julio cesar silva araujo tese
TRANSCRIPT
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P4
P4
P4
N P
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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
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P4
L(p1, . . . , pq) b
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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)
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(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
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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
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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
+
+
+
+
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(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
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p4
+
+
+
P4
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p4
+
+
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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
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(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)}
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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}
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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
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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}
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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=
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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
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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
-
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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
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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
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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 )
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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
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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
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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)
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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
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+
+
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
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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
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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)
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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)
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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
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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
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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
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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
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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
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c(N55) =c(N50) = 2 IN5(T
2, w2, c) 3