  Several algebraic properties of a binomial edge ideal $J_G$ can be
interpreted in terms of combinatorial properties of its associated graph $G$.
In particular, the so-called cut-point sets of a graph $G$, special sets of
vertices that disconnect $G$ in a minimal way, play an important role since
they are in bijection with the minimal prime ideals of $J_G$. In this paper we
establish the first graph-theoretical characterization of binomial edge ideals
$J_G$ satisfying Serre's condition $(S_2)$ by proving that this is equivalent
to having $G$ accessible, which means that $J_G$ is unmixed and the cut-point
sets of $G$ form an accessible set system. The proof relies on the
combinatorial structure of the Stanley-Reisner simplicial complex of a
multigraded generic initial ideal of $J_G$, whose facets can be described in
terms of cut-point sets. Another key step in the proof consists in proving the
equivalence between accessibility and strong accessibility for the collection
of cut sets of $G$ with $J_G$ unmixed. This result, interesting on its own,
provides the first relevant class of set systems for which the previous two
notions are equivalent.
