A subset of vertices in a graph G is total dominating set if every vertex adjacent to at least one this subset. The domination number the minimum cardinality any and denoted by ?t(G). having nonempty intersection with all independent sets maximum an transversal set. ?tt(G). Based on fact that for tree T, ?t(T) ? ?tt(T) + 1, work we give several relationships between trees T which are leading cl...

A total 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. The total domination number of a graph G, denoted by γt(G), is the minimum cardinality of a total dominating set of G. Chellali and Haynes [Total and paired-domination numbers of a tree, AKCE International Journal of Graphs and Combinatorics 1 (2004), 69– 75] established the followin...

A graph G with no isolated vertex is total domination bicritical if the removal of any pair of vertices, whose removal does not produce an isolated vertex, decreases the total domination number. In this paper we study properties of total domination bicritical graphs, and give several characterizations.

A set M of edges of a graph G is a matching if no two edges in M are incident to the same vertex. A set S of vertices in G is a total dominating set ofG if every vertex of G is adjacent to some vertex in S. The matching number is the maximum cardinality of a matching of G, while the total domination number of G is the minimum cardinality of a total dominating set of G. In this paper, we investi...

Let $G$ be a simple graph with vertex set $V$. A double Roman dominating function (DRDF) on $G$ is a function $f:Vrightarrow{0,1,2,3}$ satisfying that if $f(v)=0$, then the vertex $v$ must be adjacent to at least two vertices assigned $2$ or one vertex assigned $3$ under $f$, whereas if $f(v)=1$, then the vertex $v$ must be adjacent to at least one vertex assigned $2$ or $3$. The weight of a DR...

Let G = (V,E) be a graph without isolated vertices. A set S ⊆ V is a total dominating set if every vertex in V is adjacent to at least one vertex in S. A total dominating set S ⊆ V is a paired-dominating set if the induced subgraph G[S] has at least one perfect matching. The paired-domination number γpr(G) is the minimum cardinality of a paired-domination set of G. In this paper, we provide a c...