  Let $G$ be a simple connected non-complete graph and $J_G$ be its binomial
edge ideal in a polynomial ring $S$. Using certain invariants associated to
graphs, say $U(G)$, Banerjee and N\'{u}\~{n}ez-Betancourt gave an upper bound
for the depth of $S/J_G$, and Rouzbahani Malayeri, Saeedi Madani and Kiani
obtained a lower bound, say $L(G)$. Hibi and Saeedi Madani gave a structural
classification of graphs satisfying $L(G)=U(G)$. In this article, we give
structural classification of graphs satisfying $L(G)+1=U(G)$. We also compute
the depth of $S/J_G$ for all such graphs $G$.
