Domination by kings is oddly even

TL;DR

This paper proves a surprising parity property of dominating sets in king graphs on chessboards, showing the difference between counts of even and odd-sized dominating sets is always ±1.

Contribution

It establishes a new algebraic property of the domination polynomial evaluated at -1 for king graphs, extending to higher dimensions.

Findings

Number of even and odd dominating sets differ by ±1.

The property holds for 2D king graphs but not for cylindrical or toroidal variants.

Generalizes to d-dimensional king graphs with a similar formula.

Abstract

The $m \times n$ king graph consists of all locations on an $m \times n$ chessboard, where edges are legal moves of a chess king. %where each vertex represents a square on a chessboard and each edge is a legal move. Let $P_{m \times n}(z)$ denote its domination polynomial, i.e., $\sum_{S \subseteq V} z^{|S|}$ where the sum is over all dominating sets $S$. We prove that $P_{m \times n}(-1) = (-1)^{\lceil m/2\rceil \lceil n/2\rceil}$. In particular, the number of dominating sets of even size and the number of odd size differs by $\pm 1$. %The numbers can not be equal because the total number of dominating sets is always odd. This property does not hold for king graphs on a cylinder or a torus, or for the grid graph. But it holds for $d$-dimensional kings, where $P_{n_1\times n_2\times\cdots\times n_d}(-1) = (-1)^{\lceil n_1/2\rceil \lceil n_2/2\rceil\cdots \lceil n_d/2\rceil}$.