Computable isomorphisms for certain classes of infinite graphs

Abstract

We investigate (2,1):1 structures, which consist of a countable set [Formula: see text] together with a function [Formula: see text] such that for every element [Formula: see text] in [Formula: see text], [Formula: see text] maps either exactly one element or exactly two elements of [Formula: see text] to [Formula: see text]. These structures extend the notions of injection structures, 2:1 structures, and (2,0):1 structures studied by Cenzer, Harizanov, and Remmel, all of which can be thought of as infinite directed graphs. We look at various computability-theoretic properties of (2,1):1 structures, most notably that of computable categoricity. We say that a structure [Formula: see text] is computably categorical if there exists a computable isomorphism between any two computable copies of [Formula: see text]. We give a sufficient condition under which a (2,1):1 structure is computably categorical, and present some examples of (2,1):1 structures with different computability-theoretic properties.