Construction of trees and graphs with equal domination parameters

Dorfling, Michael; Goddard, Wayne; Henning, Michael A.; Mynhardt, C. M.

doi:10.1016/j.disc.2006.04.031

Cited by 51 publications

(18 citation statements)

References 12 publications

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“…Note that Corollary 3 can only be close to the truth if the domination number is close to 1 4 n. Our main result shows that in this case, the total domination number is smaller than guaranteed by (4). Specifically, we prove the following result.…”

Section: Introductionmentioning

confidence: 53%

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Domination and total domination in cubic graphs of large girth

Dantas

Joos

Löwenstein

et al. 2014

Discrete Applied Mathematics

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The domination number γ(G) and the total domination number γ t (G) of a graph G without an isolated vertex are among the most well studied parameters in graph theory. While the inequality γ t (G) ≤ 2γ(G) is an almost immediate consequence of the definition, the extremal graphs for this inequality are not well understood. Furthermore, even very strong additional assumptions do not allow to improve the inequality by much.In the present paper we consider the relation of γ(G) and γ t (G) for cubic graphs G of large girth. Clearly, in this case γ(G) is at least n(G)/4 where n(G) is the order of G. If γ(G) is close to n(G)/4, then this forces a certain structure within G. We exploit this structure and prove an upper bound on γ t (G), which depends on the value of γ(G). As a consequence, we can considerably improve the inequality γ t (G) ≤ 2γ(G) for cubic graphs of large girth.

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Section: Introductionmentioning

confidence: 53%

“…In [4,7] the trees that satisfy (1) or (2) with equality are characterized constructively. While numerous very deep results concerning bounds on the domination number and the total domination number under various conditions have been obtained, the relation of these two parameters is not really well understood.…”

Section: Introductionmentioning

confidence: 99%

Domination and total domination in cubic graphs of large girth

Dantas

Joos

Löwenstein

et al. 2014

Discrete Applied Mathematics

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“…This idea of labeling the vertices is introduced in [3], where trees with equal domination and independent domination numbers as well as trees with equal domination and total domination numbers are characterized.…”

Section: Resultsmentioning

confidence: 99%

“…In order to 2D-dominate the vertex u 1 , we have that v ∈ S T . Thus the set S T \ {u 3 } is a 2DD-set in T ′ , and so either S T \ {u 3 …”

Section: Lemma 8 Let T ∈ T and Let S Be A Labeling Of T So Thatmentioning

confidence: 96%

Domination versus disjunctive domination in trees

Henning

Marcon

2015

Discrete Applied Mathematics

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a b s t r a c tA dominating set in a graph G is a set S of vertices of G such that every vertex not in S is adjacent to a vertex of S. The domination number, γ (G), of G is the minimum cardinality of a dominating set of G. A set S of vertices in G is a disjunctive dominating set in G if every vertex not in S is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it in G. The disjunctive domination number, γ d 2 (G), of G is the minimum cardinality of a disjunctive dominating set in G. It is known that there exist graphs G that belong to the class of bipartite graphs or claw-free graphs or chordal graphs such that γ (G) > C γ d 2 (G) for any given constant C . In this paper, we show that if T is a tree, then γ (T ) ≤ 2γ d 2 (T )−1, and we provide a constructive characterization of the trees achieving equality in this bound.

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“…In [2,5] the trees that satisfy (1) or (2) with equality are characterized constructively. While numerous very deep results concerning bounds on the domination number and the total domination number under various conditions have been obtained, the relation of these two parameters is not really well understood.…”

Section: For a Finite Simple And Undirected Graph G A Set D Of Vermentioning

confidence: 99%

Relating ordinary and total domination in cubic graphs of large girth

Dantas

Joos

Löwenstein

et al. 2013

The Seventh European Conference on Combinatorics, Graph Theory and Applications

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Abstract. For a cubic graph G of order n, girth at least g, and domination number 1 4 + n for some ≥ 0, we show that the total domination number of G is at most 13 32For a finite, simple, andSimilarly, a set T of vertices of G is a total dominating set of G if every vertex in V (G) has a neighbor in T . Note that a graph has a total dominating set exactly if it has no isolated vertex. The minimum cardinalities of a dominating and a total dominating set of G are known as the domination number γ (G) of G and the total domination number γ t (G) of G, respectively. These two parameters are among the most fundamental and well studied parameters in graph theory [3,4,6]. In view of their computational hardness especially upper bounds were investigated in great detail. The two parameters are related by some very simple inequalities. Let G be a graph without isolated vertices. Since every total dominating set of G is also a dominating set of G, we immediately obtainSimilarly, if D is a dominating set of G, then adding, for every isolated vertex u of the subgraph G [D] of G induced by D, a neighbor of u in G

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Construction of trees and graphs with equal domination parameters

Cited by 51 publications

References 12 publications

Domination and total domination in cubic graphs of large girth

Domination and total domination in cubic graphs of large girth

Domination versus disjunctive domination in trees

Relating ordinary and total domination in cubic graphs of large girth

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