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

k-Tuple_Total_Domination_in_Inflated_Graphs

The inflated graph $G_{I}$ of a graph $G$ with $n(G)$ vertices is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $u\in X_{i}$, $v\in X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. For integer $k\geq 1$, the $k$-tuple total domination number $\gamma_{\times k,t}(G)$ of $G$ is the minimum cardinality of a $k$-tuple total dominating set of $G$, which is a set of vertices in $G$ such that every vertex of $G$ is adjacent to at least $k$ vertices in it. For existing this number, must the minimum degree of $G$ is at least $k$. Here, we study the $k$-tuple total domination number in inflated graphs when $k\geq 2$. First we prove that $n(G)k\leq \gamma_{\times k,t}(G_{I})\leq n(G)(k+1)-1$, and then we characterize graphs $G$ that the $k$-tuple total domination number number of $G_I$ is $n(G)k$ or $n(G)k+1$. Then we find bounds for this number in the inflated graph $G_I$, when $G$ has a cut-edge $e$ or cut-vertex $v$, in terms on the $k$-tuple total domination number of the inflated graphs of the components of $G-e$ or $v$-components of $G-v$, respectively. Finally, we calculate this number in the inflated graphs that have obtained by some of the known graphs