The general position number under vertex and edge removal

Abstract

Let gp(G) be the general position number of a graph G. It is proved that gp(G − x) ≤ 2gp(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) ≤ 2gp(G) holds for any edge e of a graph G. For diameter 2 graphs these results can be improved to gp(G) − 1 ≤ gp(G − e) ≤ gp(G) + 1. All these bounds are proved to be sharp.