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

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 $G$, 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 ${\rm gp}{\rm t}(G) -1 \le {\rm gp}{\rm t}(G-x) \le {\rm gp}{\rm t}(G) + {\rm deg}G(x)$. 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 to 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 are demonstrated to be sharp.

💡 Research Summary

The paper studies how three variants of the general‑position number—total (gpₜ), outer (gpₒ), and dual (gp𝒹)—behave under vertex and edge deletions. After recalling that a total general‑position set coincides with the set of simplicial vertices S(G) (so gpₜ(G)=|S(G)|), that an outer general‑position set corresponds to a clique in the strong resolving graph G_SR (gpₒ(G)=ω(G_SR)), and that a dual general‑position set is a general‑position set whose complement is convex, the authors derive sharp bounds for each invariant.

Vertex deletion.
If x is not a cut‑vertex, then the simplicial set satisfies S(G)−{x}⊆S(G−x), giving the lower bound gpₜ(G)−1 ≤ gpₜ(G−x). Since any neighbor of x that becomes simplicial after deletion contributes at most deg_G(x) new simplicial vertices, the upper bound gpₜ(G−x) ≤ gpₜ(G)+deg_G(x) follows; if x itself is simplicial the term deg_G(x) can be reduced by one. Both bounds are shown tight using stars K₁,n, complete bipartite graphs K₂,n, and edge‑deleted complete graphs Kₙ−e.

For the outer number, the authors exhibit graphs Gₙ (Fig. 1) where removing a central vertex doubles gpₒ, and fan graphs Fₙ where removing the universal vertex reduces gpₒ to 2. Hence gpₒ(G−x) can be arbitrarily larger or smaller than gpₒ(G). Nevertheless, if x belongs to some gpₒ‑set and is not a cut‑vertex, then removing x can decrease the outer number by at most one: gpₒ(G)−1 ≤ gpₒ(G−x). The bound is sharp for stars. They conjecture a universal upper bound gpₒ(G−x) ≤ gpₒ(G)+deg_G(x) (Conjecture 3.4) and prove a stronger version when x is simplicial: gpₒ(G−x) ≤ gpₒ(G)+deg_G(x)−1, again with tight examples.

For the dual number, similar phenomena occur: fan graphs show that gp𝒹 can drop from Θ(n) to 2 after deleting the hub, while the graph Gₙ shows a jump from 2n to 4n. If x lies in a gp𝒹‑set and is not a cut‑vertex, then gp𝒹(G)−1 ≤ gp𝒹(G−x). However, no upper bound in terms of gp𝒹(G) exists; the mushroom graphs Mₖ illustrate that deleting a single vertex can reduce gp𝒹 from k+2 to 0.

Edge deletion.
Let Sₑ be the set of simplicial vertices adjacent to both ends of e. Deleting e can destroy at most |Sₑ| simplicial vertices, giving gpₜ(G)−|Sₑ| ≤ gpₜ(G−e). Conversely, at most two new simplicial vertices can appear, so gpₜ(G−e) ≤ gpₜ(G)+2. Both extremes are attained by suitable stars and complete bipartite graphs.

For the outer number, the strong resolving graph changes by at most a factor of two: ω(G_SR)/2 ≤ ω((G−e)_SR) ≤ 2·ω(G_SR), which translates to gpₒ(G)/2 ≤ gpₒ(G−e) ≤ 2·gpₒ(G). The bounds are tight using paths, cycles, and complete graphs.

The dual number is the most volatile: removing a single edge may destroy the convexity of the complement of a maximal general‑position set, causing gp𝒹(G)−gp𝒹(G−e) to be arbitrarily large. The mushroom construction demonstrates this unbounded drop, and no universal upper bound exists.

Overall, the paper provides a comprehensive picture: gpₜ is relatively stable under deletions, while gpₒ and gp𝒹 can change dramatically, yet each admits meaningful lower bounds when the deleted element participates in an optimal set. All bounds are proved sharp, and several conjectures and open directions are proposed for future work.