # @ ) 3 tk 90 < F
v2
v1
1
s1
G
1
2
3
2
3
3
2
3
1
Figure 56:
v3
σ1
2
1
G
5 #
F5 (8 7E;<& ?=5 B (8 7 B 7
G
k
=
2
(
3(050 B $ 5
D&@50( 705 (E?= 4 9$ ( #% D %< 59( 3:D 5 "50:D 50 B 5
5
D&@59( 9$ D@K(E?=5
D %< 50( D(59 5
D&@50( 0$ D K 5ED%< 50( ?=5
'587 5 85 %( 3
< D B ;65 ?=5 B 797859(+" D 58 B 56 ?=5 '587
G
B 59( 78
7 5 4 5 ?=5 '587 B 59( 85 5ED
B 79785
G
1
(+ :D6
?=5 4507 65 3((E? @D 5
D D@7 v1
t = s1
Bibliography
i - 0 Am - " E -/) 6 -=< E - " < A0c Z ) 7$ " ) 0 Y - 0 # 125-R$(5 1) p - 0 1 7$ ?2 4< ?+ Fq] );- P 5$ 2d' L' P )&'D$(' 0 +;' # pJ 3 P -CN+&< F ( B 705965 5E D 4. _ a!_
12
$ 6 0
" %C.55 <%e:':) Q 7C.5-='D< ST 3g- 00 Z\'*+,+ 0 - P 'DC;ms3U- 00
- 0 # p $,-/2- +&5 ' E )&':' Z 1 0 # $ @ ) pJA0 $ +,);- 0 ' # i $('D1 # r
) %- 0E 1<7-/+ 0 $ 0 78 B ? :@D 5E R s s
$ 6 0 " %C.55 <%e:':) p <%':3g' 0 $ 1':3g':) - 0 # - 00 'O$
).-R$,$(':) ) %- 0sr
E 1<%-=+ 0 $ j 4+&5 1+ i 0 +&' # P - 00 0E );':'D$ 0 (8 78 B ( ?
7 (E?=
5E ^R V 78 5
E - " < A0cL ) # 5-=) S
- - E P -R<%- 0 - 0 # 125-R$(5 1) A0E 0sr
p ) $,$ 0E pJ0 E 1);-=+ 0 $ 0 +&5' i <7- 0 ' 0 78 B ? $K "( 5E cD VA a s
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