We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain $n \geq b_2-2$ disjoint smooth rational curves with self-intersection −2, where $b_2$ is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with $p_g = 0$ and $K^{\,2} = 8, 9$ which admit an automorphism of order 2.