To locally complement a simple graph F at one of its vertices v is to replace the subgraph induced by F on n(v)={w: vw is an edge of F} by the complementary subgraph. Graphs related by a sequence of local complementations are said to be locally equivalent. We describe invariants of locally equivalent graphs and a polynomial algorithm to recognize locally equivalent graphs. An application is given to counting the number of graphs locally equivalent to a given one.
