On the variety of general position problems under vertex and edge removal
Jing Tian, Pakanun Dokyeesun, Sandi Klavžar
TL;DR
This work investigates how vertex and edge deletions affect the three general position variants ${\rm gp}_{\rm t}$, ${\rm gp}_{\rm o}$, and ${\rm gp}_{\rm d}$ in graphs. It leverages structural characterizations, notably ${\rm gp}_{\rm t}(G) = s(G)$, ${\rm gp}_{\rm o}(G) = \omega(G_{SR})$, and the convexity criterion for ${\rm gp}_{\rm d}$, to derive tight, often sharp bounds for vertex removal, and to establish corresponding bounds for edge removal. The paper provides constructions (e.g., $G_n$, $F_n$, $M_k$, $Y_k$, $H_n$) to demonstrate extremal behavior, showing that some invariants can shift drastically or even unboundedly under deletions, while others obey near-1 or bounded changes under certain conditions. Overall, the results illuminate the stability and variability of three distinct general position notions under common graph perturbations, with implications for network robustness and metric-structure analyses.
Abstract
Let ${\rm gp}_{\rm t}(G)$, ${\rm gp}_{\rm o}(G)$, and ${\rm gp}_{\rm d}(G)$ 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 ${\rm gp}_{\rm t}(G) -1 \le {\rm gp}_{\rm t}(G-x) \le {\rm gp}_{\rm t}(G) + {\rm deg}_G(x)$. On the other hand, ${\rm gp}_{\rm o}(G-x)$ and ${\rm gp}_{\rm d}(G-x)$ can be respectively arbitrarily larger/smaller than ${\rm gp}_{\rm o}(G)$ and ${\rm gp}_{\rm d}(G)$. On the positive side, it is proved that if $x$ lies in some ${\rm gp}_{\rm o}$-set, then ${\rm gp}_{\rm o}(G)-1 \le {\rm gp}_{\rm o}(G-x)$, and that if $x$ is not a cut vertex and lies in some ${\rm gp}_{\rm d}$-set of $G$, then $ {\rm gp}_{\rm d}(G)-1 \le {\rm gp}_{\rm d}(G-x)$. For the edge removal, it is proved that (i) ${\rm gp}_{\rm t}(G) -|S(G)_{e}| \le {\rm gp}_{\rm t}(G-e) \le {\rm gp}_{\rm t}(G) +2$, where $S(G)_{e}$ is the set of simplicial vertices adjacent both endvertices of $e$, (ii) ${\rm gp}_{\rm o}(G)/2\le {\rm gp}_{\rm o}(G-e)\leq\ 2{\rm gp}_{\rm o}(G)$, and (iii) that ${\rm gp}_{\rm d}(G) - {\rm gp}_{\rm d}(G-e)$ can be arbitrarily large. All bounds proved are demonstrated to be sharp.