For the object reallocation problem, we study whether characterizations of Top Trading Cycles (TTC) based on individual rationality, efficiency, and strategyproofness on the unrestricted domain extend to restricted preference domains. We introduce the top-two condition and show that it offers a useful criterion for answering this question. The condition requires that, within every subset of objects, any two objects that can each be ranked first can also be ranked as the top two, in both possible orders. We first show that this condition is sufficient: on every domain satisfying the top-two condition, TTC is the unique rule satisfying the relevant axioms. We also provide a partial converse. For domains that fail the top-two condition within a small subset of objects and satisfy a mild extension condition, we construct a rule distinct from TTC satisfying these axioms. Our results provide a unifying perspective on existing findings for specific domains, such as the single-peaked and single-dipped domains, while also addressing several previously unexplored domains, including the circular and partial-agreement domains.

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