For a set of robots (or agents) moving in a graph, two properties are highly desirable: confidentiality (i.e., a message between two agents must not pass through any intermediate agent) and efficiency (i.e., messages are delivered through shortest paths). These properties can be obtained if the \textsc{Geodesic Mutual Visibility} (GMV, for short) problem is solved: oblivious robots move along the edges of the graph, without collisions, to occupy some vertices that guarantee they become pairwise geodesic mutually visible. This means there is a shortest path (i.e., a ``geodesic'') between each pair of robots along which no other robots reside. In this work, we optimally solve GMV on finite hexagonal grids $G_k$. This, in turn, requires first solving a graph combinatorial problem, i.e. determining the maximum number of mutually visible vertices in $G_k$.

翻译：对于在图中移动的一组机器人（或智能体），有两个特性是高度可取的：保密性（即两个智能体之间的消息不得经过任何中间智能体）和效率（即消息通过最短路径传递）。若\textsc{测地互可见性}（简称GMV）问题得以解决，则可实现这些特性：无记忆机器人沿图的边移动，避免碰撞，占据某些顶点以确保它们成对地成为测地互可见的。这意味着每对机器人之间存在一条最短路径（即“测地线”），且该路径上没有其他机器人。本工作中，我们在有限六边形网格$G_k$上最优地解决了GMV问题。这首先需要解决一个图组合问题，即确定$G_k$中互可见顶点的最大数量。