B. Bre¹ar, T.R. Hartinger, T. Kos, M. Milanič
On total domination in the Cartesian product of graphs
Discussiones Mathematicae Graph Theory
Received 03.08.2016, Revised 28.12.2016, Accepted 25.03.2017, doi: 10.7151/dmgt.2039

Ho proved in [ A note on the total domination number, Util. Math. 77 (2008) 97--100] that the total domination number of the Cartesian product of any two graphs without isolated vertices is at least one half of the product of their total domination numbers. We extend a result of Lu and Hou from [ Total domination in the Cartesian product of a graph and K2 or Cn, Util. Math. 83 (2010) 313--322] by characterizing the pairs of graphs G and H for which γtGH=\frac{1}{2}γtG γt (H), whenever γtH=2. In addition, we present an infinite family of graphs Gn with γt (Gn)=2n, which asymptotically approximate equality in \hbox{γt(Gn\Box Gn)≥ \frac{1}{2}γt(Gn)2}.
total domination, Cartesian product, total domination quotient