Let S be a closed, connected, orientable surface of genus at least 3, C(S) be the complex of curves on S and Mod_S^* be the extended mapping class group of S. We prove that a simplicial map, lambda : C(S) -> C(S), preserves nondisjointness (i.e. if alpha and beta are two vertices in C(S) such that i(alpha, beta) \noteq 0, then i(lambda(alpha), lambda(beta)) \noteq 0) iff it is induced by a homeomorphism of S. As a corollary, we prove that if K is a finite index subgroup of Mod_S^* and f : K -> Mod_S^* is an injective homomorphism, then f is induced by a homeomorphism of S and f has a unique extension to an automorphism of Mod_S^*.