Exploring the vertex and edge corona of graphs for their weakly connected 2-domination (original) (raw)

Weakly connected 2-domination in graphs

Applied Mathematical Sciences, 2021

Let G = (V (G), E(G)) be a connected graph. A set D ⊆ V (G) is a weakly connected 2-dominating set if every vertex of V (G)\D is adjacent to at least two vertices in D and the subgraph 〈D〉w weakly induced by D is connected. The weakly connected 2-domination number of G, denoted by γ2w(G), is the smallest cardinality of a weakly connected 2-dominating set of G. In this paper, we study weakly connected 2-domination in graphs and obtain some results. Mathematics Subject Classification: 05C69

On 2-Movable Domination in the Join and Corona of Graphs

Advances and Applications in Discrete Mathematics, 2024

Let GGG be a connected graph. A non-empty SsubseteqV(G)S\subseteq V(G)SsubseteqV(G) is a 222-movable dominating set of GGG if SSS is a dominating set and for every pair x,yinSx,y \in Sx,yinS, Sbackslashx,yS \backslash \{x, y\}Sbackslashx,y is a dominating set in GGG, or there exist u,vinV(G)backslashSu, v \in V(G) \backslash Su,vinV(G)backslashS such that uuu and vvv are adjacent to xxx and yyy, respectively, and (Sbackslashx,y)cupu,v(S \backslash \{x,y\}) \cup \{u,v\}(Sbackslashx,y)cupu,v is a dominating set in GGG. The 222-movable domination number of GGG, denoted by gammam2(G)\gamma_{m}^{2}(G)gammam2(G), is the minimum cardinality of a 2-movable dominating set of GGG. A 2-movable dominating set with cardinality equal to gammam2(G)\gamma_{m}^{2}(G)gammam2(G) is called gammam2\gamma_{m}^{2}gammam2-set of GGG. This paper present the 2-movable domination number in the corona and join of graphs.

Weakly connected 2-domination in the join of graphs

2021

A weakly connected 2-dominating set of a connected graph G is a set D ⊆ V (G) such that every vertex in V (G)\D is adjacent to at least two vertices in D and the subgraph, 〈D〉w, weakly induced by D is connected. In this paper, the weakly connected 2-dominating sets in the join G + H and K1 + H of graphs are characterized and their corresponding weakly connected 2-domination numbers are obtained. The necessary and sufficient conditions for the join of graphs to have weakly connected 2-domination numbers equal to 2,3 and 4 are provided. Mathematics Subject Classification: 05C69

Restrained double domination in the join and corona of graphs

International Journal of Mathematical Analysis, 2014

A double dominating set of G is a restrained double dominating set of G if for each x ∈ V (G)\S, there exists y ∈ V (G)\S such that xy ∈ E(G). In this paper, we characterized the restrained double dominating sets in the join, and corona of two graphs. We also determine sharp bounds for the restrained double domination numbers of these graphs. In particular, we show that if G and H are any graphs without isolated vertices of orders n and m, respectively, then γ r×2 (G • H) = min{n(γ r (H) + 1), nγ ×2 (H)}, where γ r , γ ×2 , and γ r×2 are, respectively, the restrained domination, double domination, and restrained double domination parameters.

On Restrained 2-Domination Number of the Join and Corona of Graphs

IOSR Journal of Mathematics, 2014

Let G be a graph. A subset S of) (G V is a restrained 2-dominating set if every vertex of r . We also characterize restrained 2-dominating sets in the corona of two graphs and give a formula for computing the restrained 2-domination number.

Restrained Weakly Connected 2-Domination in the Join of Graphs

Communications in Mathematics and Applications

Let G = (V (G), E(G)) be a connected graph. A restrained weakly connected 2-dominating (RWC2D) set in G is a subset D ⊆ V (G) such that every vertex in V (G)\D is dominated by at least two vertices in D and is adjacent to a vertex in V (G)\D, and that the subgraph 〈D〉 w weakly induced by D is connected. The restrained weakly connected 2-domination number of G, denoted by γ r2w (G), is the smallest cardinality of a restrained weakly connected 2-dominating set in G. In this paper, we characterize the RWC2D sets in the join of two graphs G and H, each of which is of order at least 3 and has no isolated vertex, and in the join K 1 ∨ F , where K 1 is the trivial graph and that at least one component of F is of order at least 3. In particular, it is shown that 2 ≤ γ r2w (G ∨ H) ≤ 4 and γ r2w (K 1 ∨ F) = min{1 + γ r (F), γ 2 (F)}, where γ r and γ 2 are the restrained domination and 2-domination parameters, respectively.

Restrained independent 2-domination in the join and corona of graphs

Applied Mathematical Sciences, 2017

A restrained independent 2-dominating set of a graph G is a set S of vertices of G such that every vertex not in S is dominated at least twice and adjacent to at least one vertex not in S, and every pair of vertices in S are not adjacent. In this paper, we characterized the restrained independent 2-dominating sets of the join and corona of graphs and calculate their restrained independent 2-domination numbers.