Optimal t-rubbling on complete graphs and paths

Given a distribution of pebbles on the vertices of a graph, a rubbling move places one pebble at a vertex and removes a pebble each at two not necessarily distinct adjacent vertices. One pebble is the cost of transportation. A vertex is t -reachable if at least t pebbles can be moved to the vertex using rubbling moves. The optimal t -rubbling number of a graph is the minimum number of pebbles in a pebble distribution that makes every vertex t -reachable. The optimal t -rubbling numbers of complete graphs and paths are determined.