Fractional global domination in graphs

Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, ${\displaystyle g\left(N\left[v\right]\right)={\sum }_{u\in N\left[v\right]}g\left(u\right)\ge 1}$ and ${\displaystyle g\left(\overline{N\left(v\right)}\right)={\sum }_{u\notin N\left(v\right)}g\left(u\right)\ge 1}$. A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number ${\displaystyle {\gamma }_{fg}\left(G\right)}$ is defined as follows: ${\displaystyle {\gamma }_{fg}\left(G\right)}$ = min{|g|:g is an MGDF of G } where ${\displaystyle \left|g\right|={\sum }_{v\in V}g\left(v\right)}$. In this paper we initiate a study of this parameter.