The integer $\{2\}$-domination number of grids

TL;DR

This paper investigates the integer 2-domination number of grid graphs, providing exact values for small grids, an upper bound for larger grids, and an algorithm for general cases, with conjectures for future work.

Contribution

It computes the integer 2-domination numbers for small grid graphs and proposes an algorithm for arbitrary sizes, advancing understanding of domination parameters in grid graphs.

Findings

Exact 2-domination numbers for G_{1,n} and G_{2,n}

Upper bounds for G_{3,n}

An algorithm for counting 2-domination numbers in G_{m,n}

Abstract

For positive integers $m$ and $n$, the grid graph $G_{m,n}$ is the Cartesian product of the path graph $P_m$ on $m$ vertices and the path graph $P_n$ on $n$ vertices. An integer $\{2\}$-dominating function of a graph is a mapping from the vertex set to $\{0,1,2\}$ such that the sum of the mapped values of each vertex and its neighbors is at least $2$; the integer $\{2\}$-domination number of a graph is defined to be the minimum sum of mapped values of all vertices among all integer $\{2\}$-dominating functions. In this paper, we compute the integer $\{2\}$-domination numbers of $G_{1,n}$ and $G_{2,n}$, attain an upper bound to the integer $\{2\}$-domination numbers of $G_{3,n}$, and propose an algorithm to count the integer $\{2\}$-domination numbers of $G_{m,n}$ for arbitrary $m$ and $n$. As a future work, we list the integer $\{2\}$-domination numbers of $G_{4,n}$ for small $n$, and conjecture on its formula.