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.  For a given $N\le \#V$ when 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 $d_{(V_n,E_n)}\ge d_G$)*.