Computable Categoricity of Graphs with Finite Components

A computable graph is computably categorical if any two computable presentations of the graph are computably isomorphic. In this paper we investigate the class of computably categorical graphs. We restrict ourselves to strongly locally finite graphs; these are the graphs all of whose components are finite. We present a necessary and sufficient condition for certain classes of strongly locally finite graphs to be computably categorical. We prove that if there exists an infinite $\Delta_2^0$-set of components that can be properly embedded into infinitely many components of the graph then the graph is not computably categorical. We outline the construction of a strongly locally finite computably categorical graph with an infinite chain of properly embedded components.