G = (V, E) ᒺ, (V, E), V , E ⊆ V×V .
, . G V(G), E(G). ʳ n(G) = |V(G)|, m(G) = |E(G)|.
ʳ n(G) .
e = (v, w) ∈ E(G), :
v w ;
v w e;
e v w.
E . - ∅. V , E. .
璺 , .
E (v, v).
г ( 璺 ), .
, 璺 , .
, .
, .
, ( ) .
, . , g = (xi, xj): xi - , a xj - . .
, , , . , .
U(X), (xi, xj) xi, xj Î X, i ¹ j. .
U0(X) , - ' .
, , , . .
:
― S = S = (g1, g2,..., gn), gk gk-1, - gk+1. . , , . . , .
, i, . , 㳺 .
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:
G (X) (gl, g2, ), . , .
, . , () .
L(s) s. L (s) = ¥.
, , .
, .
, " xi, xj , xi Î G(xj) Þ xj Î G(xi), xi, xj ' .
, " xi, xj xi Î G(xj) Þ xj Ï G(xi), ' .
() G(X), () G(X) () G(X), () Ì G(x) " Î X.
(). ³ .
³, - G(X) . () G(X) , () G(X).
ϳ GA(A) G(X), Ì X, , Ì X, G, .
, , ' . , GA(A) G(X), Ì X GA(x) = G(x)Ç " Î .
= X, GA(A) = G(X). = {} GA (a) . ϳ GA(A) G (X) -, Ì X .
ϳ .
(), Ì X G(X) , G(X), . , () G(X), Ì X () Ì G(x) Ç A " Î .
() G(X) , G(X), () G(X).
, , 璺 . , - , .
, , . , (), .
( , ) , , .
.
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.
.
G(X) 0 Î X, i ¹ 0 (i Î X) , 0 .
G(X) ij , xi xj, A = | | ij | | (i = 1,..., n; j = 1,..., n; n ) . .
= | | bij | | (i = 1,..., m; j = 1,..., m; m ), :
bij =
g1,..., gm , 1,..., n G(X). S = || sij || (I = 1,..., n , j = 1,..., m ), :
sij =
.
R = | | rij | | n m, :
rij =
.
ó , , .
, (V, E),
V ' , ,
E V, .
, ei, .
, . . ' .
1. . ³ V = . ³ = 0. .
2. ( ), . 1. , 1 , ' , .
3. 1- 2- . 1- 1- + 1- 2- , 0 + 7 = 7. 2- , 2- 7.
4, 5. 1- 3- 6-.
6. 1 . 1 (, , ). , .
7. () . 2 = 7.
2- , 2-. 2- 1, 3, 4.
8. ( ) 2 1- . . 1- .
2 4. 2-, = 2- + 2- 4- = 7 + 15 = 22. 22 < ∞, 4 22.
9. 2 3. 2-, = 7 + 10 = 17. 17 , ' 3 9, 3- .
10. 2 , .
11. 2 6 .
, .
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v, . n , m G.
, d[v] , d , O(n2). n , n . , m ( ). , O (n2 + m), , m n(n - 1), O (n2).
' . .
, () . , . , ' , .
1. .
2. .
3. dz , , .
4. 3.
5. , .
O (n2), n .