A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. We call these graphs γt-critical. In this paper, we determine the domination and the total domination number in the Circulant graphs Cn〈1, 3〉, and then study γ-criticality...

An exact lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: (×i=1Kni) ≥ t + 1, t ≥ 3. Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of tw...

A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set o...

A set S of vertices in a graph G = (V,E) is a total dominating set of G if every vertex of V is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of G. The total domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the tot...

We initiate the study of total outer-independent domination in graphs. A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V (G) \ D is independent. The total outer-independent domination number of a graph G is the minimum cardinality of a total outer-independent dominating set of G. First we discuss the ...

In this paper we introduce the concept of edge domination and total edge domination in intuitionistic fuzzy graphs. We determine the edge domination number and total edge domination number for several classes of intuitionistic fuzzy graphs and obtain bounds for them. We also obtain Nordhaus gaddum type results for the parameters.

Let G = (V, E) be a graph. A set D ⊆ V is a total restrained dominating set of G if every vertex in V has a neighbor in D and every vertex in V −D has a neighbor in V −D. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number of G. In this paper, we define the concept of total restrained domination edge critical graphs, find a lower bound for...

 Let $R$ be a commutative ring and $M$ be an $R$-module with $T(M)$ as subset, the set of torsion elements. The total graph of the module denoted by $T(Gamma(M))$, is the (undirected) graph with all elements of $M$ as vertices, and for distinct elements $n,m in M$, the vertices $n$ and $m$ are adjacent if and only if $n+m in T(M)$. In this paper we study the domination number of $T(Gamma(M))$ a...

Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertic...

The total domination number of a simple, undirected graph G is the minimum cardinality of a subset D of the vertices of G such that each vertex of G is adjacent to some vertex in D. In 2007 Graffiti.pc, a program that makes graph theoretical conjectures, was used to generate conjectures on the total domination number of connected graphs. More recently, the program was used to generate conjectur...