How to exclude all points adjacent to a given point from the feasible region of IP
You can't eliminate all points in $A(y) \setminus \{y\}$ without also eliminating $y$, as the conjunction of a bunch of linear inequalities always gives a convex region, and $A(y) \setminus \{y\}$ is non-convex.

Here I am assuming we're not allowed to introduce new variables.  If we're allowed to introduce new variables, there are many uninteresting ways to achieve it, by taking advantage of the fact that with integer variables you can express any logical predicate you like; however, it seems unlikely that this will be useful for anything in practice.