  Recent research in graph pebbling has introduced the notion of a cover
pebbling number. Along this same idea, we develop a more general pebbling
function Pi(G, t, P). This measures the minimum number of pebbles needed to
guarantee that any distribution of them on G can be transformed via pebbling
moves to a distribution with pebbles on t target vertices. Furthermore, the P
part of the function gives the ability to change how many pebbles are needed to
pebble from one vertex to another. Bounds on the Pi-pebbling function are
developed, as well as its exact value for several families of graphs.
