The \textsc{Mutual Visibility} is a well-known problem in the context of mobile robots. For a set of $n$ robots disposed in the Euclidean plane, it asks for moving the robots without collisions so as to achieve a placement ensuring that no three robots are collinear. For robots moving on graphs, we consider the \textsc{Geodesic Mutual Visibility} ($\GMV$) problem. Robots move along the edges of the graph, without collisions, so as to occupy some vertices that guarantee they become pairwise geodesic mutually visible. This means that there is a shortest path (i.e., a "geodesic") between each pair of robots along which no other robots reside. We study this problem in the context of finite and infinite square grids, for robots operating under the standard Look-Compute-Move model. In both scenarios, we provide resolution algorithms along with formal correctness proofs, highlighting the most relevant peculiarities arising within the different contexts, while optimizing the time complexity.

翻译：《相互可见性》问题是移动机器人领域的经典问题。对于分布在欧几里得平面上的$n$个机器人集合，该问题要求机器人通过无碰撞移动，最终实现任意三个机器人不共线的布局。针对在图结构上移动的机器人，我们考虑测地线相互可见性问题（$\GMV$）。机器人沿图边无碰撞移动，占据某些顶点，使得彼此之间成为成对测地线相互可见。这意味着每对机器人之间存在一条最短路径（即"测地线"），且该路径上不存在其他机器人。我们在有限网格与无限网格两种场景下研究该问题，机器人运行于标准的"观察-计算-移动"模型。针对两种场景，我们提出求解算法并给出形式化正确性证明，揭示了不同场景中最具相关性的特性，同时优化了时间复杂度。