CME 305: Discrete Mathematics and Algorithms
Instructor: Professor Amin Saberi ([email protected])
Midterm – 02/28/10
Problem 1.
Show that a graph has a unique minimum spanning tree if, for every cut of the
graph, the edge with the smallest cost across that cut is unique. Show that the converse is
not true by giving a counterexample.
Solution:
Suppose MST is not unique, i.e., there exist
T
1
and
T
2
where both of them are
MST and they are not identical. Suppose
e
1
∈
T
1
but
e
1
/
∈
T
2
, if we remove
e
1
from
T
1
, then
we will have two trees with vertex sets
V
1
and
V
2
. By problem 1 of HW #2, we know that
e
1
is a minimum cost edge in the cut between
V
1
and
V
2
. Now consider
T
2
, again problem 1 of
HW #2, we know that
T
2
contains an edge
e
2
that is a minimum cost edge of the cut between
V
1
and
V
2
. However, since
e
2
6
=
e
1
we must have:
c
(
e
1
) =
c
(
e
2
) which is contradicting with
the assumption that for every cut of the graph, the edge with the smallest cost across that
cut is unique.
Counterexample for the converse:
suppose the graph is a tree with 3 nodes, where both
edges have cost 1.
Let
v
be the node with degree two.
Clearly the only spanning tree is
the graph itself, but the cut between
v
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 Spring '09
 Graph Theory, minimum cost edge

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