  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
v and w adjacent to a vertex u and an extra pebble is added at vertex u. 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 m needed to guarantee that any vertex is reachable from any
pebble distribution of m pebbles. The optimal rubbling number is the smallest
number m needed to guarantee a pebble distribution of m pebbles from which any
vertex is reachable. We determine the rubbling and optimal rubbling number of
some families of graphs including cycles.
