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 two-valued function f defined on the vertices of a graph G (V, E), I : V -+ {-I, I}, is a signed dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v E V, f(N[v]) 2: 1, where N(v] consists of v and every vertex adjacent to v. The of a signed dominating function is ICV) = L f( v), over all vertices v E V. The signed domination...

A signed edge domination function (or SEDF) of a simple graph $G=(V,E)$ is $f: E\rightarrow \{1,-1\}$ such that $\sum_{e'\in N[e]}f(e')\ge 1$ holds for each $e\in E$, where $N[e]$ the set edges in $G$ share at least one endpoint with $e$. Let $\gamma_s'(G)$ denote minimum value $f(G)$ among all SEDFs $f$, $f(G)=\sum_{e\in E}f(e)$.In 2005, Xu conjectured $\gamma_s'(G)\le n-1$, $n$ order $G$. Thi...