Solution

G= (S, A)
u∈S
`[u]←0
π[u]←NIL
F←S
F6=∅
u∈F `[u]
v u F
`[v]←`[v]+1
π[v]←u
F←F\ {u}
π
5
71 2
3
4
68
31 2
4
6
5
c=s1, . . . , skG
{si, sj} |j−i|>1

G s G|S\{s}G
s G G0=G|S\{s}
u0, u1,...,uk=u0G s
G0G0s
u1=s s (u0, u2)∈A
G
G
x y 6=x
G− {x}G
x G T x H
T U H Q
G|S−Ux U
H
G
x y 6=x
G0=G|S−{x}G N G|N−{x}
y G0x
N G|Ny G y
G N N − {x}y
G0
d x
d≥2U d −1x
d−1
d a, b ∈U a 6=b a b
H H H a b
H c1
b a Q c2
a b
H Q
H Q H H U
(a, b) (a, b)∈A U
G x ∈S x
G|S−{x}H U
H H U z ∈H
G|U∪HG G|U∪HG
x u ∈U u
H

G s
f:S→ {1, . . . , n}
c=u, v1, . . . , vk, w ∀i∈ {1, . . . , k}
f(u)< f(vi)f(w)< f(vi)
c=u, v1,...,vk, w ∀i∈ {1,...,k}f(u)< f(vi)
f(w)< f(vi) max(f(w), f (u))
f(w)< f(u)w f (w)
`(vk)←`(vk)+1 u u
vku u vk
x(x, u)∈A(x, vk)/∈A r = max{j|(x, vj)∈A}
x, vr,...vk, w f(x)< f (w) max(f(w), f(u))
(x, w)/∈A(y, v`)
y∈ {x, w, v`0}
s u, v s f(u)< f (s)f(v)< f(s)
u, s, v (u, v)∈A s
s1,...,sn
siG|s1,...,si−1
k c G = (S, A)k c :S→
{1, . . . , k}(u, v)∈A c(u)6=c(v)k
k−1
u
k0,...,k −1u u
k+ 1 k+ 1