  The cut sets of a graph are special sets of vertices whose removal disconnect
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.
