The author introduces the annihilator \(\lceil a,b\rceil\) of an undirected graph G to be the set of all vertices reachable from a by a geodesic via b, and relates it to a similar concept for lattices. \(\lceil a,b\rceil\) is called prime if \(\lceil a,b\rceil \cap \lceil b,a\rceil =\emptyset\) and \(\lceil a,b\rceil \cup \lceil b,a\rceil\) covers all the vertices of G. G is a prime annihilator intersection graph if each annihilator is the intersection of prime annihilators. Characterizations of the class of prime annihilator graphs (a superclass of the bipartite graphs) are given, and also some applications of annihilators to center problems.