Perfect domination separation on square chessboard

Abstract

This paper focuses on reducing the perfect domination number $\left(\gamma_{p f}\right)$ of the chess pieces rooks, bishops and kings on an $n \times n$ board. Here we reduce this parameter by the separation problem which separates the board by placing a minimum number of chess pieces of a particular type with a minimum number of pawns. A subset $D$ of $V(G)$ is said to be a Perfect Dominating Set (PDS) if every vertex in $V-D$ is dominated by exactly one vertex of $D$. Among all the perfect dominating sets the cardinality of the one with the minimum number of vertices is the Perfect Domination Number $\left(\gamma_{p f}\right)$.