Irredundance and domination in kings graphs

Each king on an n×n chessboard is said to attack its own square and its neighboring squares, i.e., the nine or fewer squares within one move of the king. A set of kings is said to form an irredundant set if each attacks a square attacked by no other king in the set. We prove that the maximum size of an irredundant set of kings is bounded between (n− 1)=3 and n=3, and that the minimum size of a maximal irredundant set of kings is bounded between n=9 and (n + 2)=3 , where the latter upper and lower bounds are in fact equal when n ≡ 0 (mod 3). Results are given for related domination and independence problems. c © 2002 Elsevier Science B.V. All rights reserved.