IJPAM: Volume 35, No. 2 (2007)

TOTAL $K$-DOMINATION NUMBER IN GRAPHS

Weiliang Zhao$^1$Department of Mathematics, Shanghai University, Shanghai, 200444, P.R. CHINA, Haichao Wang$^2$, Guangjun Xu$^3$
$^{1,2,3}$Department of Mathematics
Shanghai University
Shanghai, 200444, P.R. CHINA
$^1$e-mail: zwl@shu.edu.cn
$^2$e-mail: whchao2009@shu.edu.cn
$^3$e-mail: gjxu@shu.edu.cn
$^1$Zhejiang Industry Polytechnic College
Shaoxing, 312000, P.R. CHINA


Abstract.Let $G=(V,E)$ be a graph, $k\in\mathbb{N}$. A set $D\subseteq V$ is a total $k$-dominating set of $G$ if every vertex in $V$ is adjacent to at least $k$ vertices of $D$. The total $k$-domination number $\gamma^{k}_t(G)$ of $G$ is the minimum cardinality of a total $k$-dominating set of $G$. In this paper we establish some sharp bounds on $\gamma^k_t(G)$ of a graph $G$. Moreover, we give the exact value of total 2-domination number for a class of grid graphs.

Received: December 19, 2006

AMS Subject Classification: 05C69

Key Words and Phrases: $k$-domination, total $k$-domination, bounds, grid graph

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 35
Issue: 2