k-TUPLE TOTAL DOMINATION IN INFLATED GRAPHS

The inflated graph GI of a graph G with n(G) vertices is obtained from G by replacing every vertex of degree d of G by a clique, which is isomorph to the complete graph Kd, and each edge (xi, xj) of G is replaced by an edge (u, v) in such a way that u ∈ Xi, v ∈ Xj , and two different edges of G are replaced by non-adjacent edges of GI . For integer k ≥ 1, the k-tuple total domination number γ ×k,t(G) of G is the minimum cardinality of a k-tuple total dominating set of G, which is a set of vertices in G such that every vertex of G is adjacent to at least k vertices in it. For existing this number, must the minimum degree of G is at least k. Here, we study the k-tuple total domination number in inflated graphs when k ≥ 2. First we prove that n(G)k ≤ γ ×k,t(GI ) ≤ n(G)(k + 1) − 1, and then we characterize graphs G that the k-tuple total domination number number of GI is n(G)k or n(G)k + 1. Then we find bounds for this number in the inflated graph GI , when G has a cut-edge e or cut-vertex v, in terms on the k-tuple total domination number of the inflated graphs of the components of G− e or v-components of G− v, respectively. Finally, we calculate this number in the inflated graphs that have obtained by some of the known graphs.