Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph

TL;DR

This paper introduces an algorithm to compute domination polynomials for grid-like graphs, enabling precise enumeration of dominating sets and extending known integer sequences.

Contribution

The paper presents a novel algorithm with specific complexity bounds for calculating domination polynomials of grid, cylinder, torus, and king graphs, and applies it to large graphs.

Findings

Computed domination polynomials for graphs up to 24x24 size.

Provided asymptotic growth estimates for the number of dominating sets.

Extended several sequences in the Online Encyclopedia of Integer Sequences.

Abstract

We present an algorithm to compute the domination polynomial of the $m \times n$ grid, cylinder, and torus graphs and the king graph. The time complexity of the algorithm is $O(m^2n^2 \lambda^{2m})$ for the torus and $O(m^3n^2\lambda^m)$ for the other graphs, where $\lambda = 1+\sqrt{2}$. The space complexity is $O(mn\lambda^m)$ for all of these graphs. We use this algorithm to compute domination polynomials for graphs up to size $24\times 24$ and the total number of dominating sets for even larger graphs. This allows us to give precise estimates of the asymptotic growth rates of the number of dominating sets. We also extend several sequences in the Online Encyclopedia of Integer Sequences.