Dominating functions and total dominating functions of countable graphs

Let GG be a countable graph and VV be its vertex set; for any v∈Vv∈V, let N(v)N(v) be the neighbourhood of vv and N[v]N[v] be N(v)∪{v}N(v)∪{v}. Let ff be a function from VV to the set of all nonnegative real numbers. Denote the sum ∑x∈Vf(x)∑x∈Vf(x) by ‖f‖‖f‖. If for each v∈Vv∈V, ∑x∈N[v]f(x)⩾1∑x∈N[v]f(x)⩾1, then ff is called a dominating function   of GG. If for each v∈Vv∈V, ∑x∈N(v)f(x)⩾1∑x∈N(v)f(x)⩾1, then ff is called a total dominating function   of GG. Let DD and TT be respectively the collection of all dominating functions of GG and the collection of all total dominating functions of GG. Then inf{‖f‖:f∈D}inf{‖f‖:f∈D} and inf{‖f‖:f∈T}inf{‖f‖:f∈T} are called respectively, the fractional domination number   of GG and the fractional total domination number   of GG. It is proved that for any real number α⩾1α⩾1, there exists a connected graph having both fractional domination number and fractional total domination number equal to αα.