Extremal graphs for a new upper bound on domination parameters in graphs Original Research Article

In 1958, Claude Berge studied the domination number image of a graph and showed that every graph image satisfies image, where image is the degree of vertex image. We show here that every graph G satisfies the inequality image, where image stands for any of the three parameters image (irredundance number), image (domination number) and image (size of a smallest maximal independent set), and image is the size of a largest matching in the subgraph of G induced by the non-neighbours of image. This improves on Bergeʹs and other inequalities. We then give a characterization of the trees T that achieve equality in the above inequality for the parameters image and image, and of the regular graphs G that achieve equality for image.