Visibility in graphs under edge and vertex removal
Pakanun Dokyeesun, Csilla Bujtás
TL;DR
This work studies how four graph-visibility invariants—mutual-visibility $\mu$, outer $\mu_{\rm o}$, dual $\mu_{\rm d}$, and total $\mu_{\rm t}$—change when a single edge or vertex is removed from a connected graph. It provides tight general bounds for edge removals, notably $\frac{1}{2}\mu(G) \le \mu(G-e) \le 2\mu(G)$ and $\frac{1}{6}\mu_{\rm o}(G) \le \mu_{\rm o}(G-e) \le 2\mu_{\rm o}(G)+1$, along with $\mu_t(G-e) \le \mu_t(G)+2$, while showing no universal linear bounds for $\mu_{\rm d}$ and $\mu_{\rm t}$. The paper also demonstrates that vertex removal can arbitrarily decrease invariants and, for $\mu$, increase by at most a factor of two, but $\mu_{\rm d}$ and $\mu_{\rm t}$ may grow without bound, via constructive families. Realizability results characterize which pairs $(\sigma,n)$ can occur for each invariant, linking visibility to graph order and aiding understanding of local operations on networks. Overall, the work advances the theory of visibility in graphs under local perturbations with precise extremal and realizability results, informing applications in network design and routing under failures.
Abstract
For a connected graph $G$ and $X\subseteq V(G)$, we say that two vertices $u$, $v$ are $X$-visible if there is a shortest $u,v$-path $P$ with $V(P)\cap X \subseteq \{u,v\}$. If every two vertices from $X$ are $X$-visible, then $X$ is a mutual-visibility set in $G$. The largest cardinality of such a set in $G$ is the mutual-visibility number $μ(G)$. When the visibility constraint is extended to further types of vertex pairs, we get the definitions of outer, dual, and total mutual-visibility sets and the respective graph invariants $μ_o(G)$, $μ_d(G)$, and $μ_t(G)$. This work concentrates on the possible changes in the four visibility invariants when an edge $e$ or a vertex $x$ is removed from $G$ and the graph remains connected. It is proved that $\frac{1}{2}μ(G) \le μ(G-e) \le 2μ(G)$ and $\frac{1}{6}μ_o(G) \le μ_o(G-e) \le 2μ_o(G)+1$ hold for every graph. Further general upper bounds established here are $μ_t(G-e) \leq μ_t(G)+2$ and $μ(G-x) \leq 2μ(G)$. For all but one of the remaining cases, it is shown that the visibility invariant may increase or decrease arbitrarily under the considered local operation. For example, neither $μ_d(G-e)$ nor $μ_d(G-x)$ allows lower or upper bounds of the form $a \cdot μ_d(G)+b$ with a positive constant $a$. Along the way, the realizability of the four visibility invariants in terms of the order is also characterized in the paper.