An optimal algorithm for geodesic mutual visibility on hexagonal grids

TL;DR

This work solves Geodesic Mutual Visibility (GMV) for synchronous, oblivious robots on finite hexagonal grids by first determining the mutual-visibility number $\mu(G_k)$ and a corresponding $\mu$-set $X_k$, proving $\mu(G_k)=4k$ for $k\ge4$. It then presents a time-optimal distributed algorithm $\mathcal{A}$ that uses $X_k$ as a pattern and organizes movement through guards, sectors, and row/column abstractions to ensure collision-free attainment of GMV in $\Theta(k)$ rounds. The algorithm is decomposed into a sequence of tasks (placement of guards, distribution along rows, movement toward targets, and pattern finalization) with a formal correctness proof via a transition graph and supporting lemmas. The results advance both graph-theoretic understanding of mutual visibility on hex grids and practical coordination schemes for GMV in swarm robotics, while outlining open problems for higher symmetry and broader grid generalizations.

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 $G_k$.

Figures (13)

• Figure 1: Examples of hexagonal $k\times m\times n$ grids: (left) a $2\times 2\times 2$ grid; (center) a $2\times 2\times 3$ grid; (right) a $1\times 2\times 4$ grid.
• Figure 2: (left) Visualization of $G_3$ with the three families of oriented lines used for the vertex labeling; (right) Visualization of some vertex labels;
• Figure 3: Visualization of the $\mu$-set $X_k$ when $k=5$.
• Figure 4: (left) A configuration $C$ with $\rho(C)=1$; (middle): a configuration divided into two sectors; (right) Computing the view of robots: corners are highlighted by squares, the reading starts at each corner and proceeds along the side of $G_3$, $LSS(C)=0000010$$000011111$$00001001101$$1011000000$$001100000$$0000001$ and it is obtained from $F$.
• Figure 5: The 6 sectors of $G_3$, the corner in each sector, and a special-path within one sector.

Theorems & Definitions (20)

• definition thmcounterdefinition: $\mathrm{GMV}$ problem
• theorem thmcountertheorem
• proof
• definition thmcounterdefinition
• definition thmcounterdefinition
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma