Domination by kings is oddly even
Cristopher Moore, Stephan Mertens

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 king graph consists of all locations on an 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 denote its domination polynomial, i.e., where the sum is over all dominating sets . We prove that . In particular, the number of dominating sets of even size and the number of odd size differs by . %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 -dimensional kings, where .
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Taxonomy
TopicsCulture, Economy, and Development Studies · Economic Policies and Impacts · American Constitutional Law and Politics
