AMS/MBA 546 (Spring, 2009)
Estie Arkin
Network Flows  Final
Do 4 of the following 6 problems. You are permitted to use the text (AMO), your notes and any material
handed out in class. The exam time is 2 hours and 30 minutes. GOOD LUCK!!!
Academic integrity is expected of all students at all times, whether in the presence or absence of members
of the faculty. Understanding this, I declare that I shall not give, use, or receive unauthorized aid in this
examination.
Name (PRINT CLEARLY), and Signature:
Problems to be graded:
1).
(a). A
Feedback edge set
of an undirected graph
G
= (
N,E
) is a subset of edges
E
′
⊆
E
whose
removal results in an acyclic graph
G
′
= (
N,E
\
E
′
). Given a graph with positive edge weights
w
ij
>
0,
describe a polynomial time algorithm that Fnds a minimum weight feedback edge set. Recall, this problem
is NP Complete for directed graphs.
(b). ±or a graph
G
= (
N,E
) deFne
conn
(
G
) the number of connected components of
G
. Prove that
conn
(
G
) +
m
≥
n
for every graph
G
.

N

=
n
, and

E

=
m
.
Comment: parts (a) and (b) are not related.
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 Fall '08
 Arkin,E
 Graph Theory, Estie Arkin

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