An optimal algorithm for geodesic mutual visibility on hexagonal grids

TL;DR

This paper presents an optimal algorithm for solving the Geodesic Mutual Visibility problem on finite hexagonal grids, ensuring efficient and confidential communication among robots by positioning them on mutually visible vertices.

Contribution

It introduces the first optimal solution for GMV on hexagonal grids, including a combinatorial analysis of maximum mutually visible vertices.

Findings

Optimal GMV algorithm for hexagonal grids

Maximum mutually visible vertices characterized

Enhanced confidentiality and efficiency in robot communication

Abstract

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 $G_k$.