Abstract. A graph H is computable if there is a graph G = (V, E) isomorphic to H where the set V of vertices and the edge relation E are both computable. In this case G is called a computable copy of H. The reachability problem for H in G is, given u, w ∈ V, to decide whether there is a path from u to w. If the reachability problem for H is decidable in all computable copies of H then the problem is intrinsically decidable. This paper provides syntactic-logical characterizations of certain classes of graphs with intrinsically decidable reachability relations. 