A note on double domination in graphs

Martínez, Abel Cabrera; Rodríguez-Velázquez, Juan A.

doi:10.1016/j.dam.2021.05.011

Cited by 9 publications

(2 citation statements)

References 6 publications

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“…The corollary above for the case k = 2 was given in [4], and the authors showed that the bound is achieved for the graph given in Figure 1.…”

Section: Relationships With Other Domination Parametersmentioning

confidence: 98%

Some new results on the k-tuple domination number of graphs

Martínez

2022

RAIRO-Oper. Res.

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Let $k\geq 1$ be an integer and $G$ be a graph of minimum degree $\delta(G)\geq k-1$. A set $D\subseteq V(G)$ is said to be a $k$-tuple dominating set of $G$ if $|N[v]\cap D|\geq k$ for every vertex $v\in V(G)$, where $N[v]$ represents the closed neighbourhood of vertex~$v$. The minimum cardinality among all $k$-tuple dominating sets is the $k$-tuple domination number of $G$. In this paper, we continue with the study of this classical domination parameter in graphs. In particular, we provide some relationships that exist between the $k$-tuple domination number and other classical parameters, like the multiple domination parameters, the independence number, the diameter, the order and the maximum degree. Also, we show some classes of graphs for which these relationships are achieved.

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“…The corollary above for the case k = 2 was given in [4], and the authors showed that the bound is achieved for the graph given in Figure 1.…”

Section: Relationships With Other Domination Parametersmentioning

confidence: 98%

Some new results on the k-tuple domination number of graphs

Martínez

2022

RAIRO-Oper. Res.

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“…A double dominating set of cardinality γ ×2 (G) is referred to as a γ ×2 (G)-set. The concept of double domination in graph was further studied in, for example, [1,3,7,13]. Blidia et al [1] showed that γ ×2 (G) ≤ 11n 13 if G is a graph of order n with δ(G) ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

A note on the double domination number in maximal outerplanar and planar graphs

Aziz

Rad²,

Kamarulhaili³

2022

RAIRO-Oper. Res.

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In a graph, a vertex dominates itself and its neighbors. A subset S of vertices of a graph G is a double dominating set of G if S dominates every vertex of G at least twice. The double domination number γ×2(G) of G is the minimum cardinality of a double dominating set of G. In this paper, we prove that the double domination number of a maximal outerplanar graph G of order n is bounded above by, where k is the number of pairs of consecutive vertices of degree two and with distance at least 3 on the outer cycle. We also prove that for a Hamiltonian maximal planar graph G of order n≥ 7.

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