On the algorithmic complexity of k-tuple total domination

For a fixed positive integer k, a k-tuple total dominating set of a graph G is a set D⊆V(G) such that every vertex of G is adjacent to at least k vertices in D. The k-tuple total domination problem is to determine a minimum k-tuple total dominating set of G. This paper studies k-tuple total domination from an algorithmic point of view. In particular, we present a linear-time algorithm for the k-tuple total domination problem for graphs in which each block is a clique, a cycle or a complete bipartite graph, which include trees, block graphs, cacti and block-cactus graphs. We also establish NP-hardness of the k-tuple total domination problem in undirected path graphs.

[1]  Michael A. Henning,et al.  A survey of selected recent results on total domination in graphs , 2009, Discret. Math..

[2]  Dinabandhu Pradhan.  Complexity of certain Functional Variants of total domination in Chordal bipartite Graphs , 2012, Discret. Math. Algorithms Appl..

[3]  P. Pardalos,et al.  Handbook of Combinatorial Optimization , 1998 .

[4]  Stephen T. Hedetniemi,et al.  Total domination in graphs , 1980, Networks.

[5]  James R. Walter,et al.  Representations of chordal graphs as subtrees of a tree , 1978, J. Graph Theory.

[6]  Gerard J. Chang,et al.  Algorithmic aspects of the kk-domination problem in graphs , 2013, Discret. Appl. Math..

[7]  Fanica Gavril,et al.  A recognition algorithm for the intersection graphs of paths in trees , 1978, Discret. Math..

[8]  F. Gavril.  The intersection graphs of subtrees in tree are exactly the chordal graphs , 1974 .

[9]  S. M. Hedetniemi,et al.  On the Algorithmic Complexity of Total Domination , 1984 .

[10]  S. Hedetniemi,et al.  Domination in graphs : advanced topics , 1998 .

[11]  Michael A. Henning,et al.  K-tuple Total Domination in Graphs , 2010, Discret. Appl. Math..

[12]  Dinabandhu Pradhan,et al.  Algorithmic aspects of k-tuple total domination in graphs , 2012, Inf. Process. Lett..

[13]  Michael A. Henning,et al.  Strong Transversals in Hypergraphs and Double Total Domination in Graphs , 2010, SIAM J. Discret. Math..

[14]  Chung-Shou Liao,et al.  ALGORITHMIC ASPECT OF k-TUPLE DOMINATION IN GRAPHS , 2002 .

[15]  Frank Harary,et al.  Double Domination in Graphs , 2000, Ars Comb..

[16]  Peter J. Slater,et al.  Fundamentals of domination in graphs , 1998, Pure and applied mathematics.

[17]  G. Chang.  Total domination in block graphs , 1989 .

[18]  Gerard J. Chang,et al.  Algorithmic Aspects of Domination in Graphs , 1998 .

[19]  Michael A. Henning,et al.  k-tuple total domination in cross products of graphs , 2012, J. Comb. Optim..