The general position number under vertex and edge removal

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