On the variety of general position problems under vertex and edge removal

Let gpₜ(G), gpₒ(G), and gpd(G) be the total, the outer, and the dual general position number of a graph G. This paper investigates how removing a vertex or removing an edge affects these graph invariants. It is proved that if x is not a cut vertex, then gpₜ(G) -1 ≤ gpₜ(G-x) ≤ gpₜ(G) + degG(x). On the other hand, gpₒ(G-x) and gpd(G-x) can be respectively arbitrarily larger/smaller than gpₒ(G) and gpd(G). On the positive side, it is proved that if x lies in some gpₒ-set, then gpₒ(G)-1 ≤ gpₒ(G-x), and that if x is not a cut vertex and lies in some gpd-set of G, then gpd(G)-1 ≤ gpd(G-x). For the edge removal, it is proved that (i) gpₜ(G) -|S(G)ₑ| ≤ gpₜ(G-e) ≤ gpₜ(G) +2, where S(G)ₑ is the set of simplicial vertices adjacent both endvertices of e, (ii) gpₒ(G)/2≤ gpₒ(G-e)≤\ 2 gpₒ(G), and (iii) that gpd(G) - gpd(G-e) can be arbitrarily large. All bounds proved are demonstrated to be sharp.