The general position number under vertex and edge removal

TL;DR

This paper investigates how the general position number of a graph changes under vertex and edge removal, establishing bounds and conditions, especially for diameter 2 graphs, with proofs of sharpness.

Contribution

It provides new bounds on the general position number after vertex and edge removal, including sharp bounds for diameter 2 graphs, expanding understanding of graph stability.

Findings

${ m gp}(G-x)\

bounds for ${ m gp}(G-x)$ and ${ m gp}(G-e)$

sharpness of all bounds

Abstract

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 can be improved to ${\rm gp}(G)-1\le {\rm gp}(G-e)\leq\ {\rm gp}(G) + 1$. All these bounds are proved to be sharp.