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00 f g (a) Let S = {h, c, d, g }. For each pair of vertices x, y ∈ S , ﬁnd a shortest path with endpoints x, y
and also d(x, y ).
(b) Find a shortest walk W , with endpoints p, h, that visits every edge of G at least once. What is
the weight of the walk that you found?
Solution. (a) By trials and errors (or by inspection, or by Dijkstra’s algorithm), we ﬁnd the following
table where each entry represents a shortest path and its length. Since there is a symmetry in it,
we only give the entries that are above the diagonal.
h
c
d
g h
–
–
–
– c
h, e, b, c; 8
–
–
– d
h, e, d; 7
c, b, e, d; 7
–
– g
h, i, f, g ; 8
c, b, e, i, f, g ; 9
d, e, i, f, g ; 8
– (b) In a walk that begins at h and ends at p, the degrees of all vertices, except those of h, p, must
be even while the degrees of p and h must be odd. Now since the degrees of c, d, g are odd and
that of h is even, we see that it is not possible to have a walk from p to h that uses each edge of G
exactly once. That is, in order to have a walk that uses every edge at least once, some edges have
to be repeated. Which edges to repeat? Well, the repeated edges must form paths between the
vertices c, d, g, h. In order for the walk to have a minimum weight, we see that we need two paths
between the vertices c, d, g, h so that each vertex is the endpoint of exactly one path. The graph
below represents the shortest distance between the vertices h, c, d, g .
h c 8
7 7 d 8 89
g By trials and errors (or by a minimum weight perfect matching algorithm), one ﬁnds that the
shortest path between c, d (represented by the edge {c, d}) and the shortest path between g, h
(represented by the edge {g, h}) have a minimum total distance of 15. Thus, we should repeat the
edges on the shortest paths c, b, e, d and h, i, f, g . We therefore have the following graph (where the
dotted lines represent repeated edges):
c
7 2 p 4 e i
3 1 3 2 5 61
5 a 1
f 4 h 8 3
b2 4 d 8 g 6...
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This note was uploaded on 01/13/2014 for the course MAD 5305 taught by Professor Suen during the Spring '12 term at University of South Florida  Tampa.
 Spring '12
 Suen

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