  For a graph $G$ and $a,b\in V(G)$, the shortest path reconfiguration graph of
$G$ with respect to $a$ and $b$ is denoted by $S(G,a,b)$. The vertex set of
$S(G,a,b)$ is the set of all shortest paths between $a$ and $b$ in $G$. Two
vertices in $V(S(G,a,b))$ are adjacent, if their corresponding paths in $G$
differ by exactly one vertex. This paper examines the properties of shortest
path graphs. Results include establishing classes of graphs that appear as
shortest path graphs, decompositions and sums involving shortest path graphs,
and the complete classification of shortest path graphs with girth $5$ or
greater. We also show that the shortest path graph of a grid graph is an
induced subgraph of a lattice.
