A pebble distribution places a nonnegative number of pebbles on the vertices of a graph G. In graph rubbling, the pebbles can be redistributed using pebbling and rubbling moves, typically with the goal of reaching some target pebble distribution. In graph pebbling, only the pebbling move is allowed. The cover pebbling number is the smallest k such that from any initial distribution of k pebbles, it is possible that after a series of pebbling moves there is at least one pebble on every vertex of G. The Cover Pebbling Theorem asserts that to determine the cover pebbling number of a graph, it is sufficient to consider the pebbling distributions that initially place all pebbles on a single vertex. In this paper, we prove a rubbling analogue of the Cover Pebbling Theorem, providing an answer to an open question of Belford and Sieben (2009). In addition, we prove a stronger version of the Cover Rubbling Theorem for trees.

Let G be a connected simple graph with vertex set V and a distribution of pebbles on the vertices of V. The total 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 total dominating set of G. We investigate total domination cover rubbling in graphs and determine bounds on the total domination cover rubbling number.

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.