  Let $\mathbf{CCM}$ denote the class of closed graphs with Cohen-Macaulay
binomial edge ideals and $\mathbf{PIG}$ denote the class of proper interval
graphs. Then $\mathbf{CCM}\subseteq \mathbf{PIG}$. The
$\mathbf{PIG}$-completion problem is a classical problem in molecular biology
as well as in graph theory and this problem is known to be NP-hard. In this
paper, we study the $\mathbf{CCM}$-completion problem. We give a method to
construct all possible $\mathbf{CCM}$-completion of a graph. We find the
$\mathbf{CCM}$-completion number and the set of all minimal
$\mathbf{CCM}$-completions for a large class of graphs. Moreover, for that
class, we give a polynomial-time algorithm to compute the
$\mathbf{CCM}$-completion number and a minimum $\mathbf{CCM}$-completion of a
given graph. We investigate unmixed and Cohen-Macaulay properties of binomial
edge ideals of induced subgraphs. Also, we discuss the accessible graphs
completion and the Cohen-Macaulay property of binomial edge ideals of whisker
graphs.
