(S-2)-condition and Cohen-Macaulay binomial edge ideals

We describe the simplicial complex Delta such that the initial ideal of the binomial edge ideal J(G) of G is the Stanley-Reisner ideal of Delta. By using Delta we show that if J(G) is (S-2), 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.