342
Solutions to Exercises
~
A
2
~
S
B
1 2
2
E
C
6
F
(c) As shown on the right, the minimum weight of a spanning tree after
T
is removed is 18. The two
least edges at
T
have weights 3 and 6, so we obtain 18
+
3
+
6
=
27 as a lower bound.
(d) As shown to the right, if vertex
C
is removed, the minimum
weight of a spanning tree is 26. The two least edges at
C
have
weights 1 and 2 so we get an estimate of 26
+
1
+
2
=
29 as a
lower bound. By way of comparison, the student should check
that if vertex
A
is removed, the weight of a minimum spanning
tree is 19, implying minimum Hamiltonian cycle of weight
8
A
2
~3
S
3
T
B
E
6
F
19
+
2
+
6
=
27; if vertex
D
is removed, the weight of a minimum spanning tree is 23, implying
minimum Hamiltonian cycle of weight 23
+
2
+
3
=
28; if vertex
E
is removed, the weight of
a minimum spanning tree is 20, implying minimum Hamiltonian cycle of weight 20
+
2
+
3
=
25; if vertex
F
is removed, the weight of a minimum spanning tree is 15, implying minimum
Hamiltonian cycle of weight 15
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 Summer '10
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 Graph Theory, Hamiltonian path, Spanning tree, minimum Hamiltonian cycle

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