Visibility in graphs under edge and vertex removal

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 $\mu(G)$. 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 $\mu_o(G)$, $\mu_d(G)$, and $\mu_t(G)$. 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 $\frac{1}{2}\mu(G) \le \mu(G-e) \le 2\mu(G)$ and $\frac{1}{6}\mu_o(G) \le \mu_o(G-e) \le 2\mu_o(G)+1$ hold for every graph. Further general upper bounds established here are $\mu_t(G-e) \leq \mu_t(G)+2$ and $\mu(G-x) \leq 2\mu(G)$. 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 $\mu_d(G-e)$ nor $\mu_d(G-x)$ allows lower or upper bounds of the form $a \cdot \mu_d(G)+b$ 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.