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

TL;DR

This paper proves that determining whether a simple graph can be represented as a sphere-of-influence graph in Euclidean space under the $\, ext{l}_1$ metric is an NP-Complete problem, highlighting computational complexity in geometric graph embeddings.

Contribution

It establishes the NP-Completeness of the $\, ext{l}_1$-embedding problem for simple graphs, filling a gap in understanding the complexity of sphere-of-influence graph representations.

Findings

NP-Completeness of the $\, ext{l}_1$-embedding problem proven

Completeness result contrasts with known cases for $p=\, ext{infinity}$ and $p eq 1$

Highlights computational difficulty in geometric graph embedding problems

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.