The paper considers a voting model where each voter's type is her preference. The type graph for a voter is a graph whose nodes are the possible types of the voter. A social choice function is locally strategy-proof if no type of a voter can gain by misrepresentation to a type that is connected to her true type by an edge in the type
graph. A social choice function is strategy-proof if no type of a voter can gain by misrepresentation to an arbitrary type. Local-Global equivalence (LGE) is satisfied if local strategy-proofness implies strategy-proofness. The paper identifies a condition on the graph that characterizes LGE. Our notion of ``localness'' is perfectly general
- we use this feature of our model to identify notions of localness according to which various models of multi-dimensional voting satisfy LGE. Finally, we show that LGE for deterministic social choice functions does not imply LGE for random social choice functions.