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 $reg(I(G \setminus N\_G[x]))+1 \leq reg(I(G))$?



Shedding vertex
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.


*Hà, Huy Tài; Woodroofe, Russ*, [**Results on the regularity of square-free monomial ideals**](http://dx.doi.org/10.1016/j.aam.2014.05.002), Adv. Appl. Math. 58, 21-36 (2014). [ZBL1299.13017](https://zbmath.org/?q=an:1299.13017).


