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

Cayley graph is a graph constructed out of a group Γ and a generating set A ⊆ Γ. When Γ = Zn, the corresponding Cayley graph is called as a circulant graph and denoted by Cir(n, A). In this paper, we attempt to find the total domination number of Cir(n, A) for a particular generating set A of Zn and a minimum total dominating set of Cir(n, A). Further, it is proved that Cir(n, A) is 2-total exc...

LetG = (V,E) be a finite, simple, undirected graph. A set S ⊆ V is called a total dominating set if every vertex of V is adjacent to some vertex of S. Interest in total domination began when the concept was introduced by Cockayne, Dawes, and Hedetniemi [6] in 1980. In 1998, two books on the subject appeared ([11] and [12]), followed by a survey of more recent results in 2009 [15]. The total dom...

We generalize the concept of efficient total domination from graphs to digraphs. An efficiently total dominating set X of a digraph D is a vertex subset such that every vertex of D has exactly one predecessor in X . We study graphs that permit an orientation having such a set and give complexity results and characterizations concerning this question. Furthermore, we study the computational comp...