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AMS/MBA 546 (Fall, 2010)
Estie Arkin
Network Flows: Solution sketch to homework set # 5
1). Show that the following problems are NPcomplete. You may use reductions from any of the problems
shown NPcomplete by Karp (3SAT, 3 dimensional matching, partition, vertex cover, Hamilton cycle/path,
or clique), or any other problem discussed in class. Remember to also show the problem is in NP.
(a). Given
G
= (
N,E
), is there a spanning tree
T
of
G
such that the nodes of
T
have degree at most 2?
The problem is in NP. A succinct certi±cate for “yes” instances is such a tree, which can be veri±ed in
polynomial time. Check whether the “tree” has
n

1 edges (and
n
nodes), and is it connected, thus showing
it is a tree. Then we check the degree of each node to see if it is 1 or 2. All this can be done in time
O
(
n
).
We reduce Hamilton Path to this problem. Given an instance of Hamilton Path, namely a graph
G
for which
we want to know if a Hamilton Path exists, we ask (for the same graph) whether a spanning tree exists with
all node degrees at most 2. Note that such a spanning tree is exactly a Hamilton Path!
(b). Given
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 Fall '08
 Arkin,E

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