Multiple Domination

Bull. Malays. Math. Sci. Soc.

Hernández-Mira

2021

Let G be a graph of minimum degree at least two. A set $$D\subseteq V(G)$$ D ⊆ V ( G ) is said to be a double total dominating set of G if $$|N(v)\cap D|\ge 2$$ | N ( v ) ∩ D | ≥ 2 for every vertex $$v\in V(G)$$ v ∈ V ( G ) . The minimum cardinality among all double total dominating sets of G is the double total domination number of G. In this article, we continue with the study of this parameter. In particular, we provide new bounds on the double total domination number in terms of other domination parameters. Some of our results are tight bounds that improve some well-known results.

Section: Additional Definitions and Previous Resultsmentioning

confidence: 99%

“…It is well known that γ ×2,t (G) ≥ (4 n(G)−2 m(G))/2 for any graph G of minimum degree δ(G) ≥ 2 (see [11]). The next result improves this previous lower bound whenever δ(G) ≥ 3.…”

Section: Theorem 8 For Any Graph G Withmentioning

confidence: 99%

New Bounds on the Double Total Domination Number of Graphs

Bull. Malays. Math. Sci. Soc.

Hernández-Mira

2021

“…Recently, Hansberg and Volkmann [7] put into context all relevant research results on multiple domination that have been found up to 2020. In that chapter, they posed the following open problem.…”

Section: New Bounds On the K-tuple Domination Numbermentioning

confidence: 99%

A note on the k-tuple domination number of graphs

2022

Ars Math. Contemp.

In a graph G, a vertex dominates itself and its neighbours. A set D ⊆ V (G) is said to be a k-tuple dominating set of G if D dominates every vertex of G at least k times. The minimum cardinality among all k-tuple dominating sets is the k-tuple domination number of G. In this note, we provide new bounds on this parameter. Some of these bounds generalize other ones that have been given for the case k = 2.

“…Relationships between different parameters corresponding to multiple domination have attracted the attention of several researches in the last few decades, and a high number of significant contributions are nowadays well known. Recently, Hansberg and Volkmann [9] put into context all relevant relationships concerning k-tuple domination (with emphasis in the case k = 2) that have been found up to 2020. Subsequently, Cabrera-Martínez [2,3] obtained new results in this direction.…”

Section: Relationships With Other Domination Parametersmentioning

confidence: 99%

“…Observe that the 1-tuple dominating set of G is the same as the dominating set of G. The k-tuple domination number of G, denoted by γ ×k (G), is the minimum cardinality among all k-tuple dominating sets of G. For a comprehensive survey on k-domination and k-tuple domination in graphs, we suggest the chapter [9] due to Hansberg and Volkmann. In addition, some recent results on these parameters can be found in [1,2,5,13,16].…”

Section: Introductionmentioning

confidence: 99%

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

2022

RAIRO-Oper. Res.

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.