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MATH 682
Problem Set #3
This problem set is due at the beginning of class on
February 25
. Below, “graph” means
“simple ﬁnite graph” except where otherwise noted.
1.
(15 points)
Demonstrate the following facts about a directed graph
D
.
(a)
(5 points)
Prove that
sum
v
∈
V
(
D
)
d

D
(
v
) =
∑
v
∈
V
(
D
)
d
+
D
(
v
). Recall that
d

and
d
+
represent the indegree and outdegree respectively.
(b)
(10 points)
Prove that a directed Eulerian tour (i.e. a directed closed trail
traversing every edge) exists if and only if
d

(
v
) =
d
+
(
v
) for all
v
∈
V
(
D
).
2.
(10 points)
Suppose each edge in a digraph
D
with a source
s
and sink
t
has not only
a maximum capacity
c
(
e
) but also a
minimum required ﬂow
b
(
e
), so that a ﬂow
f
must
satisfy
b
(
e
)
≤
f
(
e
)
≤
c
(
e
). Assume we know of a legal but not necessarily maximal
ﬂow
f
0
. Explain how the FordFulkerson algorithm could be modiﬁed to produce a
maximum ﬂow under these constraints.
3.
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This note was uploaded on 01/12/2012 for the course MATH 682 taught by Professor Wildstrom during the Spring '09 term at University of Louisville.
 Spring '09
 WILDSTROM
 Math

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