Algorithmic aspects of k-tuple total domination in graphs

a r t i c l e i n f o a b s t r a c t For a fixed positive integer k, a k-tuple total dominating set of a graph G = (V , E) is a subset T D k of V such that every vertex in V is adjacent to at least k vertices of T D k. In minimum k-tuple total dominating set problem (Min k-Tuple Total Dom Set), it is required to find a k-tuple total dominating set of minimum cardinality and Decide Min k-Tuple Total Dom Set is the decision version of Min k-Tuple Total Dom Set problem. In this paper, we show that Decide Min k-Tuple Total Dom Set is NP-complete for split graphs, doubly chordal graphs and bipartite graphs. For chordal bipartite graphs, we show that Min k-Tuple Total Dom Set can be solved in polynomial time. We also propose some hardness results and approximation algorithms for Min k-Tuple Total Dom Set problem.