
Matem_logika_V_13_V_18
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10.0, 3, 7, 8, 11, 13*, 14, 15*.
11.f (x1, x2, x3) = x1x2 x3 .
Вариант 9
1. ((x1 ↓ x2) x3)→ x1 x2 x3 = (x2 ≈ x3) x1 → x3 x1 .
2. f (x1, x2, x3)= (x1 x2)≈ (x3 ↓ x1x2).
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4. f (x1, x2, x3)= x1x2 ↓ x3 ↓ x1 .
5.1, 2, 4, 7, 8, 11, 13, 14.
6.1, 2, 4.
7.0, 1, 2, 3, 8, 9, 10, 11, 14, 15.
8.1, 2, 5, 7, 8, 12, 13, 14.
9.2, 4, 7, 9, 10, 14, 15.
10.0, 1, 2, 3, 8*, 9, 10, 11*, 14, 15*.
11.f (x1, x2, x3)= x1x2 ↓ x3 .
Вариант 10
1.(x1 → x2)→ (x3 x1) 1 → x3x4 = (x1x2 (x3 ≈ x1))(x3 x4).
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3. x1x2 x3 → x1x3 (x1 ≈ x2)= (x1 x2) x3 ↓ (x1 ↓ x2)→ (x1 ↓ x2).
4.f (x1, x2, x3)= x1x2 x3 → x1x3 .
5.3, 7, 11, 15.
6.0, 1, 2, 3, 4, 5, 6.
7.4, 6, 7, 8, 9, 10, 14, 15.
8.1, 5, 6, 7, 8, 9, 10, 15.
9.0, 3, 7, 8, 9, 10, 11, 12, 15.
10.1, 5*, 6, 7, 8, 9*, 10, 15.
11.f (x1, x2, x3)= x1x2 x3 .
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Вариант 11 |
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1.(x1x2 |
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↓ (x1 ↓ x2)↓ x4 |
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2. f (x1, x2, x3)= (x1 ↓ x2)≈ (x3 x2 x1).
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3. |
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5.1, 2, 3, 4, 5, 6, 7, 9, 13, 14, 15.
6.1, 2, 3, 4, 5, 6.
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7.0, 1, 2, 3, 4, 8, 9, 10 , 11, 12.
8.0, 1, 2, 3, 9, 10, 13, 14, 15.
9.0, 2, 3, 5, 11, 12, 15.
10.0, 2*, 3, 5, 11, 12*, 15*.
11.f (x1, x2, x3) = x1 x3 x2 .
Вариант 12
1.(x1x2 ↓ x3) (x1x3 x2) (x1x2 ↓ x3)= x1 ↓ (x2 x3).
2.f (x1, x2, x3)= (x1x2 x3 → x1) (x1 ≈ x2 ≈ x3).
3.(x1 ≈ x2)(x1 → x2)→ (x1 ≈ x2)(x1 ↓ x2) = (x1 x2)(x1 x2)(x1 ↓ x2).
4.f (x1, x2, x3)= (x1 ≈ x2) (x2 ≈ x3).
5.0, 3, 4, 7, 8, 11, 12, 15.
6.2, 3, 4, 5.
7.1, 3, 6, 8, 9, 10, 12, 13.
8.0, 2, 5, 6, 8, 11, 12, 13, 14.
9.0, 2, 4, 7, 8, 10, 13, 15.
10.0, 2, 5, 6*, 8, 11*, 12*, 13, 14.
11.f (x1, x2, x3)= (x1x2) (x2 x3).
Вариант 13
1.(x1 x2)((x1 ≈ x2)(x1 x2))= x1 → x2 (x1 → x2).
2.f (x1, x2, x3)= ((x1 x2) x1 ≈ (x3 ≈ x2)) x3 .
3.((x1 ↓ x2) x3)→ x1 x2 x3 = (x2 ≈ x3) x1 → x3 x1 .
4.f (x1, x2, x3)= x3x1 → x2 x1 .
5.0, 1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15.
6.0, 2, 3, 4, 5, 7.
7.2, 5, 7, 8, 11, 12, 13, 15.
8.1, 4, 6, 7, 10, 11, 12, 14.
9.0, 3, 6, 8, 9, 12, 13, 15.
10.0, 3, 6, 8, 9*, 12, 13*, 15*.
11.f (x1, x2, x3)= x3x1 x2 x1 .
Вариант 14
1.(x1x2 x3 → x1) x1 ≈ x2 ≈ x3 = x3 (x1 ≈ x2) x3((x1 ↓ x2)↓ (x1 ↓ x2 )) x1 .
2.f (x1, x2, x3)= x1 ↓ x2 x3 ≈ x1x2 .
3.(x2 x3 ≈ x1x2)↓ x3 → x2 x1 = (x2 x3 ↓ x1x2) x3 x2 x1 .
4.f (x1, x2, x3)= x3 (x1 → x2 x3) x2 .
5.0, 1, 6, 7, 8, 9, 14, 15.
6.0, 3, 4, 6, 7.
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7.1, 3, 6, 7, 9, 12, 13, 14, 15.
8.0, 2, 5, 7, 8, 9, 11, 12.
9.2, 4, 7, 9, 10, 11, 13, 15.
10.0*, 2, 5, 7, 8, 9*, 11*, 12.
11.f (x1, x2, x3) = x3 (x1 → x2 x3).
Вариант 15
1. (x1 ↓ x2) ~ ((x1 ~ x2) (x1 | x2)) = x2 x1 .
2. f (x1x2x3) = x1 | (x3↓x2) .
3. x1 ↓ x2 ~ x3 → (x1 x2 ) = x1x2→ x3 x3→ x1x2 (x1 ~ x2 ) .
4.f (x1, x2 , x3, x4 ) = x1|x2 ↓ (x3 ↓ x4 ) .
5.2, 3, 4, 5, 8, 9, 14, 15.
6.0, 1, 2, 5.
7.0, 1, 6, 7, 8, 9, 10, 11.
8.3, 5, 6, 7, 8, 9, 13, 14.
9.5, 7, 8, 9, 10, 11, 15.
10.0*, 1*, 6*, 7, 8, 9, 10, 11.
11.f (x1, x2 , x3) = x1|x2 ↓ x3 .
Вариант 16
1.(x1 ≈ x2) (x2 → x1)↓ x1 = (x1 x2) 1(x2 x1 x2 x1x2).
2.f (x1, x2, x3)= x1 → x2 (x1 → x2) x3 .
3.(x1x2 → x3) (x1 → x2 x3) (x1 x2) x3 = (x1 ↓ x2)↓ (x2 ↓ x3)↓ x3 .
4.f (x1, x2, x3)= (x1 x2) (x2 x3).
5.0, 1, 2, 3, 12, 13, 14, 15.
6.1, 2, 3, 7.
7.0, 1, 2, 3, 4, 10, 11, 14, 15.
8.0, 1, 2, 3, 7, 8, 9, 10, 11, 15.
9.2, 3, 4, 5, 9, 10, 12, 13.
10.0, 1*, 2*, 3, 7*, 8, 9, 10, 11, 15.
11.f (x1, x2, x3)= (x1 x2).
Вариант 17
1. ((x1 → x2 ) → ((x1 x2 ) (x1 ~ x2 ))) = x1 x2 . 2. f (x1x2x3) = x1→(x2|x1) .
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3.[(x1 x2) x1] → x2 ↓ x1 = x2x1 .
4.f (x1, x2, x3) = (x1~x2)&x3 .
5.1, 2, 4, 7, 9, 10, 13, 15.
6.1, 2, 4, 5.
7.2, 3, 6, 8, 9, 13, 14, 15.
8.0, 1, 2, 7, 8, 9, 10, 13, 14, 15.
9.2, 3, 4, 9, 10, 11, 14, 15.
10.0, 1*, 2, 7*, 8*, 9*, 10, 13, 14, 15.
11.f (x1, x2, x3) = (x1 x2)&x3 .
Вариант 18
1. ([(x1 ↓ x2 ) → x3] [x1 | (x2 | x3)]) 1 = (x1 x2 | x3 ) ~ (x1 x2 x3) .
2. f (x1, x2 , x3 ) = (x1 ↓ x2) ↓ [(x2 ↓ x3) ↓ x3] .
3.(x1x3 x2)(x3 → x1) ~ x2 = x1 | (x3↓x2) .
4.f (x1, x2 , x3) = (x1 x2 x3) ↓ x3 .
5.4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15.
6.0, 3, 4, 7.
7.3, 4, 5, 6, 13, 14, 15.
8.2, 6, 7, 10 ,12 ,13 14, 15.
9.0, 2, 4, 8, 12, 14, 15.
10.2*, 6, 7*, 10 ,12 ,13 14, 15.
11.f (x1, x2 , x3) = (x1 x2 x3) .
Вариант 19
1. (x1 → x2 ) (x1 x2 ) (x1 x2 ) = (x1↓x2 ) → x1x2 .
2.f (x1, x2 , x3) = (x2 ~ x3) x1→x3 x1 .
3.(x1x2 x3) ↓ (x4~(x1↓x2)) = (x1x2↓x3) ↓ (x4 ↓ (x1↓x2)) ↓ (x4 ↓ (x1 ↓ x2)) .
4.f (x1, x2 , x3) = x1x2 x3 x1x2x3 .
5.0, 10, 11, 12, 13, 14, 15.
6.1, 2, 5, 6.
7.7, 8, 9, 11, 12, 14.
8.1, 2, 3, 4, 9, 11, 12, 14.
9.3, 4, 5, 6, 11, 13, 15.
10.3*, 4*, 5, 6*, 11, 13, 15.
11.f (x1, x2, x3) = x1x2 x1x2x3 .
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Вариант 20
1.[(x1 x2) x1]→x2↓x1 = x2 x1 .
2.f (x1, x2 , x3) = (x1 x2 | x3) ~ (x1 x2 x3) .
3.(x1→x2)→[(x3 x1) 1]→x3x4 = (x1x2 (x3 ~ x1))(x1 | x4) .
4.f (x1, x2 , x3) = x1x2 x1x3 x2x3 .
5.0, 2, 4, 6, 9, 11, 13, 15.
6.0, 1, 2, 3, 5, 6, 7.
7.3, 5, 6, 7, 9, 12, 13, 15.
8.3, 4, 5, 6, 7, 8, 9, 10, 11.
9.5, 6, 7, 8, 9, 10, 11, 12, 13.
10.5*, 6, 7*, 8*, 9, 10*, 11, 12, 13.
11.f (x1, x2 , x3) = x1x2 x1x3 .
Вариант 21
1.(x1 ~ x2) [(x2 → x1) ↓ x1] = (x1 x2) 1 (x1x2 x2 x1x2) .
2.f (x1, x2, x3) = x1x2→x3 x3→x1x2 .
3.(x1 → x2 ) (x1 x2 ) (x1 x2) = (x1↓x2) → x1x2 .
4.f (x1, x2, x3) = [(x2 x3 ↓ x1x2) | x3] | x2x1 .
5.0, 1, 2, 3, 7, 11, 15.
6.0, 1, 4, 5.
7.2, 7, 8, 9, 10, 11, 12, 13, 14.
8.0, 1, 5, 6, 8, 11, 12.
9.2, 3, 7, 8, 10, 13, 14, 15.
10.2*, 3, 7*, 8*, 10, 13*, 14, 15.
11.f (x1, x2, x3) = x1x2→x3 .
Вариант 22
1.(x1x2→x3) (x1 → x2x3) (x1 x2) x3 = (x1 ↓ x2) ↓ [(x2 ↓ x3) ↓ x3] .
2.f (x1, x2, x3) = (x1→x2) (x2↓x1) x3 .
3.[x1x2x3 (x3x1 x1x2x3)] 1 = [x1 → (x2 ↓ x3)] ~ [(x1 x3)] ↓ x1x2] .
4.f (x1, x2 , x3) = (x1 x2)(x1 x2 )x3 .
5.2, 3, 4, 5, 10, 11, 12, 13.
6.1, 2, 4, 5, 6.
7.3, 4, 6, 7, 8, 9, 11, 13, 15.
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8.1, 2, 4, 6, 7, 9, 10, 14, 15.
9.2, 3, 4, 6, 8, 9, 11, 12.
10.3*, 4*, 6, 7*, 8, 9*, 11, 13, 15.
11.f (x1, x2 , x3) = (x1 x2)(x1 x2 ) .
Вариант 23
1.f (x1, x2 , x3) = [(x1x2↓x3)↓x1] ↓ x2 .
2.f (x1, x2 , x3) = x1x2 x3 x1x2x3 .
3.(x1 ~ x2 ) [(x2 → x1) ↓ x1] = (x1 x2) 1 (x1x2 x2 x1x2) .
4.f (x1, x2, x3) = [(x1x2↓x3)↓x1] ↓ x2 .
5.0, 2, 5, 7, 8, 10, 13, 15.
6.1, 2, 3, 5, 6, 7.
7.7, 9, 10, 13, 14, 15.
8.0, 1, 2, 6, 10, 12, 13, 14.
9.2, 3, 4, 8, 12, 14, 15.
10.2*, 3, 4*, 8*, 12*, 14, 15.
11.f (x1, x2 , x3) = x1x2 x3 x1x2x3 .
Вариант 24
1. (x1x2 ↓ x3) | (x1 x2 → x3) = (x1x2|x3) → (x2 ↓ x1x3) .
2.f (x1, x2, x3) = [(x2 x3 ↓ x1x2) | x3] | x2x1 .
3.(xy ↓ z) (xz | y) (xy → z) = x↓(y|z) .
4.f (x1, x2 , x3 ) = x1x2→x3 x3→x1x2 .
5.0, 1 , 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15.
6.0, 2, 3, 4, 6, 7.
7.4, 7, 8, 9, 11, 12, 15.
8.1, 2, 4, 5, 9, 10, 13, 14.
9.0, 3, 4, 6, 7, 11, 12, 15.
10.4*, 7*, 8, 9, 11*, 12, 15.
11.f (x1, x2 , x3 ) = x1x2→x3 .
Вариант 25
1.[(x1 x2 ) (x1 ~ (x1 ~ x2))] 1 = x1 | x2 .
2.f (x1, x2, x3) = (x1x2 x3) → [(x2~x1) ↓ x3] .
3.(x1x2x3→x1) (x1 ~ x2 ~ x3) = x3(x1 ~ x2) x3((x1↓x2)↓(x1↓x2)) x1x2 x3 .
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4.f (x1, x2, x3) = (x1 x2)(x1 x2) ~ x3 .
5.0, 4, 8, 11, 12, 13, 14, 15.
6.0, 1, 3, 4, 6, 7.
7.6, 7, 8, 9, 10, 14, 15.
8.1, 3, 5, 8, 9, 10, 11, 12.
9.3, 5, 7, 10, 11, 12, 13, 14.
10.1*, 3, 5*, 8, 9, 10, 11, 12.
11.f (x1, x2, x3) = (x1 x2)(x1 x3) .
Вариант 26
1.(x1 → x2 ) (x1 x2 ) (x1 x2) = (x1↓x2) → x1x2 .
2.f (x1, x2 , x3) = [(x1x2↓x3)↓x1] ↓ x2 .
3.(x1x2 x3) ↓ (x4~(x1↓x2)) = (x1x2↓x3) ↓ (x4 ↓ (x1↓x2)) ↓ (x4 ↓ (x1 ↓ x2)) .
4.f (x1, x2 , x3 ) = x1x2 x3 x1x2x3 .
5.0, 10, 11, 12, 13, 14, 15.
6.1, 2, 5, 6.
7.7, 8, 9, 11, 12, 14.
8.1, 2, 3, 4, 9, 11, 12, 14.
9.3, 4, 5, 6, 11, 13, 15.
10.1*, 2, 3*, 4, 9*, 11, 12, 14.
11.f (x1, x2 , x3 ) = x1x2 x3 x1x2x3 .
Вариант 27
1. [(x1 x2 ) (x1 ~ (x1 ~ x2))] 1 = x1 | x2 .
2.f (x1, x2 , x3) = (x2 ~ x3) x1→x3 x1 .
3.(x1x2→x3) (x1 → x2x3) (x1 x2) x3 = (x1 ↓ x2) ↓ [(x2 ↓ x3) ↓ x3] .
4.f (x1, x2, x3)= (x1 x2)≈ (x3 ↓ x1x2).
5.0, 4,8,9,10, 11, 12, 13, 14.
6.1, 3, 5, 7.
7.7, 8, 9, 10,11, 12, 14.
8.0, 2, 3, 4, 10, 11, 12, 15.
9.3, 4, 5, 9, 11, 13, 15.
10.0*, 2*, 3, 4*, 10, 11, 12, 15.
11.f (x1, x2 , x3) = (x2 ~ x3) x1→x3 .
Вариант 28
1. (x1x2 x3) ↓ (x4~(x1↓x2)) = (x1x2↓x3) ↓ (x4 ↓ (x1↓x2)) ↓ (x4 ↓ (x1 ↓ x2)) .
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2. f (x1, x2 , x3) = (x2 ~ x3) x1→x3 x1 .
3. [(x1 x2 ) (x1 ~ (x1 ~ x2))] 1 = x1 | x2 .
4.f (x1, x2 , x3) = [(x1x2↓x3)↓x1] ↓ x2 .
5.3,4, 10, 11, 12, 13, 14, 15.
6.1, 2,4,5, 6.
7.1,3,5,7, 8, 9, 11, 12, 14.
8.1, 2, 3, 4,7,8, 9, 11, 12, 14.
9.1,2,3, 4, 5, 6, 11, 13, 15.
10.1*, 2, 3*, 4*,7,8, 9*, 11, 12, 14.
11.f (x1, x2, x3)= (x1 x2)(x3 ↓ x1x2).
Вариант 29
1.(x1x2→x3) (x1 → x2x3) (x1 x2) x3 = (x1 ↓ x2) ↓ [(x2 ↓ x3) ↓ x3] .
2.f (x1, x2, x3)= (x1 x2)≈ (x3 ↓ x1x2).
3. ((x1 → x2 ) → ((x1 x2) (x1 ~ x2))) = x1 x2 .
4.f (x1, x2 , x3) = x1x2 x3 x1x2x3 .
5.1,3 , 4, 10, 12, 14, 15.
6.0, 3,4, 5, 6.
7.7, 8, 9, 10, 12, 15.
8.1, 2, 3, 4, 5, 11, 13, 14.
9.2, 4, 5, 6, 7, 10, 15.
10.1*, 2*, 3, 4*, 5, 11*, 13, 14.
11.f (x1, x2, x3)= (x1x2)(x3 ↓ x1x2).
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Вариант 30 |
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((x1 → x2 ) → ((x1 x2 ) (x1 ~ x2 ))) = x1 x2 . |
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f (x1, x2, x3)= (x1x2) (x3 ↓ |
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3.(x1x2 x3) ↓ (x4~(x1↓x2)) = (x1x2↓x3) ↓ (x4 ↓ (x1↓x2)) ↓ (x4 ↓ (x1 ↓ x2)) .
4.f (x1, x2, x3)= (x1 x2)≈ x3 → (x1 x2).
5.0, 1,10, 11, 12, 13, 14, 15.
6.3, 4, 5, 6.
7.7, 9, 11, 12, 14.
8.2, 3, 7, 9, 11, 12, 14.
9.2, 4, 5, 8, 11, 13, 15.
10.2*, 3*, 7, 9*, 11, 12*, 14.
11.f (x1, x2, x3)= (x1 x2)x3 .
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3. СБОРНИК ЗАДАНИЙ ДЛЯ СЕМЕСТРОВОЙ РАБОТЫ № 2
ПО КУРСУ «Дискретная математика»
3.1. Пример решения и оформления
Задание
Дано:
Х2 V4 Х3
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V5 |
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X1 V1 X4 V7 X5
Задание 1.
Задать граф следующими способами: перечислением, матрицами смежности и инцидентности.
Решение: |
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Перечисление: |
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Множество вершин: X = |
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Множество связей: |
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<x5x3>, <x4 x5> |
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Множество изолированных вершин: пусто. |
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Матрица инцидентности: |
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|
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|
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V1 |
V2 |
|
V3 |
V4 |
V5 |
V6 |
V7 |
|
|
X1 |
|
-1 |
-1 |
|
0 |
0 |
0 |
0 |
0 |
|
|
X2 |
|
0 |
1 |
|
1 |
1 |
0 |
0 |
0 |
|
|
X3 |
|
0 |
0 |
|
0 |
-1 |
-1 |
-1 |
0 |
|
|
X4 |
|
1 |
0 |
|
-1 |
0 |
1 |
0 |
1 |
|
|
X5 |
|
0 |
0 |
|
0 |
0 |
0 |
1 |
-1 |
|
Матрица смежности:
|
X1 |
X2 |
X3 |
X4 |
X5 |
X1 |
0 |
0 |
0 |
0 |
0 |
X2 |
1 |
0 |
1 |
1 |
0 |
X3 |
0 |
0 |
0 |
0 |
0 |
X4 |
1 |
0 |
1 |
0 |
1 |
X5 |
0 |
0 |
1 |
0 |
0 |
29

Задание 2.
Определить следующие основные характеристики графа: число ребер и дуг; число вершин; коэффициент связности графа; степени всех вершин; цикломатическое число графа.
Решение:
Число ребер – 0 ; дуг – 7. Число вершин – 5.
Коэффициент связности графа - 1.
Степени всех вершин:
|
Х1 |
Х2 |
Х3 |
Х4 |
Х5 |
Полустепень исхода |
0 |
3 |
0 |
3 |
1 |
Полустепень захода |
2 |
0 |
3 |
1 |
1 |
Степень |
2 |
3 |
3 |
4 |
2 |
Цикломатическое число графа = (число связей – число вершин) + коэффициент связности. Т.е. 7- 5+1 = 3; цикломатическое число равно 3.
Задание 3.
Определить, является ли данный граф:
-планарным или плоским графом (обосновать ответ и выполнить обратное преобразование);
-двудольным графом (обосновать ответ и, если необходимо, то достроить до двудольного графа);
-деревом (обосновать ответ и, в случае циклического графа, привести один из вариантов основного дерева);
-псевдографом или мультиграфом, или простым графом (обосновать ответ и выполнить необходимые преобразования).
Решение:
Данный граф является плоским, поскольку все его связи пересекаются только в вершинах. Преобразуем данный граф в планарный граф:
|
Х2 |
V4 |
Х3 |
|
V2 |
V3 |
V5 |
|
V6 |
X1 |
V1 |
X4 |
V7 |
X5 |
30