Total domination number of the conjunction of graphs

In this paper we discuss the total domination number with respect to the conjunction of two graphs. We estimate the total domination number of the conjunction G∧HG∧H, showing that γt(G∧H)⩽γt(G)γt(H)γt(G∧H)⩽γt(G)γt(H), for graphs having no isolated vertices. Additionally, we show that for G≅PnG≅Pn or for G and H having the domination number equal to half their orders, the equality is attained.