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We study the expressive power of the two-variable fragment of order-invariantfirst-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 thestructures under scrutiny as a linear order, provided that the truth value of asentence over a finite structure never depends on which linear order is chosenon its domain. We prove that on classes of structures of bounded degree, anyproperty expressible in this logic is definable in first-order logic. We thenshow that the situation remains the same when we add counting quantifiers tothis logic.

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