Closed Cohen-Macaulay completion of binomial edge ideals

TL;DR

This paper investigates the problem of completing graphs to have Cohen-Macaulay binomial edge ideals, providing methods, algorithms, and characterizations for a large class of graphs, with implications in graph theory and molecular biology.

Contribution

It introduces a method to construct all possible Cohen-Macaulay completions, finds the completion number, and provides polynomial algorithms for a significant class of graphs.

Findings

Characterization of Cohen-Macaulay completions for certain graphs

Polynomial-time algorithm for computing the completion number

Analysis of Cohen-Macaulay properties in induced subgraphs and whisker graphs

Abstract

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.