When Is Reachability Intrinsically Decidable?

A graph $\mathcal{H}$ is computableif there is a graph $\mathcal{G}=(V,E)$ isomorphic to $\mathcal{H}$ where the set Vof vertices and the edge relation Eare both computable. In this case $\mathcal{G}$ is called a computable copyof $\mathcal{H}$. The reachability problemfor $\mathcal{H}$ in $\mathcal{G}$ is, given u,w茂戮驴 V, to decide whether there is a path from uto w. If the reachability problem for $\mathcal{H}$ is decidable in allcomputable copies of $\mathcal{H}$ then the problem is intrinsically decidable. This paper provides syntactic-logical characterizations of certain classes of graphs with intrinsically decidable reachability relations.