An electronic device selectively couples a head with links in a graphics processing unit to a currently selected display port in a pair of display ports. During operation, control logic in the electronic device monitors a pair of configuration signals from the pair of display ports, where the pair of configuration signals correspond to physical connections to the pair of display ports. Then, the control logic determines a selection control signal based on the monitored pair of configuration signals, a policy setting and a default display port, where the selection control signal specifies the currently selected display port. Moreover, the control logic provides the selection control signal to a multiplexer in the electronic device. Next, the multiplexer selectively couples the head with the links in the graphics processing unit to the currently selected display port based on the selection control signal.