Mathematical definitions of enclave and exclave, and applications

•We presented mathematical definitions of enclave and exclave.•We presented mathematical properties of enclave and exclave.•We showed applications of the definitions to the problems of political boundaries.

This paper offers mathematical definitions of enclave and exclave using a graph theoretical framework, and gives some applications of the definitions. Some basic properties of enclave and exclave are also verified by using the definitions. Though there are numerous enclaves and exclaves in the world, they have not been analyzed mathematically so far. Giving them mathematical definitions is useful for the problems of political boundaries, e.g. the drawing of the boundaries of single member electoral districts and the economic analysis of political integration.