Flows in networks with barrier reachability

The problem of flows in networks with barrier-type reachability restrictions is considered. We introduce new definitions that allow one to describe a flow in a network with reachability constraints, in particular, a representation of a flow as a vector-valued function. Conditions for preserving the flow and restricting the maximum flow along an arc are formulated in terms of vector-valued functions. This allows one to consider flow problems without passing to an unfolding, which is a graph with connected arcs.