Edge weighting functions on dominating sets

In this article, we use edge weighting functions on dominating sets to show that if we impose a regularity condition on a graph, then upper bounds on both the upper domination number and the upper total domination number can be greatly improved. More precisely, we prove that for k≥1 if G is a k-regular graph on n vertices, then the upper domination number of G is at most n/2, and the upper total domination number of G is at most n/(2-1k). Furthermore, we show that these bounds are sharp and characterize the infinite families of graphs that achieve equality in both these bounds.