This paper considers a single location on an undirected network with unreliable edges that maximizes the lowest performance level of the network service with respect to all nodes. The problem is termed the reliable 1-center problem and finds applications in telecommunication and computer networks. The users are concerned with the network capability of establishing a route to some service provider. The objective function is formally stated as either minimizing the maximum expected number of unsuccessful responses to demand requests over all nodes, named the reli-minmax problem, or maximizing the minimum expected number of successful responses to demand requests over all nodes, named the reli-maxmin problem, as sub-problems of the most general reliable 1-center problem. Solutions are presented that solve the problem in polynomial time or reduce it to the 1-center problem when it is applied on general networks using a pre-designated route policy, e.g. the most reliable route policy. © 2008 Elsevier Ltd. All rights reserved.