Dominating functions and total dominating functions of countable graphs

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