Основы теории графов. Алейников С.М., Горяйнов В.В
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Задание 3. Операции над графами
Графы G1 и G2 заданы матрицами смежности B1 и B2 соответственно. Построить исходные графы G1 и G2 , их пересечение, объединение и симметрическую разность. Составить матрицы смежности полученных графов.
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1 |
0 |
1 |
|
|
2 |
υ3 |
|
0 |
1 |
1 |
0 |
|
|
|
|
|
|
|
|
||||||||||
|
υ4 |
0 |
1 |
1 |
1 |
|
|
|
υ4 |
0 |
1 |
0 |
1 |
|
||
20 |
|
|
υ1 υ2 υ3 υ4 |
|
|
|
|
|
υ1 υ2 υ3 υ4 |
|
||||||
υ1 |
0 1 1 1 |
|
|
|
υ1 |
1 1 0 0 |
|
|||||||||
|
=υ2 |
|
1 |
1 |
1 |
0 |
|
|
|
=υ2 |
|
1 |
1 |
1 |
0 |
|
B |
|
|
, |
B |
|
. |
||||||||||
1 |
υ3 |
|
1 |
1 |
1 |
0 |
|
|
2 |
υ3 |
|
0 |
1 |
0 |
1 |
|
|
|
|
|
|
|
|
||||||||||
|
υ4 |
1 |
0 |
0 |
0 |
|
|
|
υ4 |
0 |
0 |
1 |
0 |
|
||
21 υ1 |
|
υ1 υ2 υ3 υ4 |
|
|
|
|
|
υ1 υ2 υ3 υ4 |
|
|||||||
1 0 0 1 |
|
|
|
υ1 |
0 1 0 0 |
|
||||||||||
|
=υ2 |
|
0 |
1 |
1 |
1 |
|
|
|
=υ2 |
|
1 |
1 |
1 |
1 |
|
B |
|
|
, |
B |
|
. |
||||||||||
1 |
υ3 |
|
0 |
1 |
1 |
0 |
|
|
2 |
υ3 |
|
0 |
1 0 1 |
|
||
|
|
|
|
|
|
|
||||||||||
|
υ4 |
1 |
1 |
0 |
0 |
|
|
|
υ4 |
0 |
1 1 1 |
|
||||
22 |
|
|
υ1 υ2 υ3 υ4 |
|
|
|
|
|
υ1 υ2 υ3 υ4 |
|
||||||
υ1 |
0 1 0 1 |
|
|
|
υ1 |
0 1 0 1 |
|
|||||||||
|
=υ2 |
|
1 |
0 |
1 |
0 |
|
|
|
=υ2 |
|
1 |
1 |
1 |
1 |
|
B |
|
|
, |
B |
|
. |
||||||||||
1 |
υ3 |
|
0 |
1 |
1 |
1 |
|
|
2 |
υ3 |
|
0 |
1 1 0 |
|
||
|
|
|
|
|
|
|
||||||||||
|
υ4 |
1 |
0 |
1 |
1 |
|
|
|
υ4 |
1 |
1 0 0 |
|
||||
23 |
|
|
υ1 υ2 υ3 υ4 |
|
|
|
|
|
υ1 υ2 υ3 υ4 |
|
||||||
υ1 |
0 1 0 1 |
|
|
|
υ1 |
0 0 0 1 |
|
|||||||||
|
=υ2 |
|
1 |
0 |
1 |
0 |
|
|
|
=υ2 |
|
0 |
1 |
0 |
1 |
|
B |
|
|
, |
B |
|
. |
||||||||||
1 |
υ3 |
|
0 |
1 |
0 |
1 |
|
|
2 |
υ3 |
|
0 |
0 |
0 |
1 |
|
|
|
|
|
|
|
|
||||||||||
|
υ4 |
1 |
0 |
1 |
0 |
|
|
|
υ4 |
1 |
1 |
1 |
0 |
|
||
|
|
|
|
|
|
|
|
|
126 |
|
|
|
|
|
|
|
24 |
|
|
υ1 υ2 υ3 υ4 |
|
|
|
|
υ1 υ2 υ3 υ4 |
|
|||||||
υ |
|
1 |
0 |
1 |
0 |
|
|
υ |
1 1 0 0 |
|
||||||
|
1 |
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
B |
=υ2 |
|
0 |
1 |
0 |
1 |
|
, |
B |
=υ2 |
|
1 |
0 |
1 |
1 |
. |
1 |
υ3 |
|
1 |
0 |
1 |
0 |
|
|
2 |
υ3 |
|
0 |
1 |
0 |
1 |
|
|
|
|
|
|
|
|
||||||||||
|
υ4 |
0 |
1 |
0 |
1 |
|
|
|
υ4 |
0 |
1 |
1 |
0 |
|
||
25 |
|
|
υ1 υ2 υ3 υ4 |
|
|
|
|
υ1 υ2 υ3 υ4 |
|
|||||||
υ |
|
1 |
1 |
0 |
1 |
|
|
υ |
1 1 1 0 |
|
||||||
|
1 |
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
B |
=υ2 |
|
1 |
1 |
1 |
0 |
|
, |
B |
=υ2 |
|
1 |
0 |
0 |
1 |
. |
1 |
υ3 |
|
0 |
1 |
1 |
0 |
|
|
2 |
υ3 |
|
1 |
0 |
0 |
0 |
|
|
|
|
|
|
|
|
||||||||||
|
υ4 |
1 |
0 |
0 |
1 |
|
|
|
υ4 |
0 |
1 |
0 |
1 |
|
||
26 |
|
|
υ1 υ2 υ3 υ4 |
|
|
|
|
υ1 υ2 υ3 υ4 |
|
|||||||
υ |
|
0 |
1 |
1 |
1 |
|
|
υ |
1 0 1 0 |
|
||||||
|
1 |
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
B |
=υ2 |
|
1 |
0 |
1 |
1 |
|
, |
B |
=υ2 |
|
0 |
0 |
1 |
1 |
. |
1 |
υ3 |
|
1 |
1 |
1 |
0 |
|
|
2 |
υ3 |
|
1 |
1 |
0 |
1 |
|
|
|
|
|
|
|
|
||||||||||
|
υ4 |
1 |
1 |
0 |
1 |
|
|
|
υ4 |
0 |
1 |
1 |
1 |
|
||
27 |
|
|
υ1 υ2 υ3 υ4 |
|
|
|
|
υ1 υ2 υ3 υ4 |
|
|||||||
υ1 |
|
0 |
1 1 1 |
|
|
υ1 |
0 0 1 0 |
|
||||||||
|
=υ2 |
|
1 |
1 |
1 |
1 |
|
|
|
=υ2 |
|
0 |
0 |
1 |
1 |
|
B |
|
, |
B |
|
. |
|||||||||||
1 |
υ3 |
|
1 |
1 0 1 |
|
|
2 |
υ3 |
|
1 |
1 |
1 |
0 |
|
||
|
|
|
|
|
|
|
||||||||||
|
υ4 |
1 |
1 1 0 |
|
|
|
υ4 |
0 |
1 |
0 |
0 |
|
||||
28 |
|
|
υ1 υ2 υ3 υ4 |
|
|
|
|
υ1 υ2 υ3 υ4 |
|
|||||||
υ |
|
0 |
0 |
1 |
0 |
|
|
υ |
1 0 1 0 |
|
||||||
|
1 |
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
B |
=υ2 |
|
0 |
1 |
1 |
1 |
|
, |
B |
=υ2 |
|
0 |
1 |
1 |
0 |
. |
1 |
υ3 |
|
1 |
1 |
0 |
1 |
|
|
2 |
υ3 |
|
1 |
1 |
0 |
1 |
|
|
|
|
|
|
|
|
||||||||||
|
υ4 |
0 |
1 |
1 |
1 |
|
|
|
υ4 |
0 |
0 |
1 |
0 |
|
||
|
|
|
|
|
|
|
|
|
127 |
|
|
|
|
|
|
|
29 |
|
|
|
υ1 υ2 υ3 υ4 |
|
|
|
|
|
|
υ1 υ2 υ3 υ4 |
|
||||||
|
υ1 |
1 0 0 1 |
|
|
|
|
υ1 |
1 |
0 1 0 |
|
||||||||
|
=υ2 |
|
0 |
0 |
0 |
1 |
|
|
|
=υ2 |
|
0 |
1 |
1 1 |
|
|||
B |
|
|
, |
B |
|
. |
||||||||||||
1 |
υ3 |
|
0 |
0 |
1 |
1 |
|
|
2 |
υ3 |
|
1 |
1 |
0 0 |
|
|||
|
|
|
|
|
|
|
|
|
||||||||||
|
|
υ4 |
1 |
1 |
1 |
0 |
|
|
|
|
υ4 |
0 |
1 |
0 1 |
|
|||
30 |
|
|
|
υ1 υ2 υ3 υ4 |
|
|
|
|
|
|
υ1 υ2 υ3 υ4 |
|
||||||
|
υ1 |
1 |
1 |
0 1 |
|
|
|
|
υ1 |
0 1 1 |
0 |
|
||||||
|
=υ2 |
|
1 |
1 |
1 |
0 |
|
|
|
=υ2 |
|
1 |
1 |
1 |
0 |
|
||
B |
|
|
, |
B |
|
. |
||||||||||||
1 |
υ3 |
|
0 |
1 |
0 |
1 |
|
|
2 |
υ3 |
|
1 |
1 |
1 |
1 |
|
||
|
|
|
|
|
|
|
|
|
||||||||||
|
|
υ4 |
1 |
0 |
1 |
0 |
|
|
|
|
υ4 |
0 |
0 |
1 |
1 |
|
||
Задание 4. Кратчайший путь в орграфе
Орграф задан весовой матрицей W . Построить его, найти величину кратчайшего пути и сам путь из вершины υ1 в вершину υ6 или υ7 по алгоритму
Дейкстры.
ВАРИАНТЫ
1 |
|
|
υ1 υ2 |
υ3 |
υ4 |
υ5 |
υ6 |
|
2 |
|
|
υ1 υ2 |
υ3 |
υ4 |
υ5 |
υ6 |
|
|
|
|
|
|
|
|
|
||||||||||
|
υ1 |
0 6 |
9 |
12 |
∞ ∞ |
|
|
υ1 |
0 12 |
∞ 15 16 ∞ |
|
||||||
|
|
|
∞ 0 |
7 |
8 |
12 ∞ |
|
|
|
|
∞ 0 |
14 |
∞ ∞ ∞ |
|
|||
|
υ2 |
|
|
υ2 |
|
||||||||||||
W = |
υ3 |
|
∞ ∞ 0 |
4 |
3 |
5 |
|
W = |
υ3 |
|
∞ ∞ |
0 |
∞ ∞ 14 |
|
|||
|
|
∞ ∞ ∞ 0 |
7 |
9 |
. |
|
|
∞ 8 |
12 |
0 |
10 ∞ |
. |
|||||
|
υ4 |
|
|
υ4 |
|
||||||||||||
|
υ5 |
|
∞ ∞ ∞ ∞ |
0 |
8 |
|
|
υ5 |
|
∞ 12 11 |
∞ 0 |
15 |
|
||||
|
|
|
∞ ∞ ∞ ∞ |
∞ 0 |
|
|
|
|
∞ ∞ ∞ |
∞ ∞ 0 |
|
||||||
|
υ6 |
|
|
υ6 |
|
||||||||||||
128
3 |
|
|
υ1 υ2 |
υ3 |
υ4 |
υ5 |
υ6 |
|
4 |
|
|
υ1 υ2 |
υ3 |
υ4 |
υ5 |
υ6 |
|
|
|
|
|
|
|
|
|
||||||||||
|
υ1 |
0 6 |
9 |
7 |
19 ∞ |
|
υ1 |
0 5 |
7 |
10 |
9 |
∞ |
|
||||
|
|
|
∞ 0 |
11 |
∞ |
∞ ∞ |
|
|
|
|
∞ 0 |
∞ 8 |
6 |
14 |
|
||
|
υ2 |
|
|
υ2 |
|
||||||||||||
W = |
υ3 |
|
∞ ∞ |
0 |
∞ |
∞ 16 |
|
W = |
υ3 |
|
∞ 7 |
0 |
6 |
3 |
10 |
|
|
|
|
∞ 11 15 |
0 |
7 |
∞ |
. |
|
|
∞ ∞ ∞ 0 |
5 |
6 |
. |
|||||
|
υ4 |
|
|
υ4 |
|
||||||||||||
|
υ5 |
|
∞ 7 |
9 |
∞ |
0 10 |
|
|
υ5 |
|
∞ ∞ ∞ ∞ 0 |
8 |
|
||||
|
|
|
∞ ∞ ∞ |
∞ |
∞ 0 |
|
|
|
|
∞ ∞ ∞ ∞ ∞ 0 |
|
||||||
|
υ6 |
|
|
υ6 |
|
||||||||||||
5 |
|
υ1 υ2 |
υ3 |
υ4 |
υ5 |
υ6 υ7 |
|
6 |
|
|
|
υ1 υ2 |
υ3 |
υ4 |
υ5 |
υ6 |
|
|
|
|
|
|
|
|
|
||||||||||
υ1 |
0 ∞ 10 15 |
7 ∞ ∞ |
|
|
υ1 |
0 4 |
6 |
10 |
∞ ∞ |
||||||||
|
|
∞ 0 |
∞ ∞ 13 19 ∞ |
|
|
υ2 |
|
∞ 0 |
∞ |
4 |
∞ 15 |
|
|||||
υ2 |
|
|
|
|
|||||||||||||
υ3 |
|
∞ 8 |
0 |
12 |
6 10 21 |
|
W = |
υ |
|
∞ 3 |
0 |
4 |
5 17 |
|
|||
|
|
∞ ∞ ∞ 0 |
∞ 10 15 |
|
|
3 |
|
∞ ∞ ∞ 0 |
∞ 5 |
. |
|||||||
W =υ4 |
. |
|
υ4 |
|
|||||||||||||
υ5 |
|
∞ ∞ ∞ ∞ 0 |
9 22 |
|
|
υ |
5 |
|
∞ ∞ ∞ 4 |
0 |
9 |
|
|||||
υ6 |
|
∞ ∞ ∞ ∞ ∞ 0 17 |
|
|
|
|
∞ ∞ ∞ ∞ |
∞ 0 |
|
||||||||
|
|
|
υ6 |
|
|||||||||||||
|
|
∞ ∞ ∞ ∞ ∞ ∞ 0 |
|
|
|
|
|
|
|
|
|
|
|
||||
υ7 |
|
|
|
|
|
|
|
|
|
|
|
||||||
7 |
|
|
υ1 υ2 |
υ3 |
υ4 |
υ5 |
υ6 |
|
8 |
|
|
|
υ1 υ2 |
υ3 |
υ4 |
υ5 |
υ6 |
|
|
υ1 |
|
|
|
|
|
|
|
||||||||||
|
0 6 |
8 |
∞ 12 ∞ |
|
|
υ1 |
0 7 |
15 |
∞ |
4 ∞ |
||||||||
|
|
|
∞ 0 |
∞ |
6 |
∞ 13 |
|
|
υ2 |
|
∞ 0 |
7 |
16 |
∞ ∞ |
|
|||
|
υ2 |
|
|
|
|
|||||||||||||
W = |
υ3 |
|
∞ 6 |
0 |
6 |
7 ∞ |
|
W = |
υ |
|
∞ ∞ 0 |
19 |
∞ 2 |
|
||||
|
|
∞ ∞ |
∞ |
0 |
∞ |
8 |
. |
|
3 |
|
∞ ∞ ∞ 0 |
∞ |
7 |
. |
||||
|
υ4 |
|
|
υ4 |
|
|||||||||||||
|
υ5 |
|
∞ 5 |
∞ |
6 |
0 |
9 |
|
|
υ |
5 |
|
∞ 3 |
4 |
15 |
0 |
8 |
|
|
|
|
∞ ∞ ∞ |
∞ |
∞ 0 |
|
|
|
|
|
|
|
∞ 0 |
|
||||
|
υ6 |
|
|
υ6 |
∞ ∞ ∞ ∞ |
|
||||||||||||
129
9 |
|
υ1 υ2 |
υ3 |
υ4 |
υ5 |
υ6 υ7 |
|
10 |
|
|
υ1 υ2 |
υ3 |
υ4 |
υ5 |
υ6 |
|
|
|
|
|
|
|
|
|
|
||||||||||
υ1 |
0 8 |
18 21 ∞ 15 ∞ |
|
|
υ1 |
0 11 13 |
∞ |
∞ ∞ |
|
||||||||
|
|
∞ 0 |
∞ 12 |
7 ∞ ∞ |
|
|
υ2 |
|
∞ 0 |
12 10 |
∞ 20 |
|
|||||
υ2 |
|
|
|
|
|||||||||||||
υ3 |
|
∞ ∞ 0 |
7 |
9 ∞ 6 |
|
W = |
υ |
|
∞ ∞ 0 |
∞ 11 ∞ |
|
||||||
|
|
∞ ∞ ∞ 0 |
9 |
8 3 |
|
|
3 |
|
∞ ∞ 14 |
0 |
12 11 |
. |
|||||
W =υ4 |
. |
|
υ4 |
|
|||||||||||||
υ5 |
|
∞ ∞ ∞ ∞ 0 |
5 5 |
|
|
υ |
5 |
|
∞ ∞ ∞ |
∞ 0 |
7 |
|
|||||
υ6 |
|
∞ ∞ ∞ ∞ ∞ 0 4 |
|
|
|
|
∞ ∞ ∞ |
∞ ∞ 0 |
|
||||||||
|
|
|
υ6 |
|
|||||||||||||
|
|
∞ ∞ ∞ ∞ ∞ ∞ 0 |
|
|
|
|
|
|
|
|
|
|
|
||||
υ7 |
|
|
|
|
|
|
|
|
|
|
|
||||||
11 |
|
υ1 υ2 |
υ3 |
υ4 |
υ5 |
υ6 |
|
12 |
|
υ1 υ2 |
υ3 |
υ4 |
υ5 |
υ6 |
|
||
|
|
|
|
|
|
|
|
||||||||||
|
υ1 |
0 8 |
3 |
∞ 14 ∞ |
|
|
υ1 |
0 11 |
12 |
7 |
∞ ∞ |
|
|||||
|
|
|
∞ 0 |
∞ |
∞ 6 ∞ |
|
|
|
|
∞ 0 |
14 |
9 |
11 18 |
|
|||
|
υ2 |
|
|
υ2 |
|
||||||||||||
W = |
υ3 |
|
∞ 3 |
0 |
2 |
3 12 |
|
W = |
υ3 |
|
∞ ∞ |
0 |
6 |
7 16 |
|
||
|
|
∞ ∞ ∞ |
0 |
∞ |
6 |
. |
|
|
∞ ∞ ∞ 0 |
8 |
∞ |
. |
|||||
|
υ4 |
|
|
υ4 |
|
||||||||||||
|
υ5 |
|
∞ ∞ ∞ 4 |
0 |
6 |
|
|
υ5 |
|
∞ ∞ ∞ ∞ 0 |
10 |
|
|||||
|
|
|
∞ ∞ ∞ |
∞ ∞ 0 |
|
|
|
|
∞ ∞ ∞ ∞ ∞ 0 |
|
|||||||
|
υ6 |
|
|
υ6 |
|
||||||||||||
13 |
|
υ1 υ2 |
υ3 |
υ4 |
υ5 υ6 |
|
14 |
|
υ1 υ2 |
υ3 |
υ4 |
υ5 |
υ6 |
|||
|
|
|
|
|
|
|
||||||||||
|
υ1 |
0 7 |
∞ 9 |
13 ∞ |
|
υ1 |
0 5 |
9 |
8 |
∞ ∞ |
||||||
|
|
|
∞ 0 |
7 |
∞ |
∞ ∞ |
|
|
|
|
∞ 0 |
3 |
∞ |
|
|
|
|
υ2 |
|
|
υ2 |
∞ ∞ |
|||||||||||
W = |
υ3 |
|
∞ ∞ 0 |
∞ |
∞ 7 |
|
W = |
υ3 |
|
∞ ∞ 0 |
∞ |
∞ 4 |
|
|||
|
|
∞ 5 |
9 |
0 |
7 15 |
. |
|
|
∞ 3 |
5 |
0 |
|
|
. |
||
|
υ4 |
|
|
υ4 |
7 ∞ |
|||||||||||
|
υ5 |
|
∞ 8 |
6 |
∞ |
0 11 |
|
|
υ5 |
|
∞ 3 |
∞ |
∞ |
0 |
4 |
|
|
|
|
∞ ∞ ∞ |
∞ |
∞ 0 |
|
|
|
|
∞ ∞ ∞ |
∞ |
∞ |
0 |
|
||
|
υ6 |
|
|
υ6 |
|
|||||||||||
130