# Graphs with a polynomial number of shortest paths between any pair of vertices

Let $G$ be a simple undirected graph, and let $s$ and $t$ be two arbitrary vertices of $G$. Even for some rather restricted graph classes, the number of shortest paths between $s$ and $t$ can be exponential in the size of $G$. However, for some graph classes like block graphs, the number of shortest paths between $s$ and $t$ is 1 (such graphs are known as geodetic graphs).

For what graph classes $G$ is it true that between any pair of vertices $s$ and $t$ of $G$, the number of shortest paths between $s$ and $t$ is bounded from above by $\text{poly}(n,m)$? Here, $n$ is the number of vertices, and $m$ the number of edges.

For example, are there graphs (other than block graphs) that would have a constant number of shortest paths between any pair of vertices?