For a given connected graphG= (V ,E), a setDtr ⊆ V (G) is a total restrained dominating set if it is dominating and both 〈Dtr〉 and 〈V (G)−Dtr〉 do not contain isolate vertices. The cardinality of the minimum total restrained dominating set in G is the total restrained domination number and is denoted by tr(G). In this paper we characterize the trees with equal total and total restrained dominati...

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 S ⊆ V (G) is a restrained strong resolving hop dominating in G if for every v ∈ (G)\S, there exists w such that dG(v, w) = 2 and or (G)\S has no isolated vertex. The smallest cardinality of set, denoted by γrsRh(G), called the domination number G. In this paper, we obtained corresponding parameter graphs resulting from join, corona lexicographic product two graphs. Specifically, character...


 Let G be a connected graph. A set S ⊆ V (G) is restrained 2-resolving hop dominating of if and = or ⟨V (G)\S⟩ has no isolated vertex. The domination number G, denoted by γr2Rh(G) the smallest cardinality G. This study aims to combine concept with sets graphs. main results generated in this include characterization join, corona, edge corona lexicographic product graphs, as well their corr...

Let G = (V,E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex in V is adjacent to a vertex in S and every vertex of V −S is adjacent to a vertex in V −S. The total restrained domination number of G, denoted by γtr(G), is the minimum cardinality of a total restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We sho...

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 smallest cardinality of a restrained dominating set of G. It is known that if T is a tree of order n, then γr(T ) ≥ d(n+2)/3e. In this note we provide a simple constructive characteriz...

Given graph G = (V,E), a dominating set S is a subset of vertex set V such that any vertex not in S is adjacent to at least one vertex in S. The domination number of a graph G is the minimum size of the dominating sets of G. In this paper we study some results on domination number, connected, independent, total and restrained domination number denoted by γ(G), γc(G) ,γi(G), γt(G) and γr(G) resp...