Cohen-macaulay binomial edge ideals in terms of blocks with whiskers

For a graph G, Bolognini et al. have shown JG is strongly unmixed ? JG is Cohen-Macaulay ? G is accessible, where JG denotes the binomial edge ideals of G. 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. Also, we give an infinite class of graphs whose binomial edge ideals are Cohen-Macaulay, and from that, we classify all r-regular r-connected graphs such that attaching some special whiskers to it, the binomial edge ideals become Cohen-Macaulay. Finally, we define a new class of graphs, called \textit{strongly r-cut-connected} and prove that the binomial edge ideal of any strongly r-cut-connected accessible graph having at most three cut vertices is Cohen-Macaulay.