On $\ell_1$ embeddings of finite metric spaces, and sphere-of-influence graphs

TL;DR

This paper characterizes the pair-cut cone of metrics and provides conditions for $\, ext{ extlangle} ext{ extbackslash ell}_1$-embeddability into Euclidean space, with applications to sphere-of-influence graphs and graph embeddability.

Contribution

It introduces the pair-cut cone of metrics, characterizes its membership criteria, and establishes new conditions for $\, ext{ extlangle} ext{ extbackslash ell}_1$-embeddability of metrics and graphs.

Findings

Characterization of the pair-cut cone of metrics.

New criteria for $\, ext{ extlangle} ext{ extbackslash ell}_1$-embeddability.

Example of a graph not embeddable in the pair-cut cone.

Abstract

We introduce the {\em pair-cut cone $PCUT_n$} of metrics on sets with $n\ge 3$ elements, that correspond to linear combinations with non-negative coefficients of the cut-metrics resulting from cuts that are pairs. Given a metric, we fully characterize membership in the pair-cut cone in terms of quantities computed from the metric directly. We also prove a new result by which a metric $d$ that satisfies a system of inequalities, lies in the (full) cut cone of metrics, making it $\ell_1$-embeddable into Euclidean space. We give applications of our results to the $\ell_1$-embeddability of simple graphs into Euclidean space as {\em sphere-of-influence graphs}. We exhibit an example of a simple graph that admits no such $\ell_1$-metric in the pair-cut cone.