Abstract. This paper considers the fundamental problem of self-stabilizing leader election (SSLE) in the model of population protocols. In this model, an unknown number of asynchronous, anonymous and nite state mobile agents interact in pairs over a given communication graph. SSLE has been shown to be impossible in the original model. This impossibility can been circumvented by a modular technique augmenting the system with an oracle -an external module abstracting the added assumption about the system. Fischer and Jiang have proposed solutions to SSLE, for complete communication graphs and rings, using an oracle Ω?, called the eventual leader detector. In this work, we present a solution for arbitrary graphs, using a composition of two copies of Ω?. We also prove that the di culty comes from the requirement of self-stabilization, by giving a solution without oracle for arbitrary graphs, when an uniform initialization is allowed. Finally, we prove that there is no self-stabilizing implementation of Ω? using SSLE, in a sense we de ne precisely.