A total dominating set of a graph G = (V,E) with no isolated vertex is a set D ⊆ V (G) such that every vertex is adjacent to a vertex in D. A total dominating set D of G is a locating-total dominating set if for every pair of distinct vertices u and v in V −D, N(u) ∩D = N(v) ∩D. Let γ L(G) be the minimum cardinality of a locating-total dominating set of G. We show that for a nontrivial tree T o...

A secure (total) dominating set of a graph G = (V, E) is a (total) dominating set X ⊆ V with the property that for each u ∈ V − X , there exists x ∈ X adjacent to u such that (X − {x}) ∪ {u} is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total) domination number γs(G) (γst(G)). We characterize graphs with equal total and secure total domination...

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...

In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam [S. Arumugam, C. Sivagnanam, Neighborhood total domination in graphs, Opuscula Math. 31 (2011) 519–531]. A neighborhood total dominating set, abbreviated NTD-set, in a graph G is a dominating set S in G with the property that the subgraph induced by the open neighborhood of t...

Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V \ S. The restrained domination number of G, denoted by γr(G), is the minimum cardinality of a restrained dominating set of G. A set S ⊆ V is a total dominating set if every vertex in V is adjacent to a vertex in S. The total domination number of a graph...

A set D of vertices in a graph G = (V,E) is a total dominating set if every vertex of G is adjacent to some vertex in D. A total dominating set D of G is said to be weak if every vertex v ∈ V −D is adjacent to a vertex u ∈ D such that dG(v) ≥ dG(u). The weak total domination number γwt(G) of G is the minimum cardinality of a weak total dominating set of G. A total dominating set D of G is said ...

For a positive integer k, a total {k}-dominating function of a graph G without isolated vertices is a function f from the vertex set V (G) to the set {0, 1, 2, . . . , k} such that for any vertex v ∈ V (G), the condition ∑ u∈N(v) f(u) ≥ k is fulfilled, where N(v) is the open neighborhood of v. The weight of a total {k}-dominating function f is the value ω(f) = ∑ v∈V f(v). The total {k}-dominati...

A total Roman dominating function on a graph $G$ is a function $f: V(G) rightarrow {0,1,2}$ such that for every vertex $vin V(G)$ with $f(v)=0$ there exists a vertex $uin V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set ${xin V(G): f(x)geq 1}$ has no isolated vertices. The total Roman domination number of $G$, denoted $gamma_{tR}(G)$, is the minimum weight $omega(f)=sum_...

A subset S of vertices in a graph G is a global total dominating set, or just GTDS, if S is a total dominating set of both G and G. The global total domination number γgt(G) of G is the minimum cardinality of a GTDS of G. We present bounds for the global total domination number in graphs.