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I remember having heard once that generic spin-graphs e.g. Ising, or at least 2-local ones, can always be mapped to one another one (which will typically be larger), where every spin has at most $n$ neighbours. I'm not completely sure about the value of $n$, but seem to remember $n=3$ works. This also implies that finding ground states of 2-local Hamiltonians with max $n$ neighbours is an NP-hard problem.

Could someone confirm this and point me to a reference?

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