On the uniqueness of continuation of a partially defined metric

The problem of continuation of a partially defined metric can be efficiently studied using graph theory. Let $G=G(V,E)$ be an undirected graph with the set of vertices $V$ and the set of edges $E$. A necessary and sufficient condition under which the weight $w\text{\hspace{0.17em}⁣}:E\to {\mathbb{R}}^{+}$ on the graph $G$ has a unique continuation to a metric $d\text{\hspace{0.17em}⁣}:V\times V\to {\mathbb{R}}^{+}$ is found.