julio cesar silva araujo tese

8/21/2019 Julio Cesar Silva Araujo Tese

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9/208

P4

P4

P4

N P


10/208

P4

2

d

s

d

d= 1

gcd(n, d) 2

gcd(n, s) 2

n

d

n

s

d= s = 2

d= s = 3


11/208


12/208


13/208

P4

L(p1, . . . , pq) b


14/208

G = (V, E)

k

G

c : V

{1, . . . , k}

c

uv E

c(u) = c(v)

k c = (S1, . . . , S k)

V(G)

Si i {1, . . . , k} G k

k

G

(G)

k

G

k

(G)

G

G= (V, E)

=v1, . . . , vn

V(G)

vi(1 i n)

N(vi){v1, . . . , vi1}

G= (V, E)

(G)

G

V(G)

vi

G[{v1, . . . , vi1}] G

v1, . . . , vi1

P4

P5

P4

P4

G

P4 H G

|V(H)| 6

H

P4 H {C4, C4}

P4 P4

P5

G= (V, E)

w: V(G)

R+ k c = (S1, . . . , S k) G Si

w(Si) = maxvSiw(v) i {1, . . . , k} c

w(c) =k

i=1 w(Si)

G

w

(G, w)

w(G)

(G, w)

(G) =w(G)


15/208

(G, w)

w

(G)

G

k

k

k

k

k

w(G) k

G

k

P5

+

P4

P5

G P4 V

V(G)

|V| 5

G[V]

P4

P4

+

G= (V, E)

w : E(G) R+ t k (G, w)

k

c G

v V(G)

{u|uN(v) c(u)=c(v)}

w(uv) t.

t

k

(G, w)

(G, w)

t

t

(G, w)

t(G, w) k (G, w)

t

k

k

t

(G, w)

t

k

k

+

+

+

t

k

M

M


16/208

t

G

P

G

e E(G)

P

e E(G)

e

G(P) G(P) P

u, v

uv

G

G

l: E(G) R

G

u, v V(G)

{u, v}

G

G

G

G

C3

K2,3

C3

G


17/208

G

32 |V(G)|

32 G

G

|E(G)| = 32 |V(G)| 32 ABC {C3, K2,3}

ABC

d

Ed

S Ed

p1, p2 S p1 p2 S

S Ed

S

S

S V(G)

G

uv

u, v S

S

S

uv

S

P3

V(G)

G

P3 G

S

G

w V(G)

w S

w

P3 uv

u, v S

S

V(G)

G= (V, E)

I[u, v]

u, v V(G)

(u, v)

SV I[S] =

u,vSI[u, v]

S V

I[S] =S

SV

Ih[S]

S

S

S

G

Ih[S] =V G G

hn(G)

hn(G) k

G

k

+

+

+

+


18/208

(q, q 4)

H = (V(H), E(H))

V(H)

E(H) V(H)

E = (E, E+)

E

E+

E

H

H

H

H

v0E0 . . . vkEkv0

vi V(H) Ei E(H) vi Ei vi+1 ( k) E

+i i {1, . . . , k}

H


19/208

p4

+

+

+

P4


20/208

p4

+

+


21/208

P4

P4

G

G

G= (V, E)

= v1, . . . , vn V

vi

{v1, . . . , vi1}

G

V(G)

G

(G)

P5

P5

G

r

coNP

(G) (G) + r

(G) r (G)

(G) c (G)

(G)

G

P4 k

(q, q 4)

+

P4


22/208

(G) |V(G)| k

k

k G = (V, E)

k

(G) k

O(n2k1

)

P4

P5 P4

P4 P4

P4 P4 P4 P4 P4 P4

P4

P4

P4

G= (V, E)

S

V(G)

G[S]

G

S

NG(v) v G

N(v)

G

M V(G)

G

w

V\M

w

M w

V

{x}

x V

M

G

M

G

M M =

M M

M M

G

G

G

T(G)

G

T(G)

G

v

G

M(v)

r

T(G)

V(r) ={r1, . . . , rk}

r

T(G)

G[M(r)]

r

G[M(r1)], . . . , G[M(rk)] G[M(r)] r

G[M(r1)], . . . , G[M(rk)] G[M(r)]

G[M(r)]

G[M(r)]

r

{M(r1), . . . , M (rk)}


23/208

M(r)

G[M(r)]

G(r)

G[{v1, . . . , vk}] vi

M(ri) 1 i k r M(r)

G1 G2 S =

{s1, s2, s3, s4} K= {k1, k2, k3, k4} R= {r}

S

K

R

S

K

R

K

S

f : SK

s S

N(s) = f(s)

N(s) = K f(s)

R

G= (V =S K, E)

S

K

R

V = (S, K R)

G= (V =S K, E)

S(G) K(G)

R(G)

S

K

S(G)

K(G) K S(G)

R(G) =V\S(G) K(G)

{C5, C4, C4} {C4, C4}

G

S(G) = {s1, s2} K(G) =

{k1, k2} R(G) ={s3, s4, k3, k4}


24/208

P4

G P4 HG

|V(H)| 6

H

P4 H P4

G= (V, E)

T(G)

r

T(G)

r1, . . . , rk G

P4 G(r)

P5 P5 C5 M(ri) 1 i k

H = (SKR, E)

M(ri) 1 i k

R

H= (S K, E)

S

K

P4

G(r)

P5

P5 C5 M(ri) 1 i 5 M(r)

P4

P4 P4

P4

P4

P4

P4

P4 P4 P4

P4 P4 P4 P4 P4 P4

P4


25/208

P4

P4

G= (V, E)

P4

T(G)

T(G)

(G)

T(G)

G[M(v)]

v

T(G)

r

T(G)

r1, . . . , rk

G[M(r)]

(G[M(r)]) = (G[M(r1)]) +. . .+ (G[M(rk)]) r

G[M(r)]

(G[M(r)]) = max((G[M(r1)]), . . . , (G[M(rk)]))

G[M(r)]

r

T(G)

G

H

G

G

v V(G)

H

V(G) = {V(G)\{v}} V(H)

E(G) =

{E(G)\{uv | u NG(v)}} E(H) {uh | u NG(v) and h H}

G H1, . . . , H n V(G) ={v1, . . . , vn} G

viV(G) Hi 1 i n

G

(Hi) G

[V(Hi)]

c

G

c1, . . . , cp

G[V(Hk)] k {1, . . . , n} Si 1 i p

G[V(Hk)] ci ui Si c

ui w cj 1 j < i p

w G[V(Hk)] w /

G[V(Hk)] w V(G)\V(Hk) uj G[V(Hk)]

cj

uiw

ujw

c

w G[V(Hk)] c

G[V(Hk)] p G[V(Hk)]

p (G[Hk]) (Hk)

G = (H1 . . . H5, E)

G[V(Hi)]

Hi (G[Hi]) i i

G[Hi] (G[Hi]) i {1, . . . , 5}


26/208

H2 H3 H4 H5

P5

H1

H1

H2 H3

H4 H5

P5

H1

H2 H3

H4 H5

C5

H1, . . . , H 5

P5 = (H1 . . . H5, E)

S = (S1, . . . , S k)

P5 (P5 )

v Si u Sj j < ii, j {1, . . . , k} v (G) =

k

G

v H1 k (P

5 ) = 1+ 2

N(v) V(H1) V(H2) N(v)

S1, . . . , S k1 (P5 ) G[V(H1) V(H2)]

k = (P5 ) 1+ 2

V(P5 ) 1 2

P

5

1 + 2

v V(H5) k (P

5 ) = 4+ 5

v V(H2) k

(P5 ) =

1+ 2+ 3 , if 1 41+ 2 , if 1> 4 and3 s12+ 3+ 4 , if 1> 4 and3> s1

s1 = 1 4

N(v) V(H1) V(H2) V(H3) N(v)

S1, . . . , S k1 (P5 ) H1 H2 H3

k= (P5 ) 1+ 2+ 3

4 1 (P

5 ) = 1+ 2+ 3

V(H1) V(H4) V(H3)

V(H4) P5 4 1

3 2 P5

1+ 2+ 3

H4 H1 H3

H4 H1


27/208

P4

4 < 1 s1 = 1 4

3 s1 (P

5 ) = 1 + 2

V(P5 ) 1 4 3 2

1+ 2 H1 H4

4 H1 H4 s1

H1 3 H3

1 H1

H2 P5 2

H2

1+ 2

S = {S

1, . . . , S

p}

P

5

p > 1

+ 2

H1 H2 H3

i

H3 H1 H2

3+ 4 1 S

2+ 3+ 4+ 1

S

H1 H2 H3 4< 1

j

H1 H2 H3 H4

Si H3 j

Sj H1 i

3 > s1 (P

5 ) = 1 + 2 + 3 s1

H1 H4 H3

s1

H1

H3

V(P5 )

1 4 3 2

V(H1) V(H4) s1 H1

H4 s1

H3 H3 3 s1

V(H1) V(H2)

1+ 2+ 3 s1

(P5 ) 1+2+3s1

S

P5

1+

2+

3 s

1

3 s1 +1 H3 H1 H2 i

S

1+2+3s1+1 = 2+3+4+1

j

H1

H2 H3 H4 i j

v V(H4) k

(P5 ) =

5+ 4+ 3 , if 5 25+ 4 , if 5> 2 and3 s52+ 3+ 4 , if 5> 2 and3> s5


28/208

s5 = 5 2

v V(H3) k

(P5 ) =

2+ 3+ 4 , if 1 4 or 5 21+ 2+ 3 , if 1< 4, 5< 2 and2 s4 53+ 4+ 5 , if 1< 4, 5< 2 and2 s4< 5

s4 = 4 1

k V(H3)

(P5 ) 2+ 3+ 4

1 4 5 2 (P5 ) = 2+ 3+

4 1 4 1 4 2 3

5 2

5 2 4 3

P5 2+ 3+ 4

s2 = 2 5 1 < 4 5 < 2

1< 4 1 4 2 3

4+ 3+ 2 s4

5 < 2 5 2 4 3

2+ 3+ 4 s2

2 s4 5 (P5 ) = 2+ 3+ 4 s4

(P5 ) 4+ 3+ 2 s4

S

P5

4 + 3 + 2 s4

v V(H3) 2 s4+ 1 H2

H3 H4 5 < 2 s4+ 1

i

H2 H3 H4 H5

2+ 3+ 4 s4+ 1 = 1+ 2+ 3+ 1 j H4

H1 H2 H3

Si H2 j Sj H4

i

2 s4 < 5 (P

5 ) = 2+ 3+

4 s2 P5

H4

H1 H2

H2 H4 5

P5


29/208

P4

4

s2 1 (P5 ) = 4+ 3+ 2 s2

4 s2 1 2 s4 < 5

1, . . . , 5 (P

5 )

(P5 )

(P5 )

H1, . . . , H 5 P5 =

(H1 . . . H5, E)

S= (S1, . . . , S k) P5 (P5 )

v H1 k (P5 ) = 1+ 2+ 3

V(P5 )

1 2 3 1 +2 +3

H1 H2 H3

1+2+3

v V(H1)

v H2 k

(P5 ) =

1+ 2+ 3 , if 4 31+ 2+ 4 , if 4> 3 and1 52+ 4+ 5 , if 4> 3, 1> 5 and4 s1 31+ 2+ 3 , if 4> 3, 1> 5 and4 s1< 3

s1= 1 5

4 3 (P5 ) = 1 +2 +3

(P5 ) 1+ 2+ 3 V(P5 )

1 2 3

1+ 2+ 3

S= {S1, . . . , S

p} P

5 p 1 + 2 +3 + 1

i

Si V(H4) 4 3

S

1+ 2+ 4+ 1

j

Sj V(H3) H4 i

H3 j Si

Sj S


30/208

4 > 3 s4 = 4 3

5 1 (P5 ) = 1 + 2 + 4

5 1 (P5 ) 1 +2 +4

P5 5 1 4 2

1+ 2+ 4

P5 S

= {S1, . . . , S

p} p 1+ 2+ 4+ 1

v V(H2)

i

Si V(H3) S

i V(H3) =

Si V(H5) = 4 > 3 S

1 + 2+ 3+ 2

Sj S

l S

j S

l V(H4)

S Sj Sl i

5< 1 4 s1 3

(P5 ) = 1+ 2+ 4 s1 4> 3

5< 1 V(P5 ) 1 5 4 2

1+ 2+ 4 s1

S = {S1, . . . , S

p}

P5 p 1 + 2 + 4 s1 + 1 v V(H2)

4 s1 + 1 H3 H4

4 s1 3 i 4 s1 + 1 H4 s1= 1 5 S

1+ 2+ 4 s1+1 = 2+ 4+ 5+ 1

v V(H2) j

H1 H3 S

Si S

j

5 < 1 4 s1 < 3 (P5 ) =

1+ 2+ 4 s4 4 s1 < 3

1 s4 > 5

1 s4 > 5 (P5 ) = 1+ 2+ 4 s4 (P5 ) 1+ 2+ 4 s4 V(P5 )

4 3 1 2

1+ 2+ 4 s4

S= {S1, . . . , S

p}

P5 p 1 + 2 + 4 s4 + 1 v V(H2)

4 s4+ 1 H1 H3 H5

1 s4 >5 i H1 H3

S

1 + 2 + 4 s4 +1 = 1 + 2 + 3 + 1

v V(H2) j S

j V(H4)


31/208

P4

v V(H3) k

(P5 ) =

1+ 3+ 2 , if 5 21+ 3+ 5 , if 5> 2 and1 43+ 4+ 5 , if 5> 2, 1> 4 and5 s1 21+ 2+ 3 , if 5> 2, 1> 4 and5 s1< 2

s1= 1 4

s5 = 5 2

s4

v V(H4) k

(P5 ) =

2+ 4+ 5 , if 1 54+ 5 , if 1< 5 and s5 21+ 2+ 4 , if 1< 5 and s5< 2

s5= 5 1

2+ 4+ 5

1 5 (P5 ) = 2+ 4+ 5

V(P5 ) 1 5 2 4

P5 2+ 4+ 5

1 < 5

s5 2 (P5 ) = 4+ 5

V(P5 )

4+ 5

S = {S1, . . . , S

p} P

5 p 4 +5 +1

v V(H4) i

H2 H4 H5 s5 2 S 1+ 2+ 4+ 1

1+ 1 H5 H2 H4

j

1+ 1

Sj V(H5) Si S

j

S

1 < 5 s5 < 2

(P5 ) = 4+ 5+ 2 s5

P5 4+ 5+ 2 s5


32/208

S = {S1, . . . , S p} p 4+ 5+ 2 s5+ 1

v V(H4)

2 s5+ 1 H2 i

4+ 5+ 2 s5+ 1 = 4+ 2+ 1+ 1

j

H5 Si S

j

S

v H5 k

(P5 ) =

3+ 5+ 4 , if 1 45+ 4 , if 1< 4 and s4 31+ 3+ 5 , if 1< 4 and s4< 3

s4 = 4 1

P5

1, . . . , 5

H1, . . . , H 5 C

5 =

(H1 . . . H5, E)

S= (S1, . . . , S k) C

5 (C

5 )

v V(H1)

k

(C5 ) =

1+ 2+ 3 , if 5 2 or 4 31+ 2+ 4 , if 5< 2, 4< 3 and2 s3 51+ 3+ 5 , if 5< 2, 4< 3 and2 s3< 5

s3= 3 4

v V(H1) (C5 ) 1+ 2+ 3

5 2 or 4 3 (C

5 ) = 1+ 2+ 3

5 2

V(C5 )

5

2 3 1

1+ 2+ 3

4 3 V(C5 ) 4 3 2

1 C

5

1+ 2+ 3

5 < 2 and 4 < 3

s2 = 2 5 2 s3 5

(C5 ) = 1 + 2+ 3 s3 V(C

5 )

3 4 2 1

C5 1+ 2+ 3 s3


33/208

P4

S = {S1, . . . , S p}

C5 p 1+ 2+ 3 s3+ 1 v V(H1) 2 s3 + 1 H2

H1 H3 i H5 2 s3 5

1+ 2+ 3 s3+ 1 = 1+ 2+ 4+ 1

4+ 1 H3 H1 H2

j

H4 S

Si S

j

2 s3 < 5 (C5 ) = 1+

2+ 3 s2 2 s3 < 5 3 s2 > 4 3 s2 > 4 3 s24

3 s

2

4

(C5

) = 1

+ 2

+ 3

s2

C5 1, . . . , 5

G

P4 T(G)

v

T(G)

G(v)

H = (S(H) K(H) R(H), E)

(G[R])

G[M(v)]

H

S(H)

K(H)

R(H)

O(V(H))

S = (S1, . . . , S k)

G[M(v)]

S(H)

K(H)

S(H)

K(H)

G[M(v)]

S(H)

K(H)

G[M(v)]

V(S(H))V(K(H))

R(H)

K(H)

S(H)

M(v)

i

j

S(H)

Si Sj S

(H)

S(H)

Si

Sj S

(H)

R(H)

(M(v)) |K(H)|+ 1

V(M(v))

S(H)

K(H)

|K(H)| + 1

K(H)

S(H)

R(H)

G[M(v)]

S

R(H)

K(H)

j

R(H)

i

Si S

(H)

Si


34/208

Sj S

(M(v)) =|K(H)| + (R(H))

(M(v))

(M(v)) =

|K(H)| + (R(H))

R(H)=

|K(H)| + 1

v

T(G)

G(v)

H= (S K R, E)

fr R f2

(R)

G[M(fr)] (G[M(v)])

S = (S1, . . . , S k) G[M(v)] (G[M(v)])

f2 f2 S

f2 K

G[M(v)]

S K

R

S(G)

K(G)

R(G)

x

w

f2 S

K

G[M(v)]

S

K

f2 S x w

G[M(v)]

S

i

j

Si Sj S

i

j

x

w

x Si w Sj x w

y K

x

w

y Sl

(K R) Sl = {y} u S u

K R

i

j

i

S

w

j

(S K) Sj ={w}

w

y

l

j

S

G[M(v)]

S

f2 K x w

i

j

x Si

w Sj x w

w

i

x


35/208

j

G[M(v)]

G = (V, E)

P4 |V| = n

(G)

O(n3)

(G)

G

T(G)

O(n2)

G

P4

P4 P4 P4 P4 P4

P4 P4 P4 (G)

P4

P4

P4

P5

P5

N P

P4

P4

P4

P4 P4

P4


36/208


37/208

P4

G = (V, E)

w : V(G) R+

G

k

c = (S1, . . . , S k) G

Si

w(Si) = maxvSi

w(v),

i {1, . . . , k}.

c

w(c) =k

i=1

w(Si).

G

w

(G, w)

w(G)

(G, w)

(G, w)

(G)

(G, w)

(G)

4114

P4


38/208

w: V(G) {1}

(G) =w(G)

N P

+

w(G) G

w

G

P5 +

G

+

w(G) k

P4

k

G

(G) =k

k

k

k

a

b

G

a

b

a b

NG(a) NG(b)

G= (V, E)

u, v

G

uv E(G)

Kn n

k

k

k

G1 G2 k a1b1 E(G1)

a2b2 E(G2) G G1 G2 a1b1

a2b2 a1 a2 b1b2 k

G

k

a

b

G

a

b

k

(G) k

G

k


39/208

G1

b2b1

a2a1

G2 Gb2b1

a1 a2

k

k

k

H k

(H) k

I

H (I)< k G

(G) k

G

k

H

k

k

G = (V , E , w)

G= (V, E)

w: V R+

G = (V , E , w)

n= |V(G)|

vV w(v) =k K

kn

G = (V , E , w)

c

G

i

c

repc(i)

v V w(v) w(x) x V c(x) =

c(v) = i

G= (V , E , w)

H = (V, E, w)

H G

H

G

V V

E E

v V

w(v) w(v)

k

iNKki k


40/208

G1 G2 k

a1b1 E(G1) a2b2 E(G2) G

G1 G2 a1b1 a2b2 a1 a2 a1 a2

w(a1 a2) = max{w(a1), w(a2)} b1b2

k

G

k

a

b

G

a

b

a b

w(a b) =

{w(a), w(b)}

k

G= (V , E , w)

k

w(G) k G k H

w(G) k G k

H

G =

(V , E , w)

w(G) k G

k

G

u, v

G = G+ uv

k

G

p

P

V(G)

Ci c G

Ci P

Ci Cj P

V(G)

w(repc(i)) w(repc(j))

c

c

Ci Cj

w(G) w(repc(j)) c

Kk

p

w(G) k

p N

G

k

G

G

a

b

c

ab, bc /E(G)

ac E(G)

G1= G + ab G2= G + bc

G

G1 G2 k H1

H2 ab bc H1

H2

H1 H2 ab H1 bc H2

b

H1

H2


41/208

G

G

k

G k H

w(G) k w(G) w(H)

w(H) k

H

H

Kki K

ki i N

k

k

H

a

b

k

H

a b

H

k

w(H)< k

c

H

c

H

c

a

b

a b c H

c

i

a b

j

repc(j) =repc(j) i repc(i)

a b

a

b

c

c

k

w(H

) k

H

k

H1

H2 (a1, b1) (a2, b2) H1

H2 a1 a2 H

a1 a2 (H)< k (H1) k (H2) k

c

H

c(a1 a2)=c(b1)

c(a1 a2) = c(b2) b1 b2

c(a1 a2) = c(b1) c

c

H1

a1 a1 a2 Cj c

repc(j) repc(j) a1

a1 a1 a2 w(c) w(c)< k

w(H1) k

k

k

u

v

u

v

k

k


42/208

P4

P5

P4

P4

P5

G

P4 V(G)

P4

P4

P4

P4

P4

G

P4

T(G)

H= (S K R, E)

P4

P4

G = (V, E)

P4 w : V(G) R

+ T(G)

w(G) T(G)

T(G)

G[M(v)]

v

T(G)

v


43/208

P4

G[M(v)]

v T(G)

P4

v T(G)

v1, . . . , vp

wv(G[M(v)]) =p

i=1 wvi (G[M(vi)]) wv w

v

G[M(v)]

(G[M(vi)], wvi) i {1, . . . , p}

T(G)

P4

G= (V =S K R, E)

S

K

R

G

G

c

c

R

(G[R], wR) (G, w)

w : V(G) R+ wR

w

R

H G

cG = {S1, . . . , S k} cH = {S

1, . . . , S

l}

G

H

cG cH

f :{1, . . . , l} {1, . . . , k}

Si Sf(i) i {1, . . . , l}

K

k

c = {S1, . . . , S k} G

k

K

u K

v R

uv E(G)

Si c

K

R

CK(c)

CR(c)

Sj K= Sj R= Sj C K(c) Sj C R(c)

CK(c)

CR(c)

K

R

c

S

S

SL S


44/208

s S

s

u K R

w(u) w(s)

c

G[V(G)\SL]

c G s SL c(u) u KR

w(u) w(s)

SL

w(c) = w(c)

G[V(G)\SL] G c

G[V(G)\SL] c

G

s S

u K R

su /E(G)

w(s)> w(u)

(G, w)

c = {S1, . . . , S k}

(G, w)

R=

Si c

Si /C K(c)

c= C K(c) CR(c)

Sj c S R

Sj r

R Sj w(r) = maxrR w(r)

R =

Si Sj c

S

c

c

Si

Sj w(c)< w(c)

R = Si S R= Sj =Si Sj C R(c)

Si Sj c

w(c)< w(c)

c

CR(c)

Sj S

j Sj S

j

S

w(Sj)

w(Sj) Sj

Sj S c

c

S Sj Sj w(c)

c

Sj c S R R=

c

G

r

R

S Sj r

c

w(c)< w(c)

S

cS(c)

CK(c)

c

R=


45/208

P4

R=

cR (G[R], wR)

c

(G, w)

cR wR

w R

cR = {S

r1, . . . , S

rt } r

w(r) = maxrR w(r) Sr1

c = {S1, . . . , S y} G c =

CK(c) CR(c) CK(c) = {S1, . . . , S x}

CR(c) ={Sx+1, . . . , S y} x {1, . . . , y} Sx+1

CR(c)

S

c

c

cR c = {S1, . . . , S x+t} G Si =S

i i {1, . . . , x} Sx+1 = S

r1 {S

x+1\R} Sx+j =S

rj

j {2, . . . , t}

cR G[R] c

G

c

c

CK(c)

CK(c) w(Sx+1) = w(S

x+1)

Sx+2, . . . , S x+t cR (G[R], wR)

w(c) w(c)

c

cR

K

R

K

R

(G[R], wR)

(G, w)

S

S

S=

{s1, . . . , sm} w(s1) . . . w(sm) S(j)

S(j)+

{s1, . . . , sj1} {sj, . . . , sm} k r K

R

k

K

G

s

k

ki si ki K ki=k

G

c

G

S

j {1, . . . , m + 1} c

S(j)

cS(c)

R=

r

S(j)+

K

G

S(j)+\{s}

k

s S(j)

k

ki K S(j)

+\{si} ki=k

si S(j)

+

k

c

G

j 1

S

r


46/208

cS(c)

j 1 = 0

c

c

w(c)w(c)

s1, . . . , sj1 sj

c

G

sj, . . . , sm K G

c = c

G

c

c

c(sm) = c(k) c c

{sj, . . . , sm}\{s

}

c(k) = c(sm)

{sj, . . . , sm}\{s}

c(k

)

c

w(c

(k

)) = w(c(k

))

s

m

S

c(sm) = c(k)

c

c

w(c) w(c)

s / {sj, . . . , sm} c(s) =c(k)

c

sj

k

c

c(sm) = c(k) sm = s

c

sm k

c

sm S

c(sm) =c(k)

sm= s c(s) =c(sm) =c(ki) ki=k

c

c

{sj, . . . , sm}\{si} c(sm) si {sj, . . . , sm}

k

w(c) w(c)

G= (S K R, E)

w: V(G) R+

cR (G[R], wR) wR

w R

(G, w)

O(n3)

cR

R

c

G

c

c

cR

S

G

cR G[R]

cR

S

O(n log n)

K

R

cR


47/208

P4

G= (S K R, E) cR G[R] G

m |S| s0 sm+1 S w(s0). . .w(sm+1) k k r c, c rR

c(r) :=cR(r)

kK c

(k) := |K| K

j = 1, . . . , m + 1

i= 0, . . . , j 1 R=

c(si)c(r)

c(si)cS

G i= j, . . . , m

c(si) K c(f(si)) w(c)< w(c)

cc

i= j, . . . , m (si, k

) /E(G) c(si)c(k)

c(si)c(k)

w(c)< w(c) cc

kiK\{k} i= j, . . . , m

(si, ki) /E(G) c(si)

c(ki)

c(si)c(k)

w(c)< w(c)

cc

c


48/208

G[R]

S

O(n3)

(G, w)

w : V(G) R+ P4

T(G)

T(G)

v

M(v)

(G, w)

O(n3)

G

T(G)

T(G)

w(G[M(v)])

v

G[M(v)]

O(n2)

T(G)

w(G[M(v)]) v T(G)

P4

v

T(G)

P4

G

M(v)

P4

M(v)

M(v)

P4

P4

G= A B

cA ={S1, . . . , S 4} A

S1 ={k1} S2 ={k2} S3 ={k3} S4 ={s1, s2, s3}

cB = {S

1, S

2, S

3} B 6 S

1 = {u1}

S2= {u2} S3= {u3}

G

cA cB

G

7

cG G 6

S1 = {s1, k1, u1} S2 = {s2, k2, u2} S3 = {s3, k3, u3}

cG G A

A

cA


49/208

P4

s

u

3

1

12

k1

s1

s2 k2

k3

u2

u3

2

22

2

2 1

1

A B

c

(G, w)

i

c G

i

c

G

(G, w)

w: V(G)

R+ G= G1 . . . Gm Gi G

c= {S1, . . . , S k} (G, w) cGi = {S

i1, . . . , S

ik} Gi S

ij = V(Gi) Sj j {1, . . . , k}

w(Si1) . . . w(Sik) i {1, . . . , m}

c= {S1, . . . , S

k} k G

c= {S1, . . . , S k} c

w(c) w(c)

c

c

w(S1) . . . w(S

k) S

ij =V(Gi)

Sj j {1, . . . , k} i {1, . . . , m} S

j j

(G, w)

Sij j Gi cGi

= {Sij | j

{1, . . . , k}}

cGi Rij

j

{Sij |j {1, . . . , k}} i {1, . . . , m}

j {1, . . . , k}

w(Sj ) maxi{1,...,m}

w(Rij)

j = 1

S1

G

max{w(R11), . . . , w(R

m1 )}

j >2

w(Sj)< max{w(R

1j ), . . . , w(R

mj )}

max{w(R1j ), . . . , w(R

mj )}= w(R

1j )


50/208

w(Sj)< w(R1

j ) w(R1

j1) . . . w(R1

p) . . . w(R1

1).

p = 1, . . . , j

Sqp {S

1, . . . , S

k} c R

1p

Sqp

w(R1p) w(Sqp), p= 1, . . . , j .

w(Sj)< w(Sqp)

p= 1, . . . , j

j

w(Sj) c

w(S1) . . . w(Sk)

c

S1, . . . , S k

Sj :=R1

j . . . Rmj , j = 1, . . . , k .

c

P4

P4

P4

G

=

{S1, . . . , S q} G q 1 |Si| 2 i {1, . . . , q }

f :q

i=1 Si V q

i=1 Si

Ki = {z | z = f(s) for some s Si} i {1, . . . , q }

A

P4 G

i {1, . . . , q }

x, y Si A= {x,y ,f (x), f(y)}

S=q

i=1 Si K= V q

i=1 Si

S

K

P4


51/208

P4

HG

w(H) w(G) G P4

G

(, f)

G[S]

G[K]

G[S]

G[K]

w(G[S]) w(G) w(G[K]) w(G)

(G, w)

k

P4

P4

k

w(T) T


52/208


53/208

k

c

l

|{v N(u) | c(v) = c(u)}| l

u V

l

l

G

l(G) k G l

k

G

l

l(G) l= 0

0(G) =(G)

(G, w)

G= (V, E)

w: E R+

(G, w)

c

G

u

Iu(G,w,c) =

{vN(u)|c(v)=c(u)}w(u, v).

t

c

t

k

(G, w)

c

k

G

Iu(G,w,c) t

u V

t R+ k G

t

k

t

(G, w)

t(G, w) (G, w)


54/208

t R+ t(G, w)

t= 0

0(G, w) =(G) w(e) = 1 e E

l(G, w) =l(G) l

k

k

(G, w)

Tk(G, w) t (G, w)

t

k

(G, w)

k

Tk(G, w)

l(G) k

l 2

k 2

G

t

G

t 2

+

t

+

1

i

i

+

L(p1, . . . , pd)

G

i

pi i= 1, . . . , d


55/208

+

+

t(G, w) Tk(G, w)

t(G, w)

M

M

t

t

+

(G, w)

w :

E(G) Q+ v V(G) dw(v) =uN(v) w(u, v) G (G, w) = maxvVdw(v)

k

c : V {1, . . . , k}

G

v

V(G)

i= 1, . . . , k diw,c(v) =

{uN(v)|c(u)=i}(u, v) d

c(v)w,c(v) =

Iv(G,w,c) k c G w c

v V(G)

Iv(G,w,c) d

jw,c(v) j= 1, . . . , k

gcd(w)

w

gcd(w)> 0


56/208

gcd

Q

x

y

y/x

gcd(w)

gcd(w)

v

u

diw,c(v)> d

jw,c(u) diw,c(v) d

jw,c(u) + gcd(w)

t

gcd(w)

a Z

a gcd(w)< t

djw,c(v) v j

W

W

djw,c(v) v

j

v

diw,c(v)

i

W

diw,c(v) d

jw,c(v) gcd(w) W |E| maxeEw(e)

w

k

G

w

t

k

(G, w) (G) (G) + 1 G

w

l(G) (G)+1

l+1

(G, w)

w : E(G) Q+ t

gcd(w)

t(G, w)

(G, w) + gcd(w)

t + gcd(w)

.

t

G

t(G, ) = 1

t(G, )> 1

w

k

c

G

dw(v) =

uN(v)w(u, v) kdc(v)w,c(v).

k =

(G,w)+gcd(w)

t+gcd(w)

2

c

w

k

G

c

t

k

(G, w)

v

G

c(v) =i

diw,c(v) > t c

w

djw,c(v) > t j = 1, . . . , k


57/208

gcd(w)

t

gcd(w)

djw,c(v) t +gcd(w)

j= 1, . . . , k

(G, w) dw(v) k(t + gcd(w)),

(G, w) (G, w) + gcd(w),

G (G) = (G) + 1

G

G

w

k

(G, w)

k = 1

T1(G, w) = (G, w)

w

(G, w)

w: E(G) R+ k

Tk(G, w)(G, w)

k .

c

w

k

G

v V(G)

kTk(G, w) kdc(v)w,c(v) dw(v) =

uN(v)

w(u, v) (G, w)

T1(G, w) = (G, w) kTk(G, w) . . .

T1(G, w)

(G, w)

w: E(G) R+ k

p

Tkp(G, w)Tp(G, w)

k .

t= Tp(G, w) c t p (G, w) i= 1, . . . , p

Gi G i c

(Gi, w) t 1 i p (Gi, w)

t/k

k

ci {(i 1)k + 1, . . . , i k}

ci t/k kp (G, w)


58/208

kp

p

Gi k

kp

Gi

ki

pi=1 ki = kp T kp

(G, w)

T= max{ (Gi,w)ki , 1 i p}

k

k 1

(G, w)

v1, . . . , vn G

G

vivj w vivj

w(vivj)< w(vivj) w(vivj) =w(vivj) j < j

Ekw(v) min{|N(v)|, k 1}

v

Ekw(v) Ek(v) w

Gkw V(G)

vV(G) E

kw(v)

Ekw(v) min{|N(v)|, k 1}

v

v

v

Gkw G

k 1

(G, w) Gkw

k 1

k

x

xy

E(Gkw) \ E

kw(x) E

kw(y) \ E

kw(x)

(x1, . . . , xr) 1 i r xixi+1 Ekw(xi+1)\ E

kw(xi) xr+1 = x1 x1x2 w x2x3 w w

xrx1 w x1x2 w(x1x2) = w(x2x3) = = w(xrx1) =

w(x1x2) m xm

v1, . . . , vn j j

xm = vj xm+2 = vj

m

j > j

xmxm+1 wxm+1xm+2

(G, w)

Gkw k

Gkw x

k1

Gkw x (Gx)

kw

(G x)kw k k

Gkw x k Gkw

x

k 1

(G, w)

Tk(G, w) (G\E(G

kw), w)


59/208

11

u v

uK Kv

w(u,v)=w(u,v)1

u v

G

G

uv E(G)

k = 4

M

(G, w)

w

k

(G, w)

w(e) {1, M}

e E(G)

Tk(G, w) =

Tk(G, w)

M= 1 +

eE(G) w(e)

f(G, w) =

{eE(G)|w(e)=M}(w(e) 1).

f(G, w) = 0

M

G

G= G

G

w

Tk(G, w) =Tk(G, w) f(G, w) =f(G, w) 1

f(G, w)

(G, w)

f(G, w)> 0 e= uv E(G) 2 w(e) 1


70/208

V0

2

3

4

V0 2 N0 N1 N2

3

N05, N00, N01 N11, N12 N22 N23

IN1(T2, w2, c)> 1 N12 N22 N23

N01 N11, N12 N22 N23

IN2(T

2, w2, c)> 1

V0 2 N0 N2

N50 N00 N01 N12 N22 N23

IN0(T

2, w2, c) > 1 IN2(T2, w2, c) > 1 N3 N5

N3 IN3(T

2, w2, c) > 1

c(N3) = 3 N4 N11 N33 N34

IN2(T2, w2, c) > 1 N12 N22 N23

N33 N34

N3 N22 N11

N1

N00 N01 N11 N12 N22

c(N1) = 1 N4

c(N4) = 4 c(N5) = 3 c(N23) = 4

V0 N2 N3 N33 N34

V0 N3 N4 N23

N33

V0 2 N0 N3

N50 N00 N01 N23 N33 N34

IN0(T2, w2, c)> 1 IN3(T

2, w2, c)> 1 N1

N12 N22 N44 N45

IN3(T

2, w2, c)> 1 N44 N45

N23 N33 N34 N44 N45

N4

T

V0

V0


71/208

V0 c(N0) = c(N2) =

c(N4) = 2 c(N1) = 3

N01 N11 N12 N1

N01 N11 N12 N23 N33 N34

N3 N1 N45 N55 N50

IV0(T

2, w2, c)> 1

V0

Ni Ni+2 0 i 3 c(N0) =

c(N2) = 2 c(N1) = 3 c(N3) = c(N5) = 4 c(N4) = 3

c(N50) = 1 3 c(N01) = 1 4 c(N50) = 3 c(N01) = 4 c(N00) = 1

N50 N00 N01 N12 N22 N23

IV0(T

2, w2, c) 1 c(N34) =c(N45) = 2

N44

c(N0) =c(N3) = 2 c(N1) =c(N4) = 3 c(N2) =c(N5) = 4

{N50 N00 N01} {N12 N22 N23} {N34 N44 N45}

IV0(T

2, w2, c)> 1

(T2, w2) 0 j 4

Cj ={(j, 0) + a(5f1) + b(2f1 + f2)| a, b Z} 0 j 4 j + 1

Cj

(T2, w2)

0 j 3

Dj = {(j, 0) + a(4f1) + b(f1+ 2f2) | a, b Z}

D0 D1 D2

D3 0 j 3 D

j ={(j, 1)+a(4f1)+b(f1 +2f2)| a, b Z}

0 j 3

j+ 1

Dj

(T2, w2)

c

(T2, w2)

c(V0) = 1

v

Iv(T

2, w2, c) 2.5

V0

N0

V0

N1 N2 N3


72/208

c(N1) = 1

N2 N3 N4 N5

c(N2) = c(N3) = c(N4) = c(N5) = 2 c(N12) =

c(N50) = 3 N3 N4

N22 N23 N33 N34 N44 N45 N55

V0

c(N22) = c(N23) = c(N33) = 3 N23

N3 N223 N233 N333 N334 N34

c(N2) = c(N3) = c(N4) = 2 c(N5) = 3

N12

N12 N2 N3

N44 V0

N22 N23 N33 N34 N23 N33

N222 N223 N233

N333 N334 N44

N33 V0 N23

N34 N45 N34 N233 N333 N334

N33 N34

c(N45) = 1

V0 N22 N23

N34 N22 N23

N45 N334 N344

N3 N33 N44

N233 N333

c(N12) = 3

c(N22) = 1 V0

N23 N33 N34 V0

N44 IN34(T

2, w2, c)3 V0 c(N44) = 2 N4

c(N45) =c(N55) =c(N50) = 3 IN5(T

2, w2, c) 3

c(N22) = 2 N2 N3

N33 N34 N44 N45 V0

c(N01) = c(N11) = c(N23) = 3 N112 N122

IN12(T

2, w2, c) 3 N2 c(N112) =c(N122) = 1 IN1(T

2, w2, c) 3


73/208

c(N22) = 3

N23 N33 N34

V0 N44 N4

N45 N55 N50 c(N44) = 3

N45 N3 N4 N55 N50

N5 N34 N23

N33

N23 N223 N233 N333 N334

N45 c(N55) = 2 N23

N33 N34 N34

N44 N50 N4

N334

N344

N444 N445

c(N23) =c(N33) = 3 c(N34) = 1 c(N11) = 2

N01 N2 N22

N112 N122 N222 N223 N233

N01 N50

N5

N4 N344 N444 N445 N455

N4 N44 N45

c(N2) = c(N3) = c(N5) = 2

c(N4) = 3

N12 N11 N01 N00 N50

c(N12) = 2 N2 N01

N11

N34 N22 N23 N33

V0 N2 N23

N122 N222 N223 N233

N333 N3

c(N44) =c(N45) = 3 N4

N34

N2

N44

N45

V0 N22

N23 N33 N44

N45

N3 N4

c(N44) = 2 c(N45) = 3 N23 N33

V0 N4

N55 N50 IN5(T

2, w2, c) 3

c(N44) = 3 c(N45) = 2 N33

V0 c(N23) = 3 N4


74/208

c(N55) =c(N50) = 2 IN5(T

2, w2, c) 3

julio cesar silva araujo tese

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