Abstract: | For any graph with , a shortest path reconfiguration graph can be formed with respect to and ; we denote such a graph as . The vertex set of is the set of all shortest paths from to in while two vertices in are adjacent if and only if the vertex sets of the paths that represent and differ in exactly one vertex. In a recent paper (Asplund et al., 2018), 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. |