The Technologies of Total Domination.

Total domination and total domination subdivision numbers

DOMINATION NUMBER OF TOTAL GRAPH OF MODULE

 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...

Total domination versus paired domination

A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by γt. The maximal size of an inclusionwise minimal total dominating set, the upper total domination num...

Trees with equal total domination and game total domination numbers

In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453–1462], where the players Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a t...

Trees with Equal Total Domination and Total Restrained Domination Numbers

For a graph G = (V, E), a set S ⊆ V (G) is a total dominating set if it is dominating and both 〈S〉 has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number. A set S ⊆ V (G) is a total restrained dominating set if it is total dominating and 〈V (G) − S〉 has no isolated vertices. The cardinality of a minimum total restrained dominating set in ...