Section 12.3
341
®
A
B~5
F
1
2
C
E
6
3
C
D
D
(c) [BB] As shown on the right, the minimum weight of a spanning tree after
B
is removed is 12. The
two least edges at
B
have weights 1 and 5, so we obtain 12 + 1 + 5
=
18 as a lower bound.
(d)
It
is not possible to improve the lower bound in (c), as we can see from the cycle found in (a).
16. (a) The graph is complete, so an obvious approach is to try to choose the lowest weight available edge
at each vertex. Such a cycle is
ADEF BGA,
which has weight 2 + 36 + 21 + 55 + 13 + 51
= 178.
(b) As shown on the right, the minimum weight of a spanning tree after
D
is removed is 120. The
two least edges at
D
have weights 2 and 36, so we obtain 120 + 2 + 36
=
158 as a lower bound.
A
A
BV035 Bq,
5A5
t;¥F
13
13~
~21
C
2
E
C
E
D
(c) As shown on the left, the minimum weight of a spanning tree after F is removed is 101. The two
edges of least weight at F have weights 21 and 52, so we obtain an estimate of 101 + 21 +52
=
174
as a lower bound for the weight of any Hamiltonian cycle.
(d) As shown at the right, if vertex
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 Summer '10
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 Graph Theory, Harshad number, Leftwing politics, lower bound, Hamiltonian path

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