  For any graph $G$ with $a,b\in V(G)$, a shortest path reconfiguration graph
can be formed with respect to $a$ and $b$; we denote such a graph as
$S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths from
$a$ to $b$ in $G$ while two vertices $U,W$ in $V(S(G,a,b))$ are adjacent if and
only if the vertex sets of the paths that represent $U$ and $W$ differ in
exactly one vertex. In a recent paper [Asplund et al., \textit{Reconfiguration
graphs of shortest paths}, Discrete Mathematics \textbf{341} (2018), no. 10,
2938--2948], it was shown that shortest path graphs with girth five or greater
are exactly disjoint unions of even cycles and paths. In this paper, we extend
this result by classifying all shortest path graphs with no induced $4$-cycles.
