While I am aware of this thread and others like it: https://stackoverflow.com/questions/16399597/edit-distance-between-two-graphs, I am wondering if the following has been considered and in what parlance, if any. I have searched Archiv and Google, but my search-fu is weak, not normally working in this space.

Let $P$ be a partition of $n$ vertices, and within each element of $P$ (a subset of vertices), form a complete graph. The resulting graph $G_P$ is then composed of a union of complete graphs which are mutually disjoint. Given another (arbitrary) graph $H$ on $n$ vertices, I am interested in finding the set of such $P$ (not necessarily unique) that minimize the edit distance between $H$ and $G_P$.

While I can think of an algorithm to brute force this (but NP: take all partitions of $n$ vertices, and compare them one at a time to $H$--this is NP because the number of partitions is a Bell number ($O(e^{e^x}))$, I am wondering if there are any nicer (non-NP would be fantastic) formulations that exist, given the restriction to partitions into complete graphs.