Defining Recursive Predicates in Graph Orders

We study the first order theory of structures over graphs i.e. structures of the form (G, τ ) where G is the set of all (isomorphism types of) finite undirected graphs and τ some vocabulary. We define the notion of a recursive predicate over graphs using Turing Machine recognizable string encodings of graphs. We introduce the notion of a capable structure over graphs, which is one satisfying the conditions : (1) definability of arithmetic, (2) definability of cardinality of a graph, and (3) definability of two particular graph predicates related to vertex labellings of graphs. We then show any capable structure can define every recursive predicate over graphs. We identify capable structures which are expansions of graph orders, which are structures of the form (G,≤) where ≤ is a partial order. We show that the subgraph order i.e. (G,≤s), induced subgraph order with one constant P3 i.e. (G,≤i, P3) and an expansion of the minor order for counting edges i.e. (G,≤m, sameSize(x, y)) are capable structures. In the course of the proof, we show the definability of several natural graph theoretic predicates in the subgraph order which may be of independent interest. We discuss the implications of our results and connections to Descriptive Complexity.