The domatic number d(G) of a graph G = (V,E) is the maximum order of a partition of V into dominating sets. Such a partition Π = {D1, D2, . . . , Dd} is called a minimal dominating d-partition if Π contains the maximum number of minimal dominating sets, where the maximum is taken over all d-partitions of G. The minimal dominating d-partition number Λ(G) is the number of minimal dominating sets ...

We show how to count all minimumweighted dominating sets of a graph on n vertices in timeO(1.5535).

We give an efficient construction of a reasonably small dominating set in a circulant graph on n notes and k distinct chord lengths. This result is based on bounds on some double exponential sums.