Perfect Domination 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.

A portion of these results were presented at the 49th SE International Conference on Combinatorics, Graph Theory and Computing, March 5-9, 2018 and will appear in the conference proceedings, Congressus Numerantium (2018)