The total dominating equivalences of some graphs

Let G = (V,E) be a simple graph of order n. The total dominating set of G is a subset D of V that every vertex of V is adjacent to some vertices of D and the total domination number of G is the minimum cardinality of total dominating set in G and is denoted by γt(G). The total domination polynomial of G is the polynomial Dt(G,x) =∑n i=γt(G) dt(G,i)xi, where dt(G,i) is the number of total dominating sets of G of size i. Two graphs G and H are considered Dt-equivalent if Dt(G,x) = Dt(H,x). The equivalence class of G, denoted [G], is the set of all graphs Dt-equivalent to G. In this paper, we investigate the Dt-equivalence classes of some graphs.