Visibility in graphs under edge and vertex removal

For a connected graph G and X V (G), we say that two vertices u, v are X-visible if there is a shortest u, v-path P with V (P) X \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 (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 ₒ (G), d (G), and 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 12 (G) (G-e) 2 (G) and 16ₒ (G) ₒ (G-e) 2ₒ (G) +1 hold for every graph. Further general upper bounds established here are ₜ (G-e) ₜ (G) +2 and (G-x) 2 (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 d (G-e) nor d (G-x) allows lower or upper bounds of the form a 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.