Let G be a finite graph of order n with an eigenvalue ? of multiplicity k. (Thus the ?-eigenspace of a (0,1)-adjacency matrix of G has dimension k.) A star complement for ? in G is an induced subgraph G-X of G such that |X|=k and G-X does not have ? as an eigenvalue. An exceptional graph is a connected graph, other than a generalized line graph, whose eigenvalues lie in [-2,?). We establish some properties of star complements, and of eigenvectors, of exceptional graphs with least eigenvalue -2.