Characterizations of trees with equal domination parameters

Hattingh, Johannes H.; Henning, Michael A.

doi:10.1002/1097-0118(200006)34:2<142::aid-jgt3>3.0.co;2-v

Cited by 15 publications

(8 citation statements)

References 7 publications

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“…• or there is some triple (T, X, Y ) in T and some i ∈ [5] such that (T + , X + , Y + ) arises by applying Operation i to (T, X, Y ).…”

Section: Lemma 4 Let T Be a Tree And Let X And Y Be Sets Of Verticesmentioning

confidence: 99%

“…A typical solution for the problem posed by Chellali et al [2] would be a so-called constructive characterization, that is, a recursive constructive description of the set of all extremal trees for (1). There are many examples of such characterizations in the literature [1,4,5,8]. Usually, they involve some few small extremal trees together with a small set of simple extension operations that are applied recursively in order to create all larger extremal trees.…”

Section: Introductionmentioning

confidence: 99%

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Strong equality of Roman and weak Roman domination in trees

Alvarado

Dantas

Rautenbach

2016

Discrete Applied Mathematics

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“…• or there is some triple (T, X, Y ) in T and some i ∈ [5] such that (T + , X + , Y + ) arises by applying Operation i to (T, X, Y ).…”

Section: Lemma 4 Let T Be a Tree And Let X And Y Be Sets Of Verticesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Strong equality of Roman and weak Roman domination in trees

Alvarado

Dantas

Rautenbach

2016

Discrete Applied Mathematics

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“…Trees having unique paired dominating sets were characterized in [2]. Graphs having a unique set for other domination parameters have also been much studied, including [1,3,4,7,9,10,18,24,25]. It is worth mentioning the related topic of which vertices appear in all or in no minimum dominating sets.…”

Section: Introductionmentioning

confidence: 99%

Unique minimum semipaired dominating sets in trees

Haynes¹,

Henning²

2020

Discuss. Math. Graph Theory

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Let G be a graph with vertex set V. A subset S ⊆ V is a semipaired dominating set of G if every vertex in V \ S is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number is the minimum cardinality of a semipaired dominating set of G. We characterize the trees having a unique minimum semipaired dominating set. We also determine an upper bound on the semipaired domination number of these trees and characterize the trees attaining this bound.

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“…We denote the domination, total domination, and restrained domination numbers of G by (G), t (G), and r (G), respectively. Total domination in graphs was introduced by Cockayne, Dawes, and Hedetniemi [3] and further studied, for example, in [1,8,9,16], while restrained domination was introduced by Telle and Proskurowski [18], albeit indirectly, as a vertex partitioning problem and further studied, for example, in [5][6][7]11,15]. The concept of total restrained domination in graphs was also introduced in [18], albeit indirectly, as a vertex partitioning problem and has been studied, for example, in [12,17,19].…”

Section: Introductionmentioning

confidence: 99%

Total restrained domination in graphs with minimum degree two

Henning

Maritz

2008

Discrete Mathematics

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In this paper, we continue the study of total restrained domination in graphs, a concept introduced by Telle and Proskurowksi (Algorithms for vertex partitioning problems on partial k-trees, SIAM J. Discrete Math. 10 (1997) 529-550) as a vertex partitioning problem. A set S of vertices in a graph G = (V , E) is a total restrained dominating set of G if every vertex is adjacent to a vertex in S and every vertex of V \S is adjacent to a vertex in V \S. The minimum cardinality of a total restrained dominating set of G is the total restrained domination number of G, denoted by tr (G). Let G be a connected graph of order n with minimum degree at least 2 and with maximum degree where n − 2. We prove that if n 4, then tr (G) n − 2 − 1 and this bound is sharp. If we restrict G to a bipartite graph with 3, then we improve this bound by showing that tr (G) n − 2 3 − 2 9 √ 3 − 8 − 7 9 and that this bound is sharp.

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Characterizations of trees with equal domination parameters

Cited by 15 publications

References 7 publications

Strong equality of Roman and weak Roman domination in trees

Strong equality of Roman and weak Roman domination in trees

Unique minimum semipaired dominating sets in trees

Total restrained domination in graphs with minimum degree two

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