total domination in $k_r$-covered graphs
چکیده
the inflation $g_{i}$ 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 isomorph to the complete graph $k_{d}$, and each edge $(x_{i},x_{j})$ of $g$ is replaced by an edge $(u,v)$ in such a way that $uin x_{i}$, $vin x_{j}$, and two different edges of $g$ are replaced by non-adjacent edges of $g_{i}$. the total domination number $gamma _{t}(g)$ of a graph $g$ is the minimum cardinality of a total dominating set, which is a set ofvertices such that every vertex of $g$ is adjacent to one vertex of it. a graph is $k_{r}$-covered if every vertex of it is contained in a clique $k_{r}$. cockayne et al. in [total domination in $k_{r}$-covered graphs, ars combin. textbf{71} (2004) 289-303]conjectured that the total domination number of every $k_{r}$-covered graph with $n$ vertices and no $k_{r}$-component is at most $frac{2n}{r+1}.$ this conjecture has been proved only for $3leq rleq 6$. in this paper, we prove this conjecture for a big family of $k_{r}$-covered graphs.