  Given a finite connected simple graph $G=(V,E)$ with vertex set $V$ and edge
set $E\subseteq \binom{V}{2}$, we will show that
  $1.$ the (necessarily unique) smallest block graph with vertex set $V$ whose
edge set contains $E$ is uniquely determined by the $V$-indexed family ${\bf
P}_G:=\big(\pi_0(G^{(v)})\big)_{v \in V}$ of the various partitions
$\pi_0(G^{(v)})$ of the set $V$ into the set of connected components of the
graph $G^{(v)}:=(V,\{e\in E: v\notin e\})$,
  $2.$ the edge set of this block graph coincides with set of all $2$-subsets
$\{u,v\}$ of $V$ for which $u$ and $v$ are, for all $w\in V-\{u,v\}$, contained
in the same connected component of $G^{(w)}$,
  $3.$ and an arbitrary $V$-indexed family ${\bf P}p=({\bf p}_v)_{v \in V}$ of
partitions $\pi_v$ of the set $V$ is of the form ${\bf P}p={\bf P}p_G$ for some
connected simple graph $G=(V,E)$ with vertex set $V$ as above if and only if,
for any two distinct elements $u,v\in V$, the union of the set in ${\bf p}_v$
that contains $u$ and the set in ${\bf p}_u$ that contains $v$ coincides with
the set $V$, and $\{v\}\in {\bf p}_v$ holds for all $v \in V$.
  As well as being of inherent interest to the theory of block graphs, these
facts are also useful in the analysis of compatible decompositions and block
realizations of finite metric spaces.
