Consider a configuration of pebbles distributed on the vertices of a connected graph of order n. A pebbling step consists of removing two pebbles from a given vertex and placing one pebble on an adjacent vertex. A distribution of pebbles on a graph is called solvable if it is possible to place a pebble on any given vertex using a sequence of pebbling steps. The pebbling number of a graph, denoted f (G), is the minimal number of pebbles such that every configuration of f (G) pebbles on G is solvable. We derive several general upper bounds on the pebbling number, improving previous results.

Let G be a connected simple graph with vertex set V and a distribution of pebbles on V. The domination cover rubbling number of G is the minimum number of pebbles, so that no matter how they are distributed, it is possible that after a sequence of pebbling and rubbling moves, the set of vertices with pebbles is a dominating set of G. We begin by characterizing the graphs having small domination cover rubbling numbers and determining the domination cover rubbling number of several common graph families. We then give a bound for the domination cover rubbling number of trees and characterize the extremal trees. Finally, we give bounds for the domination cover rubbling number of graphs in terms of their domination number and characterize a family of the graphs attaining this bound.

Let G be a graph with vertex set V and a distribution of pebbles on the vertices of V . A pebbling move consists of removing two pebbles from a vertex and placing one pebble on a neighboring vertex, and a rubbling move consists of removing a pebble from each of two neighbors of a vertex v and placing a pebble on v. We seek an initial placement of a minimum total number of pebbles on the vertices in V, so that no vertex receives more than one pebble and for any given vertex v ∈ V, it is possible, by a sequence of pebbling and rubbling moves, to move at least one pebble to v. This minimum number of pebbles is the 1-restricted optimal rubbling number. We determine the 1-restricted optimal rubbling numbers for Cartesian products. We also present bounds on the 1-restricted optimal rubbling number.