NP-completeness of the $\ell_1$-embedding problem for simple graphs as sphere-of-influence graphs

Abstract

In graph theory an interesting question is whether for a fixed choice of $p\in [0,\infty]$, all simple graphs appear as sphere-of-influence graphs in some Euclidean space with respect to the $\ell_p$ metric. The answer is affirmative for $p=\infty$, negative for any $p\in (0,\infty)$, and unknown for $p=1$. The result of this work shows that for the case of $p=1$, this embeddability question is a (Promise) NP-Complete problem.