$$(S_2)$$-condition and Cohen–Macaulay binomial edge ideals

We describe the simplicial complex   such that the initial ideal of the binomial edge ideal JG of G is the Stanley-Reisner ideal of  . By using   we show that if JG is (S2), then G is accessible.We also characterize all accessible blocks with whiskers of cycle rank 3 and we define a new infinite class of accessible blocks with whiskers for any cycle rank. Finally, by using a computational approach, we show that the graphs with at most 12 vertices whose binomial edge ideal is Cohen–Macaulay are all and only the accessible ones.