An Explicit Construction of Optimal Dominating Sets in Grid

TL;DR

This paper provides an explicit construction method for optimal dominating sets in grid graphs of size at least 16x16, confirming a conjecture and showing the equivalence of domination numbers for certain sets.

Contribution

It introduces a direct construction technique for optimal dominating sets in grid graphs and proves the equality of domination and [1,2]-domination numbers for large grids.

Findings

Explicit construction method for optimal dominating sets in grid graphs.

Verification that $ ext{γ}(G_{m,n}) = ext{γ}_{[1,2]}(G_{m,n})$ for $m,n \\geq 16$.

Confirmation of Chang's conjecture for grid graphs.

Abstract

A dominating set in a graph $G$ is a subset of vertices $D$ such that every vertex in $V\setminus D$ is a neighbor of some vertex of $D$. The domination number of $G$ is the minimum size of a dominating set of $G$ and it is denoted by $\gamma(G)$. Also, a subset $D$ of a graph $G$ is a $[ 1 , 2 ] $-set if, each vertex $v \in V \setminus D$ is adjacent to either one or two vertices in $D$ and the minimum cardinality of $[ 1 , 2 ] $-dominating set of $G$, is denoted by $\gamma_{[1,2]}(G)$. Chang's conjecture says that for every $16 \leq m \leq n$, $\gamma(G_{m,n})= \left \lfloor\frac{(n+2)(m+2)}{5}\right \rfloor-4$ and this conjecture has been proven by Goncalves et al. This paper presents an explicit constructing method to find an optimal dominating set for grid graph $G_{m,n}$ where $m,n\geq 16$ in $O(\text{size of answer})$. In addition, we will show that $\gamma(G_{m,n})=\gamma_{[1,2]}(G_{m,n})$ where $m,n\geq 16$ holds in response to an open question posed by Chellali et al.