  Let ${\rm gp}(G)$ be the general position number of a graph $G$. It is proved
that ${\rm gp}(G-x)\leq 2{\rm gp}(G)$ holds for any vertex $x$ of a connected
graph $G$ and that if $x$ lies in some ${\rm gp}$-set of $G$, then ${\rm gp}(G)
- 1 \le {\rm gp}(G-x)$. Constructions are given which show that ${\rm gp}(G-x)$
can be much larger than ${\rm gp}(G)$ also when $G-x$ is connected. For
diameter $2$ graphs it is proved that ${\rm gp}(G-x) \le {\rm gp}(G)$, and that
${\rm gp}(G-x) \ge {\rm gp}(G) - 1$ when the diameter of $G-x$ remains $2$. It
is demonstrated that ${\rm gp}(G)/2\le {\rm gp}(G-e)\leq 2{\rm gp}(G)$ holds
for any edge $e$ of a graph $G$. For diameter $2$ graphs these results sharpens
to ${\rm gp}(G)-1\le {\rm gp}(G-e)\leq\ {\rm gp}(G) + 1$. All these bounds are
proved to be sharp.
