On the computational complexity of upper fractional domination

Abstract This paper studies a nondiscrete generalization of Γ ( G ), the maximum cardinality of a minimal dominating set in a graph G =( V , E ). In particular, a real-valued function ⨍: V → [ 0,1 ], is dominating if for each vertex υ ϵ V , the sum of the values assigned to the vertices in the closed neighborhood of υ , N [ υ ], is at least one, i.e., ⨍(N[υ]) ≥ 1 . The weight of a dominating function ⨍ is ⨍(V) , the sum of all values ⨍(υ) for υ ϵ V , and Γ ⨍ (G), is the maximum weight over all minimal dominating functions. In this paper we show that: (1) Γ ⨍ (G) is computable and is always a rational number; (2) the decision problems corresponding to the problems of computing Γ ( G ) and Γ ⨍ (G) are NP-complete; (3) for trees Γ ⨍ =Γ , which implies that the value of Γ ⨍ can be computed in linear time.