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This question is inspired by the Graph Reconstruction Conjecture. Suppose that $\psi$ is some graph invariant and that it is NP-Hard. There is a plethora of examples, of course. Now define $D_{\psi}(G)=\frac{\psi(G)}{\sum_{v \in V(G)}{\psi(G-v)}}$. Let's call this the "deck ratio" of $\psi$.

Is $D_{\psi}$ NP-Hard?

EDIT: Per Andrew King's suggestion, let us stipulate that is $\psi(G-v)$ takes at least two distinct values.

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# If a graph invariant is NP-Hard, is it "deck ratio" NP-Hard as well?

This question is inspired by the Graph Reconstruction Conjecture. Suppose that $\psi$ is some graph invariant and that it is NP-Hard. There is a plethora of examples, of course. Now define $D_{\psi}(G)=\frac{\psi(G)}{\sum_{v \in V(G)}{\psi(G-v)}}$. Let's call this the "deck ratio" of $\psi$.

Is $D_{\psi}$ NP-Hard?