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 \(\left(\gamma_{p f}\right)\).