Suppose we have a complete graph invariant $I$ i.e. for any two graphs $G$ and $H$ we have $ I(G)=I(H) \iff G \cong H $. Suppose now that the invariant $I$ is reconstructible from the desk of $G.$ May I deduce now that the Reconstruction Conjecture is true?
For example, consider a graph $G$ and let $G_1, G_2,\ldots, G_N$ be the set of representatives of all clases isomorphic subgraphs of $G$. Let $I(G)=[k_1(G), k_2(G), \ldots, k_N(G)]$ where $k_i(G)$ is number of subgraphs of $G$ which are isomorphic to the subgraph $G_i.$ We have that $I(G)$ is a complete invariant and each $k_i(G)$ is reconstructible (it follows from Kelly formula) then the Reconstruction Conjecture is true.
Where is my mistake?