Perfect Domination in Knights Graphs

TL;DR

This paper determines the perfect domination number for knights graphs on various chessboards, providing exact values or bounds for most cases, advancing understanding of domination in combinatorial graph theory.

Contribution

It computes the perfect domination number for knights graphs on multiple chessboard types, filling gaps in existing knowledge and offering bounds for complex cases.

Findings

Exact values or bounds for most chessboard configurations.

Identification of unresolved cases with 3 rows and specific column counts.

Enhanced understanding of domination properties in knights graphs.

Abstract

For a graph $G = (V,E),$ a subset $S$ of $V$ is a perfect dominating set of $G$ if every vertex not in $S$ is adjacent to exactly one vertex in $S.$ The perfect domination number, $\gamma_p(G),$ is the minimum cardinality of a perfect dominating set of $G.$ The perfect domination number is found for knights graphs on square, rectangular, and infinite chessboards. Indeed, exact values or bounds are given for all chessboards except those with 3 rows and number of columns congruent to 1, 2, or 3 modulo 8.