Configuration of planar electrical networks with and without double adduction

We consider power distribution networks containing source nodes and electricity consumer nodes. These nodes are interconnected by a switched network. Configuring such networks consists in deciding which switches are activated and the orientation of the links between these switches, to obtain a directed acyclic graph (DAG) from the producer nodes to the consumer nodes. This DAG is valid if the electric induced flow satisfies the demand of each consumer without exceeding the production capacity of each source and the flow capacity of each switch. Besides, unlike conventional flows, the distribution of the flow on the DAG cannot be arbitrarily chosen and in some cases, network operators prefer that consumers should only be supplied by a single source. We focus on planar graphs and show that the problem of deciding if such a valid DAG exists is NP-complete in general case and weakly NP-complete in case there is only one source.