Graphs in which all shortest paths are unique I'm looking for undirected, unweighted, connected graphs $G=(V,E)$, in which for every pair $u,v \in V$, there is a unique $u \rightarrow v$ path that realizes the distance $d(u,v)$.

Is this class of graphs well-known? What other properties does it have? For example, every tree is of this kind, as well as every graph without an even cycle. However, there are graphs containing even cycles that are of this kind.
According to Information System on Graph Classes and their Inclusions, these graphs are studied under the name “geodetic graphs.”