Proximinal sets and connectedness in graphs

Let $G$ be a graph with a vertex set $V$. The graph $G$ is path-proximinal if there are a semimetric $d\text{\hspace{0.17em}⁣}:V\times V\to [0,\mathrm{\infty }[$ and disjoint proximinal subsets of the semimetric space $(V,d)$ such that $V=A\cup B$, and vertices $u$, $v\in V$ are adjacent iff $$d(u,v)⩽\inf \{d(x,y)\text{\hspace{0.17em}⁣}:x\in A,y\in B\},$$ and, for every $p\in V$, there is a path connecting $A$ and $B$ in $G$, and passing through $p$. It is shown that a graph is path-proximinal if and only if all its vertices are not isolated. It is also shown that a graph is simultaneously proximinal and path-proximinal for an ultrametric if and only if the degree of every its vertex is equal to $1$.