Rubbling and optimal rubbling of graphs

A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move one pebble is removed at vertices vv and ww adjacent to a vertex uu and an extra pebble is added at vertex uu. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number of a graph is the smallest number mm needed to guarantee that any vertex is reachable from any pebble distribution of mm pebbles. The optimal rubbling number is the smallest number mm needed to guarantee a pebble distribution of mm pebbles from which any vertex is reachable. We determine the rubbling and optimal rubbling number of some families of graphs and we show that Graham’s conjecture does not hold for rubbling numbers.