On the computational complexity of upper fractional domination

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