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. ‎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)+۱ and \gamma _{\times k,t}(G)+k for \gamma‎ ‎_{\times k,t}(\mu _۱(G))‎, ‎and characterize graphs with \gamma‎ ‎_{\times k,t}(\mu _۱(G))=\gamma _{\times k,t}(G)+۱‎.

Keywords:

k-tuple total dominating set , k-tuple total domination number , m-Mycieleskian graph