Optimum Algorithm for the Mutual Visibility Problem

On the mutual visibility in Cartesian products and triangle-free graphs

“…In this case, the length t − 1 of the y, z-path P t must guarantee (t − 1) + (m − 1 − d ) < l − 1 − d. By using the same approach as in case (a), this relationship still leads to prove that n(C ) < n(C ). In fact, it is equivalent to showing that 2d ≤ m + n, which is trivially implied by (4).…”

“…In this case, the length t − 1 of the y, z-path P t must guarantee (t − 1) + d + 1 < d + 1. Proving n(C ) < n(C ) is equivalent to showing that 2d ≤ l + n, which is implied by (4) and by the assumption l ≥ m.…”

“…Starting with this initial work, many papers have addressed the same problem (e.g., see [2,4,23]). Later, similar visibility problems were considered in different contexts, where the entities are "fat entities" modeled as disks in the Euclidean plane (e.g., see [24]) or are points on a grid based terrain and their movements are restricted only along grid lines (e.g., see [1]).…”

On the mutual visibility in Cartesian products and triangle-free graphs

“…Namely, satisfying that for any exchanged information there should be a channel which does not pass through other entities. For some of these applied researches see for instance [1,2,8,15,31].…”