To serve the growing demand of the bidirectional information exchange, we propose a single relay selection (RS) scheme for physical-layer network coding (PNC) in a bidirectional cooperative network consisting of two sources and multiple relays. This RS scheme selects a single best relay by maximizing the bottleneck of the capacity region of both information flows in the bidirectional network. We show that the proposed RS rule minimizes the outage probability and that it can be used as a performance benchmark for any RS rules with PNC. We derive a closed-form exact expression of the outage probability for the proposed RS rule and show that it achieves full diversity gain. Finally, numerical results demonstrate the validity of our analysis.