This paper considers a diamond network with $n$ interconnected relays, namely a network where a source communicates with a destination by hopping information through $n$ communicating/interconnected relays. Specifically, the main focus of the paper is on characterizing sufficient conditions under which the $n+1$ states (out of the $2^{n}$ possible ones) in which at most one relay is transmitting suffice to characterize the approximate capacity, that is the Shannon capacity up to an additive gap that only depends on $n$. Furthermore, under these sufficient conditions, closed form expressions for the approximate capacity and scheduling (that is, the fraction of time each relay should receive and transmit) are provided. A similar result is presented for the dual case, where in each state at most one relay is in receive mode.