Double Total Domination Number on Some Chemical Nanotubes

Suppose G is a graph with the vertex set V (G). A set D ⊆ V (G) is a total k-dominating set if every vertex v ∈ V (G) has at least k neighbours in D. The total k-domination number γkt(G) is the size of the smallest total k-dominating set. When k = 2 the total 2-dominating set is referred to as a double total dominating set. In this work we compute the exact values for double total domination number on H-phenylenic nanotubes HPH(m,n), m,n ≥ 2 and H-naphtalenic nanotubes HN(m,n), n = 2k, m,n ≥ 2. As all vertices have a degree 2 or 3, there is no total k-domination for k ≥ 3 for H-phenylenic and H-naphtalenic nanotubes, and the double total domination is the maximum possible.