Block graphs are a generalization of trees that arise in areas such as metric graph theory, molecular graphs, and phylogenetics. Given a finite connected simple graph G=(V,E) with vertex set V and edge set Eβ(2Vβ), we will show that the (necessarily unique) smallest block graph with vertex set V whose edge set contains E is uniquely determined by the V-indexed family \Pp_G =\big(\pi_v)_{v \in V} of the partitions Οvβ of the set V into the set of connected components of the graph (V,{eβE:vβ/e}). Moreover, we show that an arbitrary V-indexed family \Pp=(\p_v)_{v \in V} of partitions \p_v of the set V is of the form \Pp=\Pp_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,vβV, the union of the set in \p_v that contains u and the set in \p_u that contains v coincides with the set V, and \{v\}\in \p_v 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 of finite metric spaces