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 ?


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