Remarks about disjoint dominating sets

Henning, Michael A.; Löwenstein, Christian; Rautenbach, Dieter

doi:10.1016/j.disc.2009.06.017

Cited by 31 publications

(26 citation statements)

References 7 publications

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“…Recently, several authors studied the cardinalities of pairs of disjoint dominating sets in graph [2,7,8,11,12]. The context of this research motivates the question for which graphs Theorem 1 is best-possible in the sense that the union D ∪ T of the two sets necessarily contains all vertices of the graph G. Our following main result gives a partial answer to this question.…”

Section: Introductionmentioning

confidence: 77%

Partitioning a graph into a dominating set, a total dominating set, and something else

Henning

Löwenstein²,

Rautenbach³

2010

Discuss. Math. Graph Theory

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Section: Introductionmentioning

confidence: 77%

Partitioning a graph into a dominating set, a total dominating set, and something else

Henning

Löwenstein²,

Rautenbach³

2010

Discuss. Math. Graph Theory

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“…그래프   ∈  ∈에서 에 존재하 지 않는 모든 정점  ╲이 에 속한 정점들에 적어도 하나 의 간선으로 연결 (인접, adjacent)되어 있는 의 부분집합  ⊆ 를 지배집합 (dominating set, DS)이라 한다 [1][2][3]. [1,[3][4][5].…”

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Maximum Degree Vertex Domatic Set Algorithm for Domatic Number Problem

Lee¹

2015

Journal of the Korea Society of Computer and Information

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In the absence of a polynomial time algorithm capable of obtaining the exact solutions to it, the domatic number problem (DNP) of dominating set (DS) has been regarded as NP-complete. This paper suggests polynomial-time complexity algorithm about DNP. In this paper, I select a vertex   of the maximum degree    as an element of a dominating set        ⋯   , compute     from a simplified graph of        ╲   , and verify that   is indeed a dominating set through  ╲         . When applied to 15 various graphs, the proposed algorithm has succeeded in bringing about exact solutions with polynomial-time complexity    . Therefore, the proposed domatic number algorithm shows that the domatic number problem is in fact a P-problem.

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“…서 론 그래프  ∈  ∈에서간선  가 존재하면 정점 는 정점 를 지배한다 (dominate)고 한다. 에 존재하지 않는 모든 정점  ╲이 에 속한 정점들에 적어도 하나의 간선으로 연결되어 있는 의 부분집합  ⊆ 를 지배집합 (dominating set, DS)이라 한다 [1][2][3].…”

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A Dominating Set Algorithm

Lee¹

2013

Journal of the Korea Society of Computer and Information

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This paper proposes a linear-time algorithm that has been designed to obtain an accurate solution for Dominating Set (DS) problem, which is known to be NP-complete due to the deficiency of polynomial-time algorithms that successfully derive an accurate solution to it. The proposed algorithm does so by repeatedly assigning vertex  with maximum degree among vertices adjacent to the vertex  with minimum degree  to Minimum Independent DS (MIDS) as its element and removing all the incident edges until no edges remain in the graph. This algorithm finally transforms MIDS into Minimum DS (MDS) and again into Minimum Connected DS (MCDS) so as to obtain the accurate solution to all DS-related problems. When applied to ten different graphs, it has successfully obtained accurate solutions with linear time complexity . It has therefore proven that Dominating Set problem is rather a P-problem.▸Keywords : Dominating Set, Connected DS, Independent DS, Degree•제1저자 : 이상운

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Remarks about disjoint dominating sets

Abstract: We solve a number of problems posed by Hedetniemi, Hedetniemi, Laskar, Markus, and Slater concerning pairs of disjoint sets in graphs which are dominating or independent and dominating.

Cited by 31 publications

References 7 publications

Partitioning a graph into a dominating set, a total dominating set, and something else

Partitioning a graph into a dominating set, a total dominating set, and something else

Maximum Degree Vertex Domatic Set Algorithm for Domatic Number Problem

A Dominating Set Algorithm

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