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## Abstract

Given a simple graph G, a *dominating set* in G is a set of vertices S such that every vertex not in S has a neighbor in S. Denote the *domination number*, which is the size of any minimum dominating set of G, by *γ(G)*. For any integer k ≥ 1, a function f : V (G) → {0, 1, . . ., k} 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 values over all the vertices. The *{k}-domination number of G, γ _{{k}}*(G), is defined to be the minimum weight taken over all {k}-domination functions. Brešar, Henning, and Klavžar (On integer domination in graphs and Vizing-like problems.

*Taiwanese J. Math.*

**10(5)**(2006) pp. 1317--1328) asked whether there exists an integer k ≥ 2 so that γ

_{{k}}(G □ H) ≥ γ(G)γ(H). In this note we use the Roman 2-domination number, γ

_{R2}of Chellali, Haynes, Hedetniemi, and McRae, (Roman 2-domination.

*Discrete Applied Mathematics*

**204**(2016) pp. 22-28.) to prove that if G is a claw-free graph and H is an arbitrary graph, then γ

_{{2}}(G □ H) ≥ γ

_{R2}(G □ H) ≥ γ(G)γ(H).

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

## Recommended Citation

Krop, Elliot and Davila, Randy R.
(2020)
"On a Vizing-type Integer Domination Conjecture,"
*Theory and Applications of Graphs*: Vol. 7:
Iss.
1, Article 4.

DOI: 10.20429/tag.2020.070104

Available at:
https://digitalcommons.georgiasouthern.edu/tag/vol7/iss1/4

*Supplemental Reference List*