  Stable topological invariants are a cornerstone of persistence theory and
applied topology, but their discriminative properties are often
poorly-understood. In this paper we study a rich homology-based invariant first
defined by Dey, Shi, and Wang, which we think of as embedding a metric graph in
the barcode space. We prove that this invariant is locally injective on the
space of metric graphs and globally injective on a GH-dense subset. Moreover,
we show that is globally injective on a full measure subset of metric graphs,
in the appropriate sense.
