  The persistence diagram (PD) is an increasingly popular topological
descriptor. By encoding the size and prominence of topological features at
varying scales, the PD provides important geometric and topological information
about a space. Recent work has shown that well-chosen (finite) sets of PDs can
differentiate between geometric simplicial complexes, providing a method for
representing complex shapes using a finite set of descriptors. A related
inverse problem is the following: given a set of PDs (or an oracle we can query
for persistence diagrams), what is underlying geometric simplicial complex? In
this paper, we present an algorithm for reconstructing embedded graphs in
$\mathbb{R}^d$ (plane graphs in $\mathbb{R}^2$) with $n$ vertices from $n^2 - n
+ d + 1$ directional (augmented) PDs. Additionally, we empirically validate the
correctness and time-complexity of our algorithm in $\mathbb{R}^2$ on randomly
generated plane graphs using our implementation, and explain the numerical
limitations of implementing our algorithm.
