Question: Suppose we have an integer program $\min\{c^\top{x}\mid{Ax\leq{b}},x\in\mathbb{Z}_+^n\}$, and suppose that $x^*$ is a feasible solution for this IP (or even that $x^*$ is an extreme point of the convex hull of the feasible region). Is there a method to eliminate $x^*$ (and only $x^*$) from the feasible region, so that if we re-solve the IP, we do not obtain the solution $x^*$?
Background: If the integer program contains only binary variables, then we can use a no-good inequality of the form $$ \sum_{i:x^*_i=0}x_i+\sum_{i:x^*_i=1}(1-x_i)\geq1 $$ to eliminate $x^*$ and only $x^*$ from the feasible region. Is there an analogue of this (possibly using multiple inequalities) for general integer variables? I imagine that if $x^*$ is an arbitrary feasible point, the answer is no (unless we use an extended formulation of some kind, or nonlinear inequalities)--one can imagine if $x^*$ is a single point in the middle of the integer lattice, there would be no way to simply punch only that point out of the middle. But if $x^*$ is an extreme point of the convex hull, it seems that it should be possible.
One possible solution: One clear solution is to use a binary expansion of the integer variables, and replace all the general integer variables with binary variables. I am wondering if there is a way to do this without a binary expansion (i.e. hopefully in the original space of variables).