Suppose G = (V , E) is a simple graph and k is a fixed positive integer. A subset D ⊆ V is a distance k-dominating set of G if for every u ∈ V , there exists a vertex v ∈ D such that dG(u, v) ≤ k, where dG(u, v) is the distance between u and v in G. A set D ⊆ V is a distance k-paired-dominating set of G if D is a distance k-dominating set and the induced subgraph G[D] contains a perfect matchin...

A set S is a 1-movable strong resolving hop dominating of G if for every v ∈ S, either S\{v} or there exists vertex u (V (G)\S)∩NG(v) such that (S \ {v}) ∩ {u} G. The minimum cardinality denoted by γ 1 msRh(G). In this paper, we obtained the corresponding parameter in graphs resulting from join, corona and lexicographic product two graphs. Specifically, characterize sets these types determine b...

Abstract Given a graph G and subset of vertices $$D\subseteq V(G)$$ D ⊆ V ( G ) , the external neighbourhood D is defined as $$N_e(D)=\{u\in V(G){\setminus } D:\, N(u)\cap D\ne \varnothing \}$$ <mml:msu...

Several of the best known problems and conjectures in graph theory arise in studying the behavior of a graphical invariant on a graph product. Examples of this are Vizing’s conjecture, Hedetniemi’s conjecture and the calculation of the Shannon capacity of graphs, where the invariants are the domination number, the chromatic number and the independence number on the Cartesian, categorical and st...

Let G be a connected graph. A subset S ⊆ V (G) is strong resolving dominating set of if and for every pair vertices u, v ∈ (G), there exists vertex w such that u IG[v, w] or IG[u, w]. The smallest cardinality called the domination number G. In this paper, we characterize sets in lexicographic product graphs determine corresponding number.