I am interested in the following problem (A) and its related formulation (B). (A) Suppose that $G = (V,E,w)$ is an unknown weighted graph on the vertex set $V$ and that one has access to $d_G(v,v'), \forall v,v'\in V$, where $d_G(\cdot,\cdot)$ is the shortest path metric with respect to $G$. Can one recover the structure (i.e., the edge set $E$ and the weights $w$) of $G$ from just this distance information?
(B) Given a finite metric $(V,d)$, find the sparsest weighted graph $G=(V,E,w)$ that is consistent with $d$ in the sense that $d_G$ is the same as $d$.
I am curious to know if these problems or similar ones have been studied and have interesting answers.