# Shedding vertex

Let $G$ be a finite simple graph on the vertex set $\{x_1, \ldots, x_n\}$ and $I(G) := (\{x_ix_j \mid \{i,j\} \in E(G)\}) \subset R=K[x_1, \ldots, x_n]$ be the edge ideal corresponding to the graph $G$, where $K$ is a field. The Castelnuovo–Mumford regularity (or simply, regularity) $reg(I(G))$ of $I(G)$ is defined as $$reg(I(G))=\max\{j-i \mid Tor^R_i(I(G),K) \neq 0\}$$

(I) no independent set in $G\setminus N_G[x]$ is a maximal independent set in $G \setminus x$.

A vertex $x$ which satisfies Condition (I) is called a shedding vertex.

Suppose $x$ is a shedding vertex. Is $reg(I(G \setminus N_G[x]))+1 \leq reg(I(G))$?

In the case of a shedding vertex $v$, $\operatorname{reg} I(G) = \max \{ \operatorname{reg} I(G \setminus N[v]) + 1, \operatorname{reg} I(G\setminus v) \}$, by a theorem of myself and Tài Hà. So your inequality is true. See Theorem 1.5 of the following.