  A vertex k-labeling of graph G is distinguishing if the only automorphism
that preserves the labels of G is the identity map. The distinguishing number
of G, D(G), is the smallest integer k for which G has a distinguishing
k-labeling. In this paper, we apply the principle of inclusion-exclusion and
develop recursive formulas to count the number of inequivalent distinguishing
k-labelings of a graph. Along the way, we prove that the distinguishing number
of a planar graph can be computed in time polynomial in the size of the graph.}
