Given a finite connected simple graph G=(V,E) with vertex set V and edge
set Eβ(2Vβ), 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 PGβ:=(Ο0β(G(v)))vβVβ of the various partitions
Ο0β(G(v)) of the set V into the set of connected components of the
graph G(v):=(V,{eβE:vβ/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 wβVβ{u,v}, contained
in the same connected component of G(w),
3. and an arbitrary V-indexed family Pp=(pvβ)vβVβ of
partitions Οvβ of the set V is of the form Pp=PpGβ for some
connected simple graph G=(V,E) with vertex set V as above if and only if,
for any two distinct elements u,vβV, the union of the set in pvβ
that contains u and the set in puβ that contains v coincides with
the set V, and {v}βpvβ holds for all vβV.
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