The exact domination number of the generalized Petersen graphs

Yan, Hong; Kang, Liying; Xu, Guangjun

doi:10.1016/j.disc.2008.04.026

Cited by 24 publications

(10 citation statements)

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“…Petersen graphs are among the most interesting examples when considering nontrivial graph invariants. The domination and its variations (such as vertex domination, exact domination, rainbow domination, double Roman domination and other) of generalized Petersen graphs have been extensively studied in recent years, see for example [14,[30][31][32][33][34][35][36].…”

Section: Generalized Petersen Graphsmentioning

confidence: 99%

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

Poklukar

Žerovnik

2022

Mathematics

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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).

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Section: Generalized Petersen Graphsmentioning

confidence: 99%

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

Poklukar

Žerovnik

2022

Mathematics

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“…Let G be a graph with the In this paper, we consider two different generalizations of the Petersen graph. Various types of domination in the class of generalized Petersen graphs have been extensively studied in the literature (see [28][29][30][31][32]). Referring to this research, we will consider (2-d)kernels for two different generalizations of the Petersen graph.…”

Section: Introductionmentioning

confidence: 99%

On (2-d)-Kernels in Two Generalizations of the Petersen Graph

Bednarz

Paja

2021

Symmetry

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A subset J is a (2-d)-kernel of a graph if J is independent and 2-dominating simultaneously. In this paper, we consider two different generalizations of the Petersen graph and we give complete characterizations of these graphs which have (2-d)-kernel. Moreover, we determine the number of (2-d)-kernels of these graphs as well as their lower and upper kernel number. The property that each of the considered generalizations of the Petersen graph has a symmetric structure is useful in finding (2-d)-kernels in these graphs.

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“…Finding a minimum dominating set for general graphs is widely know to be NP-hard [11] and hence it is a challenge to determine classes of graphs for which γ(G) can be exactly computed. Indeed, a closed formula for the domination number has been exactly determined only for few specific classes of graphs such as directed de Bruijn graphs [6], directed Kautz graphs [16], generalized Petersen graphs [24], Cartesian product of two directed paths [21] and graphs defined by two levels of the n-cube [3]. Furthermore, close bounds are provided for some generalizations of the previous classes [9,4].…”

Section: Introductionmentioning

confidence: 99%

On the domination number of $t$-constrained de Bruijn graphs

Calamoneri¹,

Monti²,

Sinaimeri³

2021

Preprint

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Motivated by the work on the domination number of directed de Bruijn graphs and some of its generalizations, in this paper we introduce a natural generalization of de Bruijn graphs (directed and undirected), namely t-constrained de Bruijn graphs, where t is a positive integer, and then study the domination number of these graphs.Within the definition of t-constrained de Bruijn graphs, de Bruijn and Kautz graphs correspond to 1-constrained and 2-constrained de Bruijn graphs, respectively. This generalization inherits many structural properties of de Bruijn graphs and may have similar applications in interconnection networks or bioinformatics.We establish upper and lower bounds for the domination number on t-constrained de Bruijn graphs both in the directed and in the undirected case. These bounds are often very close and in some cases we are able to find the exact value.

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The exact domination number of the generalized Petersen graphs

Cited by 24 publications

References 1 publication

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

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

On (2-d)-Kernels in Two Generalizations of the Petersen Graph

On the domination number of $t$-constrained de Bruijn graphs

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