Solutions to Practice Problems for Midterm 2
Peng Du
University of California, San Diego
1 Exercise 5.5
1.1 Part (a)
No, since every spanning tree gets the same proﬁt
n

1.
1.2 Part (b)
Yes, use the same idea as problem 4.8 to give an example.
2 Exercise 5.9
2.1 Part (a)
False, see Fig. 1 for a counterexample where the heaviest edge (
b,d
) is part of
every MST.
a
b
d
c
1
1
10
1
Fig.1.
The counterexample for exercise 5.9(a).
2.2 Part (b)
True. Suppose
e
∈
T
for some MST
T
. We remove
e
and split
T
into disjoint
sets of vertices
S
1
and
S
2
. Denote the lightest edge between
S
1
and
S
2
by
e
0
.
Since there is a cycle where
e
is the unique heaviest edge, we have
w
e
0
< w
e
.
Therefore,
T
\ {
e
} ∪ {
e
0
}
is lighter than
T
, a contradiction.
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2.3 Part (c)
True. Take any cut containing
e
, then use cut property by setting
X
=
∅
.
2.4 Part (d)
True. Denote the lightest edge by
e
. Suppose some MST
T
doesn’t contain
e
. By
adding
e
to
T
, we get a cycle
C
. Since there exists
e
0
∈
C
such that
w
e
< w
e
0
,
we can get a lighter tree
T
0
=
T
\ {
e
0
} ∪ {
e
}
, a contradiction.
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 Fall '09
 Freund,Yoav
 UCI race classifications, Tour de Georgia, San Diego, lightest edge, Peng Du

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