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

Abstract We describe the simplicial complex $$\Delta $$ Δ such that initial ideal of binomial edge $$J_\textrm{G}$$ J G G is Stanley-Reisner . By using we show if $$(S_2)$$ ( S 2 ) , then accessible. also characterize all accessible blocks with whiskers cycle rank 3 and define a new infinite class for any rank. Finally, by computational approach, graphs at most 12 vertices whose Cohen–Macaulay are only ones.