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.

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 Outer independent rainbow dominating functions in graphs

2020 ◽  
Vol 40 (5) ◽  
pp. 599-615
Author(s):  
Zhila Mansouri ◽  
Doost Ali Mojdeh
Keyword(s):  
Minimum Weight ◽  
Vertex Cover ◽  
Domination Number ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Rainbow Domination ◽  
Dominating Function

A 2-rainbow dominating function (2-rD function) of a graph \(G=(V,E)\) is a function \(f:V(G)\rightarrow\{\emptyset,\{1\},\{2\},\{1,2\}\}\) having the property that if \(f(x)=\emptyset\), then \(f(N(x))=\{1,2\}\). The 2-rainbow domination number \(\gamma_{r2}(G)\) is the minimum weight of \(\sum_{v\in V(G)}|f(v)|\) taken over all 2-rainbow dominating functions \(f\). An outer-independent 2-rainbow dominating function (OI2-rD function) of a graph \(G\) is a 2-rD function \(f\) for which the set of all \(v\in V(G)\) with \(f(v)=\emptyset\) is independent. The outer independent 2-rainbow domination number \(\gamma_{oir2}(G)\) is the minimum weight of an OI2-rD function of \(G\). In this paper, we study the OI2-rD number of graphs. We give the complexity of the problem OI2-rD of graphs and present lower and upper bounds on \(\gamma_{oir2}(G)\). Moreover, we characterize graphs with some small or large OI2-rD numbers and we also bound this parameter from above for trees in terms of the order, leaves and the number of support vertices and characterize all trees attaining the bound. Finally, we show that any ordered pair \((a,b)\) is realizable as the vertex cover number and OI2-rD numbers of some non-trivial tree if and only if \(a+1\leq b\leq 2a\).

On Double Domination Numbers of Generalized Petersen Graphs

2016 ◽  
Vol 13 (10) ◽  
pp. 6514-6518
Author(s):  
Minhong Sun ◽  
Zehui Shao
Keyword(s):  
Programming Model ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Minimum Size ◽  
Generalized Petersen Graphs ◽  
Small Parameters ◽  
Petersen Graphs ◽  
Double Domination ◽  
Domination Numbers

A (total) double dominating set in a graph G is a subset S ⊆ V(G) such that each vertex in V(G) is (total) dominated by at least 2 vertices in S. The (total) double domination number of G is the minimum size of a (total) double dominating set of G. We determine the total double domination numbers and give upper bounds for double domination numbers of generalized Petersen graphs. By applying an integer programming model for double domination numbers of a graph, we have determined some exact values of double domination numbers of these generalized Petersen graphs with small parameters. The result shows that the given upper bounds match these exact values.

scholarly journals On the Double Roman Domination in Generalized Petersen Graphs P(5k,k)

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 119
Author(s):  
Darja Rupnik Poklukar ◽  
Janez Žerovnik
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Upper Bounds ◽  
Roman Domination ◽  
Generalized Petersen Graphs ◽  
Roman Domination Number ◽  
Double Roman Domination ◽  
Petersen Graphs ◽  
Roman Dominating Function ◽  
Dominating Function

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and every vertex u with f(u)=1 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 equals the minimum weight of a double Roman dominating function of G. We obtain closed expressions for the double Roman domination number of generalized Petersen graphs P(5k,k). It is proven that γdR(P(5k,k))=8k for k≡2,3mod5 and 8k≤γdR(P(5k,k))≤8k+2 for k≡0,1,4mod5. We also improve the upper bounds for generalized Petersen graphs P(20k,k).

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}.

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.

Signed domination number of hypercubes

2021 ◽  
Author(s):  
B. ShekinahHenry ◽  
Y. S. Irine Sheela
Keyword(s):  
Domination Number ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Signed Domination ◽  
Vertex Set ◽  
Number Formula ◽  
Signed Domination Number

The [Formula: see text]-cube graph or hypercube [Formula: see text] is the graph whose vertex set is the set of all [Formula: see text]-dimensional Boolean vectors, two vertices being joined if and only if they differ in exactly one co-ordinate. The purpose of the paper is to investigate the signed domination number of this hypercube graphs. In this paper, signed domination number [Formula: see text]-cube graph for odd [Formula: see text] is found and the lower and upper bounds of hypercube for even [Formula: see text] are found.

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