m-route flows in a network

The m-route flow between two vertices is defined in a flow network which is an extension of the flow between two vertices. In the communication network, the m-route flow corresponds to the set of m-route structured communication paths. In a flow network, the set of m-edge disjoint paths connecting the two vertices considered and composed of edges with the same capacity is defined as the elementary edge-m-route flows between those two vertices. For the cut of the flow network, the edge-m-route capacity is defined as an extension of the ordinary capacity and it is shown that a theorem similar to the max-flow min-cut theorem as in the ordinary theory of flows applies to the maximum edge-m-route flow. It is shown also that, when the set of m internally disjoint paths connecting the two vertices considered and which is composed of edges of the same capacity is defined as the elementary vertex-m-route flow, the vertex-m-route flow between two vertices is defined as the sum of elementary vertex-m-route flows between those two vertices; and a similar theorem holds. In the communication network, the maximum m-route flow corresponds to the maximum number of communication paths that can be composed of the m-route structure between the two terminals.