  In a graph G with a distribution of pebbles on its vertices, a pebbling move
is the removal of two pebbles from one vertex and the addition of one pebble to
an adjacent vertex. A weight function on G is a non-negative integer-valued
function on the vertices of G. A distribution of pebbles on G covers a weight
function if there exists a sequence of pebbling moves that gives a new
distribution in which every vertex has at least as many pebbles as its weight.
In this paper we give some necessary and some sufficient conditions for a
distribution of pebbles to cover a given weight function on a connected graph
G. As a corollary, we give a simple formulation for the `weighted cover
pebbling number' of a weight function W and a connected graph G, defined by
Crull et al. to be the smallest number m such that any distribution on G of m
pebbles is a cover for W. Also, we prove a cover pebbling variant of Graham's
Conjecture for pebbling.
