  We describe the simplicial complex $\Delta$ such that the initial ideal of
$J_G$ is the Stanley-Reisner ideal of $\Delta$. By $\Delta$ 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.
