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# k-Rainbow Domination Number of P3□Pn

1
Department of network technology, South China Institute of Software Engineering, Guangzhou 510990, China
2
Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China
3
South China Business College, Guang Dong University of Foreign Studies, Guangzhou 510545, China
4
Department of Mathematics and Computer Science, Sirjan University of Technology, Sirjan 7813733385, Iran
5
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 5375171379, Iran
6
School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(2), 203; https://doi.org/10.3390/math7020203
Received: 12 January 2019 / Revised: 5 February 2019 / Accepted: 19 February 2019 / Published: 21 February 2019
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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PDF [249 KB, uploaded 21 February 2019]
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# Abstract

Let k be a positive integer, and set $[ k ] : = { 1 , 2 , … , k }$ . For a graph G, a k-rainbow dominating function (or kRDF) of G is a mapping $f :$ $V ( G ) → 2 [ k ]$ in such a way that, for any vertex $v ∈ V ( G )$ with the empty set under f, the condition $⋃ u ∈ N G ( v ) f ( u ) = [ k ]$ always holds, where $N G ( v )$ is the open neighborhood of v. The weight of kRDF f of G is the summation of values of all vertices under f. The k-rainbow domination number of G, denoted by $γ r k ( G )$ , is the minimum weight of a kRDF of G. In this paper, we obtain the k-rainbow domination number of grid $P 3 □ P n$ for $k ∈ { 2 , 3 , 4 }$ . View Full-Text
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Wang, Y.; Wu, X.; Dehgardi, N.; Amjadi, J.; Khoeilar, R.; Liu, J.-B. k-Rainbow Domination Number of P3Pn. Mathematics 2019, 7, 203.

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