Total Domination on Some Graph Operators

Abstract

Let G=(V,E) be a graph; a set D⊆V is a total dominating set if every vertex v∈V has, at least, one neighbor in D. The total domination number γt(G) is the minimum cardinality among all total dominating sets. Given an arbitrary graph G, we consider some operators on this graph; S(G),R(G), and Q(G), and we give bounds or the exact value of the total domination number of these new graphs using some parameters in the original graph G.

The Perfect Roman Domination Number of the Cartesian Product of Some Graphs

“…Roman domination has been studied well and there are many research papers on this subject, such as [4,5]. ere are some variations on domination number and Roman domination number have been appeared in the literature such as total, week, and perfect [6][7][8][9][10]. In this paper, we continue the investigation of perfect Roman domination.…”

Total Domination on Tree Operators

“…Next, we give the exact value for the total domination number of S(T ). For that, we will need the following result, which was proved in [12].…”

“…Many large graphs can be obtained by applying graph operators on smaller ones, thus some of their properties are strongly related. Motivated by the above works and by [12,13], where the total domination number is studied for some graph operators acting on general graphs, we study here the total domination number of several graph operators acting on trees. Other parameters have also been studied in this type of graphs (see [1,2,9]).…”

On the Quasi-Total Roman Domination Number of Graphs

“…, and only if, x ∈ D. The (total) domination number of G, denoted by (γ t (G)) γ(G), is the minimum cardinality among all (total) dominating sets of G. For more information on domination and total domination see the books [1,2,7], the survey [8] and the recent works [9][10][11].…”