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ENGRI 115
Engineering Applications of OR
Fall 2007
Solution to Question 4
Homework 1
Question 4:
Decide whether the following statements are TRUE or FALSE. If you think a statement is
true, you need to justify it (in a few sentences). If you think a statement is false, give a speciﬁc example to
show this.
1. Let
G
= (
V, E
) be an undirected graph,
v
∈
V
any node in
G
, and
e
∈
E
any “cheapest” (i.e., lowest
cost) edge incident to
v
. Then,
some
minimum spanning tree contains
e
.
2. Let
G
= (
V, E
) be an undirected graph,
v
∈
V
any node in
G
, and
T
any minimum spanning tree in
G
. Then,
T
must contain some cheapest edge incident to
v
.
Answer:
1. TRUE. The answer needs two arguments. We proved that Prim’s algorithm produces a minimum
spanning tree (argument 1). Let’s focus on one of the cheapest edges incident to
v
, call it
e
(note:
there can be several cheapest edges). Prim’s algorithm can be started at node
v
, and it can choose
e
as ﬁrst
edge (since ties can be broken arbitrarily) (argument 2). Thus, the MST produced by Prim
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This note was uploaded on 06/01/2008 for the course ENGRI 1101 taught by Professor Trotter during the Fall '05 term at Cornell University (Engineering School).
 Fall '05
 TROTTER

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