Coding in graphs and linear orderings

There is a Turing computable embedding (I) of directed graphs A in undirected graphs (see [15]). Moreover, there is a fixed tuple of formulas that give a uniform effective interpretation; i.e., for all directed graphs A, these formulas interpret A in Phi(A). It follows that A is Medvedev reducible to Phi(4) uniformly; i.e., A <=(s) Phi(4) with a fixed Turing operator that serves for all A. We observe that there is a graph G that is not Medvedev reducible to any linear ordering. Hence, G is not effectively interpreted in any linear ordering. Similarly, there is a graph that is not interpreted in any linear ordering using computable Sigma(2) formulas. Any graph can be interpreted in a linear ordering using computable Sigma(3) formulas. Friedman and Stanley [4] gave a Turing computable embedding L of directed graphs in linear orderings. We show that there is no fixed tuple of L-omega vertical bar omega -formulas that, for all G, interpret the input graph G in the output linear ordering L(G). Harrison-Trainor and Montalban [7] have also shown this, by a quite different proof.