CME305 Sample Midterm I
1.
Matchings and Independent Sets
Assume that you are given graph
G
(
V,E
), a matching
M
in
G
and inde
pendent
S
in
G
. Show that

M

+

S
 ≤ 
V

.
Solution:
Let
M
∗
be a maximum matching and
S
∗
be a maximum in
dependent set. For each of the

M
∗

matched pairs,
S
∗
can include at
most one of the vertex, and therefore

S
∗
 ≤ 
V
 − 
M
∗

, or equivalently

M
∗

+

S
∗
 ≤ 
V

. Since
M
∗
and
S
∗
are maximum, we know

M

+

S
 ≤ 
V

holds for all
M
and
S
.
2.
Unique Minimum st Cut
Given a network
G
(
V,E,s,t
), give a polynomial time algorithm to deter
mine whether
G
has a unique minimum st cut.
Solution:
First compute a minimum
s

t
cut
C
, and deﬁne its volume by

C

. Let
e
1
,e
2
,...,e
k
be the edges in
C
. For each
e
i
, try increasing the
capacity of
e
i
by 1 and compute a minimum cut in the new graph. Let
the new minimum cut be
C
i
, and denote its volume (in the new graph) as

C
i

. If

C
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 Winter '09
 Graph Theory, minimum cut, polynomial time algorithm, minimum st cut, Eulerian.

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