ON LOCATING AND DIFFERENTIATING-TOTAL K-DOMINATING SET IN GRAPHS

LetGbe a graph with vertex setV=V(G) and edge setE=E(G). A subsetSofVis a totalk-dominating set ofGif every vertex inVis within distancekfrom some vertex inS. The totalk-domination number,kt(G) is the minimum cardinality of a totalk-dominating set ofG. We areinterested in to introduce two types of totalk-dominating sets, namely locating-totalk-dominatingset of a graphG(Lt;k(G)-set) and di erentiating-totalk-dominating set of a graphG(Dt;k(G)-set).We show that, for any graphGof ordern, andd(G)≥k+ 1,Lt;k(G)≥log2(n5) + 2. This bound issharp. We obtain sharp upper and lower bounds forLt;k(T) for every treeT. We also show that,Dt;2(T)≥5(d+1)9for every treeTof ordern≥6 and diameterd≥5, without strong support vertex.