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arXiv · 2606.13271

Euclidean vs Graph Metric: The Fixed-Source Problem

Itai Benjamini
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Abstract

We prove that two fixed sources in the Euclidean plane can be realized by a bounded-degree planar unit-edge graph on a 10-net, with graph distance from each source agreeing with Euclidean distance up to a universal additive constant. We ask whether the analogous statement holds for three non-collinear sources, and prove a logarithmic obstruction for large ordered source sets in the coordinate-planar setting.

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Itai Benjamini. 2026-06-11. Euclidean vs Graph Metric: The Fixed-Source Problem. https://arxiv.org/abs/2606.13271

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