On the Complexity of {k}-domination for Chordal Graphs

In this work we obtain a new graph class where {k}-DOM is NP-complete: the class of chordal graphs. We also identify some maximal subclasses for which it is polynomial time solvable. By relating this problem with k-DOM, we prove that {k}-DOM is polynomial time solvable for strongly chordal graphs. Besides, by expressing the property involved in k-DOM in Counting Monadic Second- order Logic, we obtain that both problems are linear time solvable for bounded tree-width graphs. In this way we enlarge the family of graphs for which k-DOM is polynomial time solvable.Sociedad Argentina de Informática e Investigación Operativ

On the Complexity of {k}-domination for Chordal Graphs

In this work we obtain a new graph class where {k}-DOM is NP-complete: the class of chordal graphs. We also identify some maximal subclasses for which it is polynomial time solvable. By relating this problem with k-DOM, we prove that {k}-DOM is polynomial time solvable for strongly chordal graphs. Besides, by expressing the property involved in k-DOM in Counting Monadic Second- order Logic, we obtain that both problems are linear time solvable for bounded tree-width graphs. In this way we enlarge the family of graphs for which k-DOM is polynomial time solvable.Sociedad Argentina de Informática e Investigación Operativ

On the Complexity of {k}-domination for Chordal Graphs

In this work we obtain a new graph class where {k}-DOM is NP-complete: the class of chordal graphs. We also identify some maximal subclasses for which it is polynomial time solvable. By relating this problem with k-DOM, we prove that {k}-DOM is polynomial time solvable for strongly chordal graphs. Besides, by expressing the property involved in k-DOM in Counting Monadic Second- order Logic, we obtain that both problems are linear time solvable for bounded tree-width graphs. In this way we enlarge the family of graphs for which k-DOM is polynomial time solvable.Sociedad Argentina de Informática e Investigación Operativ

The k-tuple domination number revisited

The following fundamental result for the domination number γ (G) of a graph G was proved by Alon and Spencer, Arnautov, Lovász and Payan: γ (G) ≤ frac(ln (δ + 1) + 1, δ + 1) n, where n is the order and δ is the minimum degree of vertices of G. A similar upper bound for the double domination number was found by Harant and Henning [J. Harant, M.A. Henning, On double domination in graphs, Discuss. Math. Graph Theory 25 (2005) 29-34], and for the triple domination number by Rautenbach and Volkmann [D. Rautenbach, L. Volkmann, New bounds on the k-domination number and the k-tuple domination number, Appl. Math. Lett. 20 (2007) 98-102], who also posed the interesting conjecture on the k-tuple domination number: for any graph G with δ ≥ k - 1, γ× k (G) ≤ frac(ln (δ - k + 2) + ln (over(d, ̂)k - 1 + over(d, ̂)k - 2) + 1, δ - k + 2) n, where over(d, ̂)m = ∑i = 1n ((di; m)) / n is the m-degree of G. This conjecture, if true, would generalize all the mentioned upper bounds and improve an upper bound proved in [A. Gagarin, V. Zverovich, A generalised upper bound for the k-tuple domination number, Discrete Math. (2007), in press (doi:10.1016/j.disc.2007.07.033)]. In this paper, we prove the Rautenbach-Volkmann conjecture. © 2007 Elsevier Ltd. All rights reserved