For a graph G and the binomial edge ideal JG of G, Bolognini et al. have proved the following: JG is strongly unmixed ⇒JG is Cohen–Macaulay ⇒G is accessible. Moreover, they have conjectured that the converse of these implications is true. Accessible and strongly unmixed properties are purely combinatorial. We give some motivations to focus only on blocks with whiskers for the characterization of all G with Cohen–Macaulay JG. We show that accessible and strongly unmixed properties of G depend only on the corresponding properties of its blocks with whiskers and vice versa. We give a new family of graphs whose binomial edge ideals are Cohen–Macaulay, and from that family, we classify all r-regular r-connected graphs, with the property that, after attaching some special whiskers to it, the binomial edge ideals become Cohen–Macaulay. To prove the Cohen–Macaulay conjecture, it is enough to show that every non-complete accessible graph G has a cut vertex v such that G / {v} is accessible. We show that any non-complete accessible graph G having at most three cut vertices has a cut vertex v for which G / {v} is accessible.