Closed Cohen-Macaulay Completion of Binomial Edge Ideals

Let [Formula: see text] denote the class of closed graphs with Cohen-Macaulay binomial edge ideals and [Formula: see text] denote the class of proper interval graphs. Then [Formula: see text] The [Formula: see text]-completion problem is a classical problem in graph theory as well as in molecular biology, and this problem is known to be NP-hard. In this paper, we study the [Formula: see text]-completion problem. We give a method to construct all possible [Formula: see text]-completions of a graph. We find the [Formula: see text]-completion number and the set of all minimal [Formula: see text]-completions for a large class of graphs. Moreover, for this class, we give a polynomial-time algorithm to compute the [Formula: see text]-completion number and a minimum [Formula: see text]-completion of a given graph. The unmixedness and Cohen-Macaulay properties of binomial edge ideals of induced subgraphs are investigated. Also, we discuss the accessible graph completion and the Cohen-Macaulay property of binomial edge ideals of whisker graphs.