  Many applications in pattern recognition represent patterns as a geometric
graph. The geometric graph distance (GGD) has recently been studied as a
meaningful measure of similarity between two geometric graphs. Since computing
the GGD is known to be $\mathcal{NP}$-hard, the distance measure proves an
impractical choice for applications. As a computationally tractable
alternative, we propose in this paper the Graph Mover's Distance (GMD), which
has been formulated as an instance of the earth mover's distance. The
computation of the GMD between two geometric graphs with at most $n$ vertices
takes only $O(n^3)$-time. Alongside studying the metric properties of the GMD,
we investigate the stability of the GGD and GMD. The GMD also demonstrates
extremely promising empirical evidence at recognizing letter drawings from the
{\tt LETTER} dataset \cite{da_vitoria_lobo_iam_2008}.
