Feasible graphs with standard universe Original Research Article

A computable graph is constructed which is not computably isomorphic to any polynomial-time graph with a standard universe (that is, the natural numbers represented either in tally or binary notation). Conditions are given under which a computable graph is computably isomorphic to a polynomial-time graph with a standard universe — for example, if every vertex has finite degree. Two special types of graphs are studied. It is shown that any computable tree is recursively isomorphic to a p-time tree with standard universe and that any computable equivalence relation is computably isomorphic to a p-time equivalence relation with standard universe.