352
Solutions to Exercises
k
+
1
~
r
+
1. Thus the number of edges on the path
UVl V2
...
VrV
is
r
2':
k,
which is precisely the
fact we wanted to show.
14. Let the bipartition sets of
JC2
,n
be {Ul'
U2}
and
{v!,
V2, .
..
,v
n }.
To show that all spanning trees are
obtained as depthfirst search spanning trees with
n
=
2,3,4 we could simply list the possible trees.
For example, the four spanning
trees of
JC2 ,2
are shown to the right,
and these are all depthfirst search
trees.
This approach is laborious, however, especially when
n
=
4, so we will give a general argument
instead.
Any spanning tree of
JC
2,n
must have some
Vi
adjacent to both
Ul and
U2,
and the remaining
Vj
are partitioned into those adjacent to
Ul and
those adjacent to
U2.
If
n
=
3 and the two other vertices (different from
Vi)
are both adjacent to the same
Ui,
say
U2,
we get a tree of the type
shown. This tree comes from a depthfirst search starting at Ul.
If
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 Summer '10
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 Graph Theory, Sets, Backtracking, Vertex, Depthfirst search, Spanning tree, connected component

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