Definability in First Order Theories of Graph Orderings

We study definability in the first order theory of graph order: that is, the set of all simple finite graphs ordered by either the minor, subgraph or induced subgraph relation. We show that natural graph families like cycles and trees are definable, as also notions like connectivity, maximum degree etc. This naturally comes with a price: bi-interpretability with arithmetic. We discuss implications for formalizing statements of graph theory in such theories of order.

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