Quelques rappels sur la théorie des graphes

G G = (S, A)
S
A{si, sj} ∈ S2
{si, sj}sisj
sisjsi∈S Adj(si) = {sj∈
S, {si, sj} ∈ A}
1
2 34
5 6
G= (S, A)S={1,2,3,4,5,6}
A={{1,2},{1,5},{5,2},{3,6}}
A
S
A
G G = (S, A)
S
A(si, sj)∈S2
(si, sj)si→sjsi
sja= (si, sj)sisj
sjsisisj
si∈S Succ(si) = {sj∈S, (si, sj)∈A}si∈S
P red(si) = {sj∈S, (sj, si)∈A}
1
2
3
4
5
6
G= (S, A)S={1,2,3,4,5,6}
A={(1,2),(2,4),(2,5),(4,1),(4,4),(4,5),(5,4),(6,3)}
p
1

d(s) = |Adj(s)|
s d+(s)
s(s, v)v∈S1
d+(s) = |Succ(s)|
s d−(s)
s(v, s)v∈S1d−(s) = |P red(s)|
s d(s) = d+(s) + d−(s)
G= (S, A, ν) (S, A)
ν:A→Rν
∀(x, y)∈S2ν(x, y)=+∞(x, y)/∈A
1
(si, sj) (sj, si)si, sj∈S
{si, sj}si, sj∈S
Knn
K3K5
1
2
3
1 2
3
4
5
G= (S, A)
G G0S0
G G S0
G0= (S0, A0)S0⊂S A0={(x, y)∈A , x ∈S0y∈S0}.
G G0S0
G G S0
G0= (S0, A0)S0⊂S A0⊂ {(x, y)∈A , x ∈S0y∈S0}.
A0=A\ {(2,2),(3,2),(3,4)}

GG0
1
2
3
4 1
2
3
4
G S0={1,2,4}
GG0
1
2
3
4 1
2
4
G= (S, A)
G
G
k
1
2
3 4
5 6
7 8
G{1,5,7} {2,4,5}

G= (S, A)n S 1n
G T n S
si∈S T [si]sj
(si, sj)∈A{si, sj} ∈ A
{si, sj}sj
T[si]siT[sj]
G= (S, A)
1
2
3
4
1→
2→2 1
3→1 4 3 2
4→1 2 3
G
A(si, sj)sj
T[si]
{si, sj}si
T[sj]
n m O(n+m)
T[si]sj(si, sj){si, sj}
p p
T
G= (S, A)n S 1n
G M n ×n
M[i][j] = 1 (i, j)∈A M[i][j] = 0

%1 2 3 4
10000
21 1 0 0
31 1 1 1
41 1 1 0
n
O(n2)n2
n2n2
G= (S, A, ν)
W∈Mn(R)Wi,j =(∞(si, sj)/∈A
ν((si, sj)) (si, sj)∈A
A
B C
EF
G
−2
62
4
−1
8
9
2
−3
W=
∞6∞∞∞−2
∞∞∞∞∞ 2
∞4∞ ∞ −1∞
∞2 8 ∞9∞
∞ ∞ ∞ ∞ ∞ ∞
∞∞−3∞ ∞ ∞
G= (S, A)
u v < s0, s1, s2, . . . , sk>
u=s0v=sk(si−1, si)∈A i ∈ {1, . . . , k}
s0, s1, . . . , sk(s0, s1),(s1, s2),...,(sk−1, sk)
k
u v v u
< s0, s1, ..., sk> s0=skk≥1
s1, s2, . . . , sk
1