Total domination in Kr-covered graphs

The inflation GI of a graph G with n(G) vertices and m(G) edges is obtained from G by replacing every vertex of degree d of G by a clique, which is isomorphic 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 . The total domination number γt(G) of a graph G is the minimum cardinality of a total dominating set, which is a set of vertices such that every vertex of G is adjacent to one vertex of it. A graph is Kr-covered if every vertex of it is contained in a clique Kr. Cockayne et al. in [Total domination in Kr-covered graphs, Ars Combin. 71 (2004) 289-303] conjectured that the total domination number of every Kr-covered graph with n vertices and no Kr-component is at most 2n r+1 . This conjecture has been proved only for 3 ≤ r ≤ 6. In this paper, we prove this conjecture for a big family of Kr-covered graphs.