Total 2-rainbow domination  numbers in trees

We obtain new results on independent 2- and 3-rainbow domination numbers of generalized Petersen graphs P ( n , k ) for certain values of n , k ∈ N . By suitably adjusting and applying a well established technique of tropical algebra (path algebra) we obtain exact 2-independent rainbow domination numbers of generalized Petersen graphs P ( n , 2 ) and P ( n , 3 ) thus confirming a conjecture proposed by Shao et al. In addition, we compute exact 3-independent rainbow domination numbers of generalized Petersen graphs P ( n , 2 ) . The method used here is developed for rainbow domination and for Petersen graphs. However, with some natural modifications, the method used can be applied to other domination type invariants, and to many other classes of graphs including grids and tori.

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General Bounds on Rainbow Domination Numbers

Graphs and Combinatorics ◽

10.1007/s00373-013-1394-9 ◽

2013 ◽

Vol 31 (3) ◽

pp. 601-613 ◽

Cited By ~ 9

Author(s):

Shinya Fujita ◽

Michitaka Furuya ◽

Colton Magnant

Keyword(s):

Rainbow Domination ◽

Domination Numbers

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Strong Equality Between the 2-Rainbow Domination and Independent 2-Rainbow Domination Numbers in Trees

Bulletin of the Malaysian Mathematical Sciences Society ◽

10.1007/s40840-015-0284-0 ◽

2015 ◽

Vol 39 (S1) ◽

pp. 205-218 ◽

Cited By ~ 3

Author(s):

J. Amjadi ◽

M. Falahat ◽

S. M. Sheikholeslami ◽

N. Jafari Rad

Keyword(s):

Rainbow Domination ◽

Domination Numbers

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On 3-Rainbow Domination Number of Generalized Petersen Graphs P(6k,k)

Symmetry ◽

10.3390/sym13101860 ◽

2021 ◽

Vol 13 (10) ◽

pp. 1860

Author(s):

Rija Erveš ◽

Janez Žerovnik

Keyword(s):

Domination Number ◽

Upper Bounds ◽

Lower And Upper Bounds ◽

Generalized Petersen Graphs ◽

Rainbow Domination ◽

Petersen Graphs ◽

Special Case ◽

Domination Numbers

We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs P(6k,k). In some cases, for some infinite families, exact values are established; in all other cases, the lower and upper bounds with small gaps are given. We also define singleton rainbow domination, where the sets assigned have a cardinality of, at most, one, and provide analogous results for this special case of rainbow domination.

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On rainbow domination numbers of graphs

Information Sciences ◽

10.1016/j.ins.2013.07.020 ◽

2014 ◽

Vol 254 ◽

pp. 225-234 ◽

Cited By ~ 19

Author(s):

Zehui Shao ◽

Meilian Liang ◽

Chuang Yin ◽

Xiaodong Xu ◽

Polona Pavlič ◽

...

Keyword(s):

Rainbow Domination ◽

Domination Numbers

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The 2-Rainbow Domination Numbers of $${\boldsymbol{C}}_4\Box {\boldsymbol{C}}_n$$ C 4 □ C n and $${\boldsymbol{C}}_8\Box {\boldsymbol{C}}_n$$ C 8 □ C n

National Academy Science Letters ◽

10.1007/s40009-018-0779-y ◽

2019 ◽

Vol 42 (5) ◽

pp. 411-418 ◽

Cited By ~ 1

Author(s):

Zehui Shao ◽

Zepeng Li ◽

Rija Erveš ◽

Janez Žerovnik

Keyword(s):

Rainbow Domination ◽

Domination Numbers

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Bounds on weak roman and 2-rainbow domination numbers

Discrete Applied Mathematics ◽

10.1016/j.dam.2014.06.016 ◽

2014 ◽

Vol 178 ◽

pp. 27-32 ◽

Cited By ~ 23

Author(s):

Mustapha Chellali ◽

Teresa W. Haynes ◽

Stephen T. Hedetniemi

Keyword(s):

Rainbow Domination ◽

Domination Numbers

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Independent 2-rainbow domination in trees

Asian-European Journal of Mathematics ◽

10.1142/s1793557115500357 ◽

2015 ◽

Vol 08 (02) ◽

pp. 1550035 ◽

Cited By ~ 2

Author(s):

J. Amjadi ◽

N. Dehgardi ◽

N. Mohammadi ◽

S. M. Sheikholeslami ◽

L. Volkmann

Keyword(s):

Minimum Weight ◽

Domination Number ◽

Sharp Bound ◽

Rainbow Domination ◽

Vertex Set ◽

Domination In Graphs ◽

Dominating Function ◽

Domination Numbers

A 2-rainbow dominating function (2RDF) on a graph G is a function f from the vertex set V(G) to the set of all subsets of the set {1, 2} such that for any vertex v ∈ V(G) with f(v) = ∅ the condition ⋃u∈N(v)f(u) = {1, 2} is fulfilled. A 2RDF f is independent 2-rainbow dominating function (I2RDF) if no two vertices assigned nonempty sets are adjacent. The weight of a 2RDF f is the value ω(f) = ∑v∈V |f(v)|. The 2-rainbow domination number γr2(G) (respectively, the independent 2-rainbow domination number ir2(G)) is the minimum weight of a 2RDF (respectively, I2RDF) on G. M. Chellali and N. Jafari Rad [Independent 2-rainbow domination in graphs, to appear in J. Combin. Math. Combin. Comput.] have studied the independent 2-rainbow domination numbers in graphs and posed the following problem: Find a sharp bound for ir2(T) in terms of the order of a tree T. In this paper we prove that for every tree T of order n ≥ 3, [Formula: see text].

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