Average distances and distance domination numbers Original Research Article

Let image be a positive integer and image be a simple connected graph with order image. The average distance image of image is defined to be the average value of distances over all pairs of vertices of image. A subset image of vertices in image is said to be a image-dominating set of image if every vertex of image is within distance image from some vertex of image. The minimum cardinality among all image-dominating sets of image is called the image-domination number image of image. In this paper tight upper bounds are established for image, as functions of image, image and image, which generalizes the earlier results of Dankelmann [P. Dankelmann, Average distance and domination number, Discrete Appl. Math. 80 (1997) 21–35] for image.