Currently, software-defined networking (SDN) platforms leverage the link-layer discovery protocol (LLDP) to discover the underlying topology. However, the LLDP is suboptimal in terms of message load. In this letter, we present the tree exploration discovery protocol (TEDP), proving that shortest paths can be built at the same time that the topology information is gathered, without extra messages compared with LLDP. We also analyze two alternative implementations for the TEDP and give insights into some features that SDN platforms should ideally provide for an efficient topology discovery service.