Let $G=(V,E)$ be a finite undirected graph which we equip with its usual graph geodesic distance $d_G$ making $(G,d_G)$ into a metric space; let $1<\#V<\infty$. For a given $1<N< \#V$ what conditions do I need on $G$ so that does there exist disjoint subsets $V_1,\dots,V_N\subseteq V$ such that

• $\biguplus_{n=1}^N\, V_n = V$,
• $d_{(V_n,E_n)}(x,y)=d_{G}(x,y)$ for every $x,y\in V_n$,

Here $E_n:=\{(v,w):\,v,w\in V_n\}$ denotes the collection of edges connecting any two vertices in the "part" $V_n$ and where $d_{(V_n,E_n)}$ denotes the graph geodesic defined on the graph $(V_n,E_n)$ (note, for arbitrary choices of $\{V_n\}_{n=1}^N$ we always have $d_{(V_n,E_n)}\ge d_G$).