Optimal pebbling of graphs

Consider a distribution of pebbles on the vertices of a graph image. A pebbling move consists of the removal of two pebbles from a vertex and then placing one pebble at an adjacent vertex. The optimal pebbling number of image, denoted image, is the least number of pebbles, such that for some distribution of image pebbles, a pebble can be moved to any vertex of image. We give sharp lower and upper bounds for image for image of diameter image. For graphs of diameter two (respectively, three) we characterize the classes of graphs having image equal to a value between 2 and 4 (respectively, between 3 and 8).