Graphs with large distinguishing chromatic number

The distinguishing chromatic number chi(D)(G) of a graph G is the minimum number of colours required to properly colour the vertices of G so that the only automorphism of G that preserves colours is the identity. For a graph G of order n, it is clear that 1 <= chi(D)(G) <= n, and it has been shown that chi(D)(G) = n if and only if G is a complete multipartite graph. This paper characterizes the graphs G of order n satisfying chi(D)(G) = n - 1 or chi(D)(G) = n - 2.