The integer $\{2\}$-domination number of grids
Jia-Ying Lee, Chia-An Liu
TL;DR
This paper investigates the integer $\{2\}$-domination number on grid graphs $G_{m,n}$, defined as the minimum total weight of a function $f:V_{m,n}\to\{0,1,2\}$ with $\sum_{y\in N[x]} f(y)\ge 2$ for every vertex $x$. It introduces a labeling-based representation and a dynamic programming algorithm that computes $\gamma_{\{2\}}(G_{m,n})$ for fixed $m$ and arbitrary $n$, enabling linear-time computation in $n$. The authors obtain exact values for $G_{1,n}$ and $G_{2,n}$, derive an upper bound for $G_{3,n}$, and demonstrate the approach by determining many $G_{4,n}$ values up to $n\le 100$ while proposing a closed form for larger $n$. The work advances understanding of domination-type parameters on grids and provides scalable methods for exact computation, with promising directions to generalize to larger $m$ and higher $k$-domination.
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.