  The cut sets of a graph are special sets of vertices whose removal
disconnects the graph. They are fundamental in the study of binomial edge
ideals, since they encode their minimal primary decomposition. We introduce the
class of accessible graphs as the graphs with unmixed binomial edge ideal and
whose cut sets form an accessible set system. We prove that the graphs whose
binomial edge ideal is Cohen-Macaulay are accessible and we conjecture that the
converse holds. We settle the conjecture for large classes of graphs, including
chordal and traceable graphs, providing a purely combinatorial description of
Cohen-Macaulayness. The key idea in the proof is to show that both properties
are equivalent to a further combinatorial condition, which we call strong
unmixedness.
