We limit our discussion to graphs that are simple and finite of order . Although 8 we often identify a graph with its set of vertices, in cases where we need to be K explicit we write . A set of vertices of is said to Z ÐKÑ Q K dominate K provided each vertex of is either in or adjacent to a vertex of . K Q Q The domination number of is the minimum order of a dominating set. A K dominating prov...

A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set o...

A total dominating set of a graph G with no isolated vertices is subset S the vertex such that every adjacent to in S. The domination number minimum cardinality set. In this paper, we study middle graphs. Indeed, obtain tight bounds for terms order graph. We also compute some known families graphs explicitly. Moreover, Nordhaus-Gaddum-like relations are presented

For a connected graph G = (V,E) of order at least two, a total restrained monophonic set S of a graph G is a restrained monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total restrained monophonic set of G is the total restrained monophonic number of G and is denoted by mtr(G). A total restrained monophonic set of cardinality mtr(G) is ...