During the past decade, there has been an extensive investigation of the computational complexity of the consistent answers of Boolean conjunctive queries under primary key constraints. Much of this investigation has focused on self-join-free Boolean conjunctive queries. In this paper, we study the consistent answers of Boolean conjunctive queries involving a single binary relation, i.e., we consider arbitrary Boolean conjunctive queries on directed graphs. In the presence of a single key constraint, we show that for each such Boolean conjunctive query, either the problem of computing its consistent answers is expressible in first-order logic, or it is polynomial-time solvable, but not expressible in first-order logic.