King’s Total Domination Number on the Square of Side \(n\)

Abstract

A set \(S \subseteq V\) is a dominating set of a graph \(G = (V, E)\) if each vertex in \(V\) is either in \(S\) or is adjacent to a vertex in \(S\). A vertex is said to dominate itself and all its neighbors. The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\). In terms of a chess board problem, let \(X_n\) be the graph for chess piece \(X\) on the square of side \(n\). Thus, \(\gamma(X_n)\) is the domination number for chess piece \(X\) on the square of side \(n\). In 1964, Yaglom and Yaglom established that \(\gamma(K_n) = \left\lceil \frac{n+2}{2} \right\rceil^2\). This extends to \(\gamma(K_{m,n}) = \left\lceil \frac{m+2}{3} \right\rceil \left\lceil \frac{n+2}{3} \right\rceil\) for the rectangular board. A set \(S \subseteq V\) is a total dominating set of a graph \(G = (V, E)\) if each vertex in \(V\) is adjacent to a vertex in \(S\). A vertex is said to dominate its neighbors but not itself. The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). In 1995, Garnick and Nieuwejaar conducted an analysis of the total domination numbers for the king’s graph on the \(m \times n\) board. In this paper, we note an error in one portion of their analysis and provide a correct general upper bound for \(\gamma_t(K_{m,n})\). Furthermore, we state improved upper bounds for \(\gamma_t(K_n)\).