Two fixed sources admit bounded-degree planar unit-edge 10-net graphs whose graph distances approximate Euclidean distances up to additive constant; large ordered source sets admit a logarithmic obstruction in the coordinate-planar case.
Benjamini,Euclidean vs
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Euclidean vs Graph Metric: The Fixed-Source Problem
Two fixed sources admit bounded-degree planar unit-edge 10-net graphs whose graph distances approximate Euclidean distances up to additive constant; large ordered source sets admit a logarithmic obstruction in the coordinate-planar case.