  Graph pebbling is a game played on a connected graph G. A player purchases
pebbles at a dollar a piece, and hands them to an adversary who distributes
them among the vertices of G (called a configuration) and chooses a target
vertex r. The player may make a pebbling move by taking two pebbles off of one
vertex and moving one pebble to a neighboring vertex. The player wins the game
if he can move k pebbles to r. The value of the game (G,k), called the
k-pebbling number of G, is the minimum cost to the player to guarantee a win.
That is, it is the smallest positive integer m of pebbles so that, from every
configuration of size m, one can move k pebbles to any target. In this paper,
we use the block structure of graphs to investigate pebbling numbers, and we
present the exact pebbling number of the graphs whose blocks are complete. We
also provide an upper bound for the k-pebbling number of diameter-two graphs,
which can be the basis for further investigation into the pebbling numbers of
graphs with blocks that have diameter at most two.
