Paired-domination in graphs

Haynes, Teresa W.; Slater, Peter J.

doi:10.1002/(sici)1097-0037(199810)32:3<199::aid-net4>3.0.co;2-f

Cited by 398 publications

(134 citation statements)

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“…The literature on this subject has been surveyed and detailed in the two books by Haynes et al [5,6]. Paired-domination in graphs was introduced by Haynes and Slater [7] as a model for assigning backups to guard for security purposes. This concept of domination is well studied (see [1][2][3][4][8][9][10]).…”

Section: Introductionmentioning

confidence: 89%

“…In [7] we have the following notation and statements. Let F be the collection of graphs C 3 ,C 5 and the subdivided stars…”

Section: Graphs With Equal γ P and γ Pmentioning

confidence: 99%

“…Since γ p (P n ) = γ p (C n ) = 2 n/4 (see [7]), by Proposition 3.1 and by Proposition 3.2 one can obtain the following statements. Proposition 3.3. γ p (P n ) = Γ p (P n ) if and only if n = 2, 3, 4, 5, 6, 7 or 9.…”

mentioning

confidence: 98%

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The paired-domination and the upper paired-domination numbers of graphs

Ulatowski¹

2015

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Abstract. In this paper we continue the study of paired-domination in graphs. A paired-dominating set, abbreviated PDS, of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired--domination number of G, denoted by γp(G), is the minimum cardinality of a PDS of G. The upper paired-domination number of G, denoted by Γp(G), is the maximum cardinality of a minimal PDS of G. Let G be a connected graph of order n ≥ 3. Haynes and Slater in [Paired-domination in graphs, Networks 32 (1998), [199][200][201][202][203][204][205][206], showed that γp(G) ≤ n − 1 and they determine the extremal graphs G achieving this bound. In this paper we obtain analogous results for Γp(G). Dorbec, Henning and McCoy in [Upper total domination versus upper paired-domination, Questiones Mathematicae 30 (2007), 1-12] determine Γp(Pn), instead in this paper we determine Γp(Cn). Moreover, we describe some families of graphs G for which the equality γp(G) = Γp(G) holds.

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Section: Introductionmentioning

confidence: 89%

“…In [7] we have the following notation and statements. Let F be the collection of graphs C 3 ,C 5 and the subdivided stars…”

Section: Graphs With Equal γ P and γ Pmentioning

confidence: 99%

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The paired-domination and the upper paired-domination numbers of graphs

Ulatowski¹

2015

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“…Note that strong equality between two parameters was considered first by Haynes and Slater [6]. Later Haynes, Henning and Slater gave in [4] and [5] constructive characterizations of trees with strong equality between some domination parameters.…”

Section: Introductionmentioning

confidence: 99%

A note on the independent Roman domination in unicyclic graphs

Chellali¹,

Rad²

2012

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“…{|S 1 | + |S 2 | : S 1 , S 2 are 2DN-set of G}, [7] & [9]. Further, a neighborhood set S ⊆ V is called an independent neighborhood set, if S is an independent and neighborhood set of G, [8] /paired neighborhood set, if S contains at least one perfect matching of G, [6] & [11] /maximal neighborhood set, if V −S does not contain a neighborhood set of G, [12]/inverse neighborhood set, if V − S contain a neighborhood set of G, [2]/ dual neighborhood set, if union of minimum 2DN-set of G, [3]. The minimum cardinality taken over all independent/maximal / inverse/dual neighborhood set in G is called an independent/paired /maximal / inverse/dual neighborhood number of G and is denoted by (ii) If G has no triangles, then η(G) = α 0 (G).…”

Section: Introductionmentioning

confidence: 99%