Distinguishing Maps II: General Case

A group A acting faithfully on a set X has distinguishing number k, written D(A,X) = k, if there is a coloring of the elements of X with k colors such that no nonidentity element of A is color-preserving, and no such coloring with fewer than k colors exists. Given a map M with vertex set V and automorphism group Aut(M), let D(M) = D(Aut(M), V ). If M is orientable, let D+(M) = D(Aut+(M), V ), where Aut+(M) is the group of orientation-preserving automorphisms. In a previous paper, the author showed there are four maps M with D+(M) > 2. In this paper, a complete classification is given for the graphs underlying maps with D(M) > 2. There are 31 such graphs, 22 having no vertices of valence 1 or 2, and all have at