Edge domination and total edge domination in the join of graphs (original) (raw)

On k-Cost Effective Domination Number in the Join of Graphs

European Journal of Pure and Applied Mathematics, 2021

In this paper, we characterize the k-cost effective domination in the join of graphs.Further, we investigate the k-cost effective domination, cost effective domination index, maximal cost effective domination in the join of graphs.

Some Properties of Whole Edge Domination in Graphs

Applied Mathematics & Information Sciences, 2022

Throughout this paper we initiate a new definition of edge domination in graphs that is called whole edge domination. Some bounds and results for edge whole domination number are established. Moreover, calculate this number of some certain graphs. Also, the effect of deletion, addition, and contraction of an edge on the domination of this definition has been determined.

Independent Semitotal Domination in the Join of Graphs

Asian Research Journal of Mathematics

A subset W \(\subseteq\) V (G) of a graph G is an independent semitotal dominating set of G, abbreviated ISTd-set of G, if W is an independent dominating set of G and every element of W is exactly of distance 2 from at least one other element of W. The independent semitotal domination number of G, denoted by \(\gamma\)it2(G), is the minimum cardinality of an ISTd-set of G. In this paper, we study the concept of independent semitotal domination in graphs and investigate the conditions for graphs on which the ISTd-sets exist. Further, the ISTd-sets of the join of any two graphs are examined. Consequently, the corresponding independent semitotal domination number of these graphs are obtained.

Some results on matching and total domination in graphs

2010

The total domination number γt(G) and the matching number α (G) of G are the cardinalities of the minimum total dominating set and the maximum matching of G, respectively. In this paper, we introduce an upper bound of the difference between γt(G) and α (G). We also characterize every tree T with γt(T) ≤ α (T), and give a family of graphs with γt(G) ≤ α (G).

Further Results on Maximal Domination in Graphs

Journal of Ultra Scientist of Physical Sciences Section A

In11, Kulli and Janakiram initiate the concept of maximal domination in graphs. In this paper, we obtained some bounds and characterizations. Also, we estimate the value of the maximal domination number of some graph products such as join of graphs, corona product, cartesian product and strong product.

On Connected Partial Domination in Graphs

European Journal of Pure and Applied Mathematics, 2021

This paper introduces and investigates a variant of partial domination called the connected α-partial domination. For any graph G = (V (G), E(G)) and α ∈ (0, 1], a set S ⊆ V (G) is an α-partial dominating set in G if |N[S]| ≥ α |V (G)|. An α-partial dominating set S ⊆ V (G) is a connected α-partial dominating set in G if ⟨S⟩, the subgraph induced by S, is connected. The connected α-partial domination number of G, denoted by ∂Cα(G), is the smallest cardinality of a connected α-partial dominating set in G. In this paper, we characterize the connected α-partial dominating sets in the join and lexicographic product of graphs for any α ∈ (0, 1] and determine the corresponding connected α-partial domination numbers of graphs resulting from the said binary operations. Moreover, we establish sharp bounds for the connected α-partial domination numbers of the corona and Cartesian product of graphs. Furthermore, we determine ∂Cα(G) of some special graphs when α...

Double total domination of graphs

Discrete Mathematics, 1997

We are interested in a notion of domination related to both vertices and edges of graphs. We present results about Q(G), the total domination number of a graph G and study ad(G), which is the minimum value of IX*(S) where S is any total dominating set of G. In particular, we prove relations between these two parameters and give lower and upper bounds for Q(G).

Locating Total Dominating Sets in the Join, Corona and Composition of Graphs

2014

which is also a total dominating set is called a locating total dominating set of G. The minimum cardinality of a locating total dominating set of G is called the locating total domination number of G. In this paper, we determine the locating total domination numbers of the join, corona and composition of graphs. 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.

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.