  Conca and Varbaro (Invent. Math. 221 (2020), no. 3) showed the equality of
depth of a graded ideal and its initial ideal in a polynomial ring when the
initial ideal is square-free. In this paper, we give some beautiful
applications of this fact in the study of Cohen-Macaulay binomial edge ideals.
For a square-free monomial ideal $I$ and a variable $x$, we give a condition
under which $I$ is Cohen-Macaulay if and only if $\big<I,x\big>$ is
Cohen-Macaulay. Using this fact and computing the initial ideal, we prove that
for the characterization of Cohen-Macaulay binomial edge ideals, it is enough
to consider only ``biconnected graphs with some whisker attached". As an
application of our results, we solve some open problems. At the end, we show
that a graph with Cohen-Macaulay binomial edge ideal has girth less than $5$ or
equal to infinity.

Commutative Algebra