k-tuple total domination and mycieleskian graphs

let $k$ be a positive integer. a subset $s$ of $v(g)$ in a graph $g$ is a $k$-tuple total dominating set of $g$ if every vertex of $g$ has at least $k$ neighbors in $s$. the $k$-tuple total domination number $gamma _{times k,t}(g)$ of $g$ is the minimum cardinality of a $k$-tuple total dominating set of $g$. if$v(g)=v^{0}={v_{1}^{0},v_{2}^{0},ldots ,v_{n}^{0}}$ and $e(g)=e_{0}$, then for any integer $mgeq 1$ the $m$-emph{mycieleskian} $mu _{m}(g)$ of $g$ is the graph with vertex set $v^{0}cup v^{1}cup v^{2}cup cdots cup v^{m}cup {u}$, where $v^{i}={v_{j}^{i}mid v_{j}^{0}in v^{0}}$ is the $i$-th distinct copy of $v^{0}$, for $% i=1,2,ldots ,m$, and edge set $e_{0}cup left( bigcup _{i=0}^{m-1}{v_{j}^{i}v_{j^{prime }}^{i+1}mid v_{j}^{0}v_{j^{prime }}^{0}in e_{0}}right) cup {v_{j}^{m}umid v_{j}^{m}in v^{m}}$. in this paper for a given graph $g$ with minimum degree at least $k$, we find some sharp lower and upper bounds on the $k$-tuple total domination number of the $m$-mycieleskian graph $mu _{m}(g)$ of $g$ in terms on $k$ and $gamma_{times k,t}(g)$. specially we give the sharp bounds $gamma _{times k,t}(g)+1$ and $gamma _{times k,t}(g)+k$ for $gamma_{times k,t}(mu _1(g))$, and characterize graphs with $gamma_{times k,t}(mu _1(g))=gamma _{times k,t}(g)+1$.