On Total Domination Number of Cartesian Product of Directed Cycles

Abstract

Let \(\gamma_t(D)\) denote the total domination number of a digraph \(D\), and let \(C_m \Box C_n\) denote the Cartesian product graph of \(C_m\) and \(C_n\), where \(C_m\) denotes the directed cycle of length \(m\), \(m \leq n\). In [On domination number of Cartesian product of directed cycles, Information Processing Letters 110 (2010) 171-173], Liu et al. determined the domination number of \(C_2 \Box C_n\), \(C_3 \Box C_n\), and \(C_4 \Box C_n\). In this paper, we determine the exact values of \(\gamma_t(C_m \Box C_n)\) when at least one of \(m\) and \(n\) is even, or \(n\) is odd and \(m = 1, 3, 5,\) or \(7\).