In this article, we consider the cooperative output regulation of singular multiagent systems under jointly connected switching topologies. The original problem is transformed into a stabilization problem by model transformation with the proposed distributed controller. Different from most existing controllers, a new graph-based switching compensator and its corresponding communication protocol are synthesized into the controller. The communication network does not remain connected at any time. And a triggered mechanism is designed to ensure that the communication between each pair of agents does not need to take place continuously. In the triggered mechanism, a novel design pattern is adopted to avoid Zeno behavior. The method in this article can reduce the computation and communication, and obtain a better tracking effect at the same time compared to previous results. Finally, a numerical simulation is given to show the effectiveness of the controller and the triggered mechanism.