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

Abstract

Let ${\mathrm{gp}}_{\mathrm{t}}(G)$, ${\mathrm{gp}}_{\mathrm{o}}(G)$, and ${\mathrm{gp}}_{\mathrm{d}}(G)$ be the total, the outer, and the dual general position number of a graph $G$, respectively. 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 ${\mathrm{gp}}_{\mathrm{t}}(G)-1\le {\mathrm{gp}}_{\mathrm{t}}(G-x)\le {\mathrm{gp}}_{\mathrm{t}}(G)+{\mathrm{deg}}_G(x)$. On the other hand, ${\mathrm{gp}}_{\mathrm{o}}(G-x)$ and ${\mathrm{gp}}_{\mathrm{d}}(G-x)$ can be respectively arbitrarily larger/smaller than ${\mathrm{gp}}_{\mathrm{o}}(G)$ and ${\mathrm{gp}}_{\mathrm{d}}(G)$. On the positive side, it is proved that if $x$ lies in some ${\mathrm{gp}}_{\mathrm{o}}$-set, then ${\mathrm{gp}}_{\mathrm{o}}(G)-1\le {\mathrm{gp}}_{\mathrm{o}}(G-x)$, and that if $x$ is not a cut vertex and lies in some ${\mathrm{gp}}_{\mathrm{d}}$-set of $G$, then ${\mathrm{gp}}_{\mathrm{d}}(G)-1\le {\mathrm{gp}}_{\mathrm{d}}(G-x)$. For the edge removal, it is proved that (i) ${\mathrm{gp}}_{\mathrm{t}}(G)-|S(G{)}_e|\le {\mathrm{gp}}_{\mathrm{t}}(G-e)\le {\mathrm{gp}}_{\mathrm{t}}(G)+2$, where $S(G{)}_e$ is the set of simplicial vertices adjacent to both endvertices of $e$, (ii) ${\mathrm{gp}}_{\mathrm{o}}(G)/2\le {\mathrm{gp}}_{\mathrm{o}}(G-e)\le 2{\mathrm{gp}}_{\mathrm{o}}(G)$, and (iii) that ${\mathrm{gp}}_{\mathrm{d}}(G)-{\mathrm{gp}}_{\mathrm{d}}(G-e)$ can be arbitrarily large. All bounds are demonstrated to be sharp.