The accessible-graph characterization of Cohen–Macaulay binomial edge ideals

Let $G$ be a finite simple graph on vertex set $[n]$. For $S\subseteq[n]$, let $c_G(S)$ be the number of connected components of $G-S$, and call $S$ a cut set if $c_G(S)>c_G(S\setminus\{i\})$ for every $i\in S$; write $\mathcal{C}(G)$ for the collection of cut sets. Let $J_G\subseteq R=K[x_i,y_i:i\in[n]]$ be the binomial edge ideal generated by $x_i y_j-x_j y_i$ for the edges $\{i,j\}$ of $G$. The graph $G$ is accessible if $\mathcal{C}(G)$ is an accessible set system and $J_G$ is unmixed, equivalently if $c_G(S)=|S|+c$ for every $S\in\mathcal{C}(G)$, where $c$ is the number of connected components of $G$. The accessible-graph conjecture. $J_G$ is Cohen–Macaulay if and only if $G$ is accessible. This conjecture seeks a combinatorial characterization of Cohen–Macaulayness for binomial edge ideals. The forward implication is known, while the converse was established for several graph classes, including chordal and traceable graphs under unmixedness, graphs with a small number of vertices, and further classes; it remains open in general.