A new lower bound on the double domination number of a graph

Hajian, Majid; Rad, Nader Jafari

doi:10.1016/j.dam.2018.06.009

Cited by 15 publications

(5 citation statements)

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“…We also prove that the problem of computing γ ×k for k ≥ 2 is NP-hard for comparability graphs of diameter two. Some bounds on the double domination number in this paper improve the main results in [3] and [12]. An upper bound on the k-tuple domatic number of graphs with characterization of the extremal graphs for the bound is given in this paper.…”

Section: Introductionsupporting

confidence: 60%

Revisiting $k$-tuple dominating sets with emphasis on small values of $k$

Samadi,

Soltankhah,

Mojdeh

2021

Preprint

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For any graph G of order n with degree sequence d 1 ≥ • • • ≥ d n , we define the double Slater number sℓ ×2 (G) as the smallest integer t such that t + d 1 + • • • + d t−e ≥ 2n − p in which e and p are the number of end-vertices and penultimate vertices of G, respectively. We show that γ ×2 (G) ≥ sℓ ×2 (G), where γ ×2 (G) is the well-known double domination number of a graph G with no isolated vertices. We prove that the problem of deciding whether the equality holds for a given graph is NP-complete even when restricted to 4-partite graphs. We also prove that the problem of computing γ ×2 (G) in NP-hard even for comparability graphs of diameter two. Some results concerning these two parameters are given in this paper improving and generalizing some earlier results on double domination in graphs. We give an upper bound on the k-tuple domatic number of graphs with characterization of all graphs attaining the bound. Finally, we characterize the family of all full graphs, leading to a solution to an open problem given in a paper by Cockayne and Hedetniemi (1977).

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Section: Introductionsupporting

confidence: 60%

Revisiting $k$-tuple dominating sets with emphasis on small values of $k$

Samadi,

Soltankhah,

Mojdeh

2021

Preprint

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“…A geodetic dominating set of size γ g (G) is said to be a γ gset. Geodetic number was introduced in [8] and further studied in [2,3,4,6,15,12]. Moreover recently geodetic number was studied in [3,7,10].…”

Section: Introductionmentioning

confidence: 99%

The Geodetic Fault Tolerant Domination Number of a Graph

Stalin¹,

John²

2023

SEAJMMS

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For a connected graph G = (V,E), a set F ⊆ V of vertices in G is called dominating set if every vertex not in F has at least one neighbor in F. A dominating set F ⊆ V is called fault tolerant dominating set if F − {v} is dominating set for every v ∈ F. A fault tolerant dominating set is said to be geodetic fault tolerant dominating set if I[F] = V . The minimum cardinality of a geodetic fault tolerant dominating set is called geodetic fault tolerant domination number and is denoted by γgft(G). The minimum geodetic fault tolerant dominating set is denoted by γgft-set. The geodetic fault tolerant domination number of certain classes of graphs are determined. Some general properties satisfied by this concept are studied. It is shown that for every positive integer 2 < a ≤ b there is a connected graph G such that γ(G) = a, γg(G) = b and γgft(G) = a + b − 2, where γ(G) and γg(G) are the domination number and geodetic domination number of G respectively.

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“…For instance, we cite the following books [19,20,21]. The double domination number, which has been less studied, was introduced in [18] by Harary and Haynes, and studied further in a number of works including [4,10,15,17,23].…”

Section: Introductionmentioning

confidence: 99%

Double domination in lexicographic product graphs

Martínez

García

Rodríguez-Velázquez

2020

Discrete Applied Mathematics

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In a graph G, a vertex dominates itself and its neighbours. A subset S ⊆ V (G) is said to be a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality among all double dominating sets of G is the double domination number. In this article, we obtain tight bounds and closed formulas for the double domination number of lexicographic product graphs G • H in terms of invariants of the factor graphs G and H.

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A new lower bound on the double domination number of a graph

Cited by 15 publications

References 1 publication

Revisiting $k$-tuple dominating sets with emphasis on small values of $k$

Revisiting $k$-tuple dominating sets with emphasis on small values of $k$

The Geodetic Fault Tolerant Domination Number of a Graph

Double domination in lexicographic product graphs

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