Critical cyclic patterns related to the domination number of the torus Original Research Article

The concern here is the domination number of the torus, image. Directly, this paper closes out a significant subset of cases, not only calculating periodic values of image, but also providing dominating sets with minimal cardinality. The work here builds from a 1994 Livingston and Stout result: For any fixed value of image, the existence of a closed-form formula in image, cyclic in nature, is assured. With that expression the value of image can be calculated in constant time relative to image. Unfortunately, given image as a parameter, algorithms known to produce the closed-form expression in image run in exponential time relative to image. In brief, the related problem has an unknown complexity.F(m)=limn⟶∞γ(Tm,n)n The nature of a closed-form formula for the two-parameter case has been a matter of some conjecture. With respect to the case when image, the best bounds here suggest that such a closed-form expression for image would not be cyclic in the usual, simple sense.