Cohen–Macaulay binomial edge ideals in terms of blocks with whiskers

For a graph Formula: see text and the binomial edge ideal Formula: see text of Formula: see text, Bolognini et al. have proved the following: Formula: see text is strongly unmixed Formula: see textFormula: see text is Cohen–Macaulay Formula: see textFormula: see text 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 Formula: see text with Cohen–Macaulay Formula: see text. We show that accessible and strongly unmixed properties of Formula: see text 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 Formula: see text-regular Formula: see text-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 Formula: see text has a cut vertex Formula: see text such that Formula: see text is accessible. We show that any non-complete accessible graph Formula: see text having at most three cut vertices has a cut vertex Formula: see text for which Formula: see text is accessible.