On 2-rainbow domination in generalized Petersen graphs

2017 ◽  
Vol 2 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Kuo-Hua Wu ◽  
Yue-Li Wang ◽  
Chiun-Chieh Hsu ◽  
Chao-Cheng Shih
Keyword(s):  
Generalized Petersen Graphs ◽  
Rainbow Domination ◽  
Petersen Graphs

scholarly journals Independent Rainbow Domination Numbers of Generalized Petersen Graphs P(n,2) and P(n,3)

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 996
Author(s):  
Boštjan Gabrovšek ◽  
Aljoša Peperko ◽  
Janez Žerovnik
Keyword(s):  
Path Algebra ◽  
Generalized Petersen Graphs ◽  
Tropical Algebra ◽  
Rainbow Domination ◽  
Petersen Graphs ◽  
Established Technique ◽  
Domination Numbers

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.

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

Symmetry ◽  
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.

scholarly journals 2-rainbow domination of generalized Petersen graphs P(n,2)

2009 ◽  
Vol 157 (8) ◽  
pp. 1932-1937 ◽  
Author(s):  
Chunling Tong ◽  
Xiaohui Lin ◽  
Yuansheng Yang ◽  
Meiqin Luo
Keyword(s):  
Generalized Petersen Graphs ◽  
Rainbow Domination ◽  
Petersen Graphs

scholarly journals On 2-Rainbow Domination Number of Generalized Petersen Graphs P(5k,k)

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 809
Author(s):  
Rija Erveš ◽  
Janez Žerovnik
Keyword(s):  
Domination Number ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Generalized Petersen Graphs ◽  
Rainbow Domination ◽  
Petersen Graphs

We obtain new results on 2-rainbow domination number of generalized Petersen graphs P(5k,k). In some cases (for some infinite families), exact values are established, and in all other cases lower and upper bounds are given. In particular, it is shown that, for k>3, γr2(P(5k,k))=4k for k≡2,8mod10, γr2(P(5k,k))=4k+1 for k≡5,9mod10, 4k+1≤γr2(P(5k,k))≤4k+2 for k≡1,6,7mod10, and 4k+1≤γr2(P(5k,k))≤4k+3 for k≡0,3,4mod10.

On 2-rainbow domination of generalized Petersen graphs

2019 ◽  
Vol 257 ◽  
pp. 370-384 ◽  
Author(s):  
Zehui Shao ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Shaohui Wang ◽  
Janez Žerovnik ◽  
...  
Keyword(s):  
Generalized Petersen Graphs ◽  
Rainbow Domination ◽  
Petersen Graphs

scholarly journals Double Roman Graphs in P(3k, k)

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 336
Author(s):  
Zehui Shao ◽  
Rija Erveš ◽  
Huiqin Jiang ◽  
Aljoša Peperko ◽  
Pu Wu ◽  
...  
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Generalized Petersen Graphs ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Petersen Graphs ◽  
Roman Dominating Function ◽  
Sharp Lower Bound ◽  
Dominating Function

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} with the properties that if f(u)=0, then vertex u is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and if f(u)=1, then vertex u is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G is the minimum weight of a double Roman dominating function of G. A graph is said to be double Roman if γdR(G)=3γ(G), where γ(G) is the domination number of G. We obtain the sharp lower bound of the double Roman domination number of generalized Petersen graphs P(3k,k), and we construct solutions providing the upper bounds, which gives exact values of the double Roman domination number for all generalized Petersen graphs P(3k,k). This implies that P(3k,k) is a double Roman graph if and only if either k≡0 (mod 3) or k∈{1,4}.

scholarly journals On the 2-extendability of the generalized Petersen graphs

1989 ◽  
Vol 78 (1-2) ◽  
pp. 169-177 ◽  
Author(s):  
Gerald Schrag ◽  
Larry Cammack
Keyword(s):  
Generalized Petersen Graphs ◽  
Petersen Graphs
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