Total Domination of the \(m \times n\) Chessboard by Kings, Crosses, and Knights

For graph \(G\), a total dominating set \(S\) is a subset of the vertices in \(G\) such that every vertex in \(G\) is adjacent to at least one vertex in \(S\). The total domination number of \(G\) is the cardinality of a smallest total dominating set of \(G\). We consider the total domination number of graphs formed from an \(m\times n\) chessboard by letting vertices represent the squares, and letting two vertices be adjacent if a given chess piece can move between the associated squares. In particular, we bound from above and below the total domination numbers of the graphs induced by the movement of kings, knights, and crosses (a hypothetical piece that moves as does a king, except that it cannot move diagonally). We also provide some results of computer searches for the total domination numbers of small square boards.