Quasi-efficient domination in grids

TL;DR

This paper investigates independent dominating sets in grid graphs, proving their existence for all grids and developing a dynamic programming method to compute minimum sizes, with results for small to large grids.

Contribution

It introduces the concept of independent [1,2]-sets in grids, proves their existence universally, and provides an efficient algorithm to determine their minimum size.

Findings

Every grid has an independent [1,2]-set.

Developed a dynamic programming algorithm using min-plus algebra.

Calculated minimum sizes for grids up to 13xN and identified patterns for larger grids.

Abstract

Domination of grids has been proved to be a demanding task and with the addition of independence it becomes more challenging. It is known that no grid with $m,n \geq 5$ has an efficient dominating set, also called perfect code, that is, an independent vertex set such that each vertex not in it has exactly one neighbor in that set. So it is interesting to study the existence of independent dominating sets for grids that allow at most two neighbors, such sets are called independent $[1,2]$-sets. In this paper we prove that every grid has an independent $[1,2]$-set, and we develop a dynamic programming algorithm using min-plus algebra that computes $i_{[1,2]}(P_m\Box P_n)$, the minimum cardinality of an independent $[1,2]$-set for the grid graph $P_m\square P_n$. We calculate $i_{[1,2]}(P_m\Box P_n)$ for $2\leq m\leq 13, n\geq m$ using this algorithm, meanwhile the parameter for grids with $14\leq m\leq n$ is obtained through a quasi-regular pattern that, in addition, provides an independent $[1,2]$-set of minimum size.