This paper concerns the problems of finding optimal trajectories between nodes on a network, which must be periodically surveyed, and probably serviced. It is shown, that such trajectories may be generated if optimal Hamiltonian cycles are used between the separate network nodes under inspection. It is known that the Hamiltonian path problem is NP-complete, but an edge decomposition of the network is proposed. This is performed by reducing in a particular way to network flow circulations. The requirements and the equations for describing such circulation are pointed out. Defining of the optimal circulations of the mobile agents is reduced to network flow programming problems. A numerical example is presented for solving a similar class of monitoring problems by mobile agents.
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