Total domination in KrKr-covered graphs

The inflation GIGI of a graph GG with n(G)n(G) vertices and m(G)m(G) edges is obtained from GG by replacing every vertex of degree dd of GG by a clique, which is isomorph to the complete graph KdKd, and each edge (xi,xj)(xi,xj) of GG is replaced by an edge (u,v)(u,v) in such a way that uinXiuinXi, vinXjvinXj, and two different edges of GG are replaced by non-adjacent edges of GIGI. The total domination number gammat(G)gammat(G) of a graph GG is the minimum cardinality of a total dominating set, which is a set of vertices such that every vertex of GG is adjacent to one vertex of it. A graph is KrKr-covered if every vertex of it is contained in a clique KrKr. Cockayne et al. in [Total domination in KrKr-covered graphs, Ars Combin. textbf{71} (2004) 289-303] conjectured that the total domination number of every KrKr-covered graph with nn vertices and no KrKr-component is at most frac2nr+1.frac2nr+1. This conjecture has been proved only for 3leqrleq63leqrleq6. In this paper, we prove this conjecture for a big family of KrKr-covered graphs.