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Section
14.4
391
are edge disjoint. Thus, there are at least m edge disjoint paths from
s
to t. But there cannot be
more since deg
s
=
m and edge disjoint paths must start with different edges. We claim that m
is also the minimum number of edges which must be removed to sever all paths from
s
to
t
(in
accordance with Menger's Theorem). To sever all such paths, we must remove at least m edges
since for every
v
E
V2,
either
sv
or
vt
must be removed. On the other hand, removing all m edges
incident with
s
clearly severs all paths from
s
to
t
so the minimum number of edges that have to
be removed is m.
(b) Employing the notation of (a), suppose
s
is in VI and
t
is in
V2.
The maximum number of edge
disjoint paths between
s
and
t
and the minimum number of edges that must be removed in order
to sever all paths from
s
to
t
is mini m,
n}
(in accordance with Menger's Theorem).
Without loss of generality, we assume that m ::;
n,
so that the minimum of m and
n
is m. Let
VI, V2, .
.. ,
VmI
be the vertices of
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 Summer '10
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 Graph Theory

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