  A graph polynomial $P$ is weakly distinguishing if for almost all finite
graphs $G$ there is a finite graph $H$ that is not isomorphic to $G$ with
$P(G)=P(H)$. It is weakly distinguishing on a graph property $\mathcal{C}$ if
for almost all finite graphs $G\in\mathcal{C}$ there is $H \in \mathcal{C}$
that is not isomorphic to $G$ with $P(G)=P(H)$. We give sufficient conditions
on a graph property $\mathcal{C}$ for the characteristic, clique, independence,
matching, and domination and $\xi$ polynomials, as well as the Tutte polynomial
and its specialisations, to be weakly distinguishing on $\mathcal{C}$. One such
condition is to be addable and small in the sense of C. McDiarmid, A. Steger
and D. Welsh (2005). Another one is to be of genus at most $k$.
