The general position number under vertex and edge removal

Abstract

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