A class name for series-parallel graphs of same length

I'm currently working on graphs classes where the distance between two specific vertices is the same in every connected spanning subgraphs, and I am looking for a name for this class.

Given a connected non-directed graph $$G$$ and two of its vertices $$u,v$$ such that $$G$$ is a two-terminal series-parallel graph (TTSPG) from $$u$$ to $$v$$ and every path from $$u$$ to $$v$$ is of the same length (equal to $$d(u,v)$$), is there a name for such graphs ?

Given a graph $$G$$ and $$u,v$$ two vertices, $$H \subseteq G$$ the induced subgraph formed of all possible paths from $$u$$ to $$v$$ (or excluding any vertex that is not in a path from $$u$$ to $$v$$), the distance between $$u$$ and $$v$$ is preserved in spanning connected subgraphs of $$G$$ if and only if $$H$$ is a TTSPG with fixed length between $$u$$ and $$v$$. Does this graph class (or the definition of $$H$$) has a known name ?