The logic programming encoding of the set-theoretic graph property known as bisimulation is analyzed. This notion is of central importance in non-well-founded set theory, semantics of concurrency, model checking, and coinductive reasoning. From a modeling point of view, it is particularly interesting since it allows two alternative high-level characterizations. We analyze the encoding style of these modelings in various dialects of Logic Programming. Moreover, the notion also admits a polynomial-time maximum fixpoint procedure that we implemented in Prolog. Similar graph problems which are instead NP hard or not yet perfectly classified (e.g., graph isomorphism) can inherit most from the declarative encodings presented.