Section 12.5
(c) The final backtracking is 17, 13,
1.
(d) The final backtracking is 15, 14, 12,
11, 10, 3, 2,
1.
14
349
15
16
15
1
7
12
13
3
9
8
3. We use the strong form
of
mathematical induction on
k,
a vertex label, the result being clearly true
for
k
=
1.
Suppose
k
>
1 and the result is true for all
f
in the interval 1
~
f
<
k;
thus, there is a
path from 1 to
f
which uses only edges required by the algorithm. We must prove the result for
k:
We
must prove that there is a path from 1 to
k
which uses only edges required by the algorithm. Vertex
k
acquires its label because the algorithm finds this vertex unlabeled but adjacent to a previously labeled
vertex
f.
Since at the time a vertex is labeled, the algorithm always picks the smallest unused label, we
must have
f
<
k.
So, by the induction hypothesis, there is a path from 1 to
f
using edges required by
the algorithm; also, the algorithm uses the edge
fk
to label
k.
Thus, there is a path from 1 to
k
along
edges which the algorithm requires.
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 Summer '10
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 Graph Theory, Mathematical Induction, Backtracking, Depthfirst search

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