Definability in first-order theories of graph orderings ⋆

Abstract

Abstract We study definability in the first-order theory of graph order: i.e. the set of all isomorphism types of simple finite graphs ordered by either the minor, subgraph or induced subgraph relation. Natural graph families like cycles and trees are definable in these orders, as also notions like connectivity, maximum degree, etc. This machinery allows us to show mutual interpretability with arithmetic for all orders. We discuss implications for formalizing statements of graph theory in such theories of order. 1