Optimal and quasi-optimal locating-dominating densities in the infinite hexagonal grid with a finite number of rows

Abstract

A set of vertices $S$ of a graph $G$ is locating-dominating if $S$ is dominating and, for each pair of distinct vertices not in $S$, their neighborhoods in $S$ are distinct. We present results on the minimum density of such sets in the infinite hexagonal grid with a finite number of rows $k$, also known as the hexagonal strip of width $k$, which we denote by $H_k$. For each $k\ge 2$, we present either an optimal solution or a quasi-optimal solution for $H_k$ that is within $1.3\%$ of the optimum. We describe an exact exponential-time algorithm for fixed k, which we implemented to find optimal solutions for $k\le 5$. As the infinite grid $H_k$ always admits a periodic optimal solution, to deal with larger values of $k$, we present an integer linear program that finds an optimal periodic solution for $H_k$ for each fixed period. This program yields high-quality feasible solutions for $H_7$ and $H_8$, which we then combine with an optimal solution for $H_3$ to obtain quasi-optimal solutions for all $k\ge 6$. All these solutions admit a very short description.

Comments: AMS-LaTeX, 18 pages with 10 figures