The pebbling number 
of a nonempty finite connected graph 
is the smallest nonnegative
integer 
such that every placement of 
pebbles can supply one pebble to any specified graph
vertex by a sequence of pebbling
moves (Hurlbert 2011). A pebbling move removes
two pebbles from one graph vertex, places one on
an adjacent vertex, and discards the other.
Deciding whether 
is NP-hard (Milans and Clark 2006).
This parameter belongs to graph pebbling. The splitting rule of chessboard pebbling defines a different puzzle.
The values of the pebbling number for various classes of graphs are given in the table below (Hurlbert).
| graph | pebbling number |
complete bipartite
graph  |  |
complete graph  |  |
cycle
graph  |  |
hypercube graph  |  |
path
graph  |  |
The pebbling number satisfies a number of bounds. Let 
be the vertex count, 
the graph
diameter, and 
the domination number of a graph 
.
Breadth lower bounds:
 |
(1)
|
Cut lower bound (which 
contained a cut vertex 
):
 |
(2)
|
Depth lower bound:
 |
(3)
|
Pigeonhole upper bound:
 |
(4)
|
Sharper bounds:
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
(Hurlbert).
For a graph with 
,
 |
(8)
|
where 
is the vertex count of 
(Hurlbert 2011).
