Rainbow Domination in Graphs

Brešar, Boštjan

doi:10.1007/978-3-030-51117-3_12

Cited by 16 publications

(39 citation statements)

References 61 publications

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“…A linear algorithm for determining a 2-rD function of minimum weight of an arbitrary tree has been presented in [2]. The algorithm was based on the related concept of the so-called weak 2-domination.…”

Section: -Rainbow and Weak {3}-domination Numbersmentioning

confidence: 99%

“…In 2008, Bresar et al [2] introduced the k-rainbow domination as a generalization of domination in graphs. The minimum weight of a k-rDF of G is called the k-rainbow domination number of G and is denoted by γ rk (G).…”

Section: Introductionmentioning

confidence: 99%

“…In 2008, Bresar et al [2] introduced the k-rainbow domination as a generalization of domination in graphs. A k-rDF f is an independent k-rainbow dominating function (Ik-rD function) if no two vertices assigned nonempty sets are adjacent.…”

Section: Introductionmentioning

confidence: 99%

“…Also, the edge or vertex critical graphs with respect to the rainbow domination have been investigated in literature. For reading the results for special families of graphs such as paths, cycles and the generalized Petersen graphs the reader can consult [1,2,4,10]. The rainbow domination numbers are studied for digraphs and also Cartesian product of some digraphs [5].…”

Section: Introductionmentioning

confidence: 99%

“…Pai and Chiu [8] developed an exact algorithm and a heuristic for 3-rainbow domination. Recently, Chang et al showed that the k-rainbow domination number is equal to the so-called weak k-domination number for strongly chordal graphs (see [2,3]).…”

Section: Introductionmentioning

confidence: 99%

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(Independent) $k$-Rainbow Domination of a Graph

Mojdeh

Mansouri

2020

Turkish Journal of Mathematics and Computer Science

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Let k be a positive integer and γ rk (G) (γ i rk (G)) be k-rainbow domination (independent k-rainbow domination) number of a graph G. In this paper, we study the k-rainbow domination and independent k-rainbow domination numbers of graphs. We obtain bounds for γ rk (G − e) (γ i rk (G − e)) in terms of γ rk (G) (γ i rk (G)). Finally, the relation between weak 3-domination and 3-rainbow domination number of graphs will be investigated.

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Section: -Rainbow and Weak {3}-domination Numbersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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(Independent) $k$-Rainbow Domination of a Graph

Mojdeh

Mansouri

2020

Turkish Journal of Mathematics and Computer Science

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On k-rainbow independent domination in graphs

Šumenjak

Rall

Tepeh

2018

Applied Mathematics and Computation

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In this paper, we define a new domination invariant on a graph G, which coincides with the ordinary independent domination number of the generalized prism G ✷ K k , called the k-rainbow independent domination number and denoted by γ rik (G). Some bounds and exact values concerning this domination concept are determined. As a main result, we prove a Nordhaus-Gaddum-type theorem on the sum for 2-rainbow independent domination number, and show if G is a graph of order n ≥ 3, then 5 ≤ γ ri2 (G) + γ ri2 (G) ≤ n + 3, with both bounds being sharp.

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Domination parameters with number 2: Interrelations and algorithmic consequences

Bonomo

Brešar

Grippo

et al. 2018

Discrete Applied Mathematics

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In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak 2-domination number, γ w2 (G), the 2-domination number, γ 2 (G), the {2}-domination number, γ {2} (G), the double domination number, γ ×2 (G), the total {2}-domination number, γ t{2} (G), and the total double domination number, γ t×2 (G), where G is a graph in which a corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, γ R (G), and two classical parameters, the domination number, γ(G), and the total domination number, γ t (G), we consider 13 domination invariants in graphs. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, a large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain new complexity results regarding the existence of approximation algorithms for the studied invariants, matched with tight or almost tight inapproximability bounds, which hold even in the class of split graphs.

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Rainbow Domination in Graphs

Cited by 16 publications

References 61 publications

(Independent) $k$-Rainbow Domination of a Graph

(Independent) $k$-Rainbow Domination of a Graph

On k-rainbow independent domination in graphs

Domination parameters with number 2: Interrelations and algorithmic consequences

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