303 research outputs found
A Graph Theoretical Approach to Network Encoding Complexity
Consider an acyclic directed network with sources and
distinct sinks . For , let denote the
min-cut between and . Then, by Menger's theorem, there exists a
group of edge-disjoint paths from to , which will be referred
to as a group of Menger's paths from to in this paper. Although
within the same group they are edge-disjoint, the Menger's paths from different
groups may have to merge with each other. It is known that by choosing Menger's
paths appropriately, the number of mergings among different groups of Menger's
paths is always bounded by a constant, which is independent of the size and the
topology of . The tightest such constant for the all the above-mentioned
networks is denoted by when all 's are
distinct, and by when all 's are in
fact identical. It turns out that and are closely
related to the network encoding complexity for a variety of networks, such as
multicast networks, two-way networks and networks with multiple sessions of
unicast. Using this connection, we compute in this paper some exact values and
bounds in network encoding complexity using a graph theoretical approach.Comment: 44 pages, 22 figure
Menger's Theorem in bidirected graphs
Bidirected graphs are a generalisation of directed graphs that arises in the
study of undirected graphs with perfect matchings. Menger's famous theorem -
the minimum size of a set separating two vertex sets and is the same as
the maximum number of disjoint paths connecting them - is generally not true in
bidirected graphs. We introduce a sufficient condition for and which
yields a version of Menger's Theorem in bidirected graphs that in particular
implies its directed counterpart.Comment: 23 pages, 6 figure
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