Algorithmic aspects of k-tuple total domination in graphs

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

[1]  Christian Laforest,et al.  Hardness results and approximation algorithms of k-tuple domination in graphs , 2004, Inf. Process. Lett..

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

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

[4]  G. Nemhauser,et al.  The k-Domination and k-Stability Problems on Sun-Free Chordal Graphs , 1984 .

[5]  Feodor F. Dragan,et al.  Dually Chordal Graphs , 1998, SIAM J. Discret. Math..

[6]  Martin Farber,et al.  Characterizations of strongly chordal graphs , 1983, Discret. Math..

[7]  R. Steele.  Optimization , 2005 .

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

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

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

[11]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[12]  Chung-Shou Liao,et al.  K-tuple Domination in Graphs , 2003, Inf. Process. Lett..

[13]  A. Hoffman,et al.  Totally-Balanced and Greedy Matrices , 1985 .

[14]  Giorgio Gambosi,et al.  Complexity and Approximation , 1999, Springer Berlin Heidelberg.

[15]  Richard M. Karp,et al.  Reducibility among combinatorial problems" in complexity of computer computations , 1972 .

[16]  Giorgio Gambosi,et al.  Complexity and approximation: combinatorial optimization problems and their approximability properties , 1999 .

[17]  M. Chleb ´ õk,et al.  Approximation Hardness of Dominating Set Problems in Bounded Degree Graphs , 2008 .

[18]  D. R. Fulkerson,et al.  Incidence matrices and interval graphs , 1965 .

[19]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[20]  Andreas Brandstiidt.  Classes of bipartite graphs related to chordal graphs , 1991 .

[21]  Viggo Kann,et al.  Hardness of Approximating Problems on Cubic Graphs , 1997, CIAC.