Restricted optimal pebbling and domination in graphs

For a graph G=(V,E), we consider placing a variable number of pebbles on the vertices of V. A pebbling move consists of deleting two pebbles from a vertex u∈V and placing one pebble on a vertex v adjacent to u. We seek an initial placement of a minimum total number of pebbles on the vertices in V, so that no vertex receives more than some positive integer t pebbles and for any given vertex v∈V, it is possible, by a sequence of pebbling moves, to move at least one pebble to v. We relate this minimum number of pebbles to several other well-studied parameters of a graph G, including the domination number, the optimal pebbling number, and the Roman domination number of G.