## Taiwanese Journal of Mathematics

### ON INTEGER DOMINATION IN GRAPHS AND VIZING-LIKE PROBLEMS

#### Abstract

We continue the study of $\{k\}$-dominating functions in graphs (or integer domination as we shall also say) started by Domke, Hedetniemi, Laskar, and Fricke [5]. For $k \ge 1$ an integer, a function $f \colon V(G) \rightarrow \{0,1,\ldots,k\}$ defined on the vertices of a graph $G$ is called a $\{k\}$-dominating function if the sum of its function values over any closed neighborhood is at least~$k$. The weight of a $\{k\}$-dominating function is the sum of its function values over all vertices. The $\{k\}$-domination number of $G$ is the minimum weight of a $\{k\}$-dominating function of $G$. We study the $\{k\}$-domination number on the Cartesian product of graphs, mostly on problems related to the famous Vizing's conjecture. A connection between the $\{k\}$-domination number and other domination type parameters is also studied.

#### Article information

Source
Taiwanese J. Math., Volume 10, Number 5 (2006), 1317-1328.

Dates
First available in Project Euclid: 20 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500557305

Digital Object Identifier
doi:10.11650/twjm/1500557305

Mathematical Reviews number (MathSciNet)
MR2253381

Zentralblatt MATH identifier
1109.05081

Subjects
Primary: 05C69: Dominating sets, independent sets, cliques

#### Citation

Brešar, Boštjan; Henning, Michael A.; Klavžar, Sandi. ON INTEGER DOMINATION IN GRAPHS AND VIZING-LIKE PROBLEMS. Taiwanese J. Math. 10 (2006), no. 5, 1317--1328. doi:10.11650/twjm/1500557305. https://projecteuclid.org/euclid.twjm/1500557305

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