Total perfect codes‎, ‎OO-irredundant and total subdivision in graphs

Let G = (V (G),E(G)) be a graph, γt(G). Let ooir(G) be the total domination and OO-irredundance number of G, respectively. A total dominating set S of G is called a total perfect code if every vertex in V (G) is adjacent to exactly one vertex of S. In this paper, we show that if G has a total perfect code, then γ t(G) = ooir(G). As a consequence, we determine the value of ooir(G) for some classes of graphs. Finally, we prove some new bounds for the total subdivision number.