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

A graph with no isolated vertices is edge critical with respect to total restrained domination if for any non-edge e of G, the total restrained domination number of G+ e is less than the total restrained domination number of G. We call these graphs γtr-edge critical. In this paper, we characterize all γtr-edge critical unicyclic graphs.

A subset S ⊆ V in a graph G = (V,E) is a total [1, 2]-set if, for every vertex v ∈ V , 1 ≤ |N(v) ∩ S| ≤ 2. The minimum cardinality of a total [1, 2]-set of G is called the total [1, 2]-domination number, denoted by γt[1,2](G). We establish two sharp upper bounds on the total [1,2]-domination number of a graph G in terms of its order and minimum degree, and characterize the corresponding extrema...