  Given a set $\Sg$ of bipartitions of some finite set $X$ of cardinality at
least 2, one can associate to $\Sg$ a canonical $X$-labeled graph $\B(\Sg)$,
called the Buneman graph. This graph has several interesting mathematical
properties - for example, it is a median network and therefore an isometric
subgraph of a hypercube. It is commonly used as a tool in studies of DNA
sequences gathered from populations. In this paper, we present some results
concerning the {\em cut vertices} of $\B(\Sg)$, i.e., vertices whose removal
disconnect the graph, as well as its {\em blocks} or 2-{\em connected
components} - results that yield, in particular, an intriguing generalization
of the well-known fact that $\B(\Sg)$ is a tree if and only if any two splits
in $\Sg$ are compatible.
