Order-Invariance in the Two-Variable Fragment of First-Order Logic

We study the expressive power of the two-variable fragment of order-invariant first-order logic. This logic departs from first-order logic in two ways: first, formulas are only allowed to quantify over two variables. Second, formulas can use an additional binary relation, which is interpreted in the structures under scrutiny as a linear order, provided that the truth value of a sentence over a finite structure never depends on which linear order is chosen on its domain. We prove that on classes of structures of bounded degree, any property expressible in this logic is definable in first-order logic. We then show that the situation remains the same when we add counting quantifiers to this logic.