A technique configures an intermediate network node to automatically determine whether a route advertised by a routing protocol is important for fast convergence in a computer network. As used herein, an important route needed for fast convergence is a route advertised by the routing protocol, such as an exterior gateway routing protocol (EGP) process executing on the node, as a next-hop address, since external connectivity relies on such a route. Notably, the EGP process interacts with an interior gateway routing protocol (IGP) process executing on the node to identify the route as an important route. Identification of an important route, in turn, allows IGP to process the route in a high priority fashion, thereby facilitating fast convergence.