Root Cover Pebbling on Graphs

Consider a graph, G, with pebbles on its vertices. A pebbling move is defined to be the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The cover pebbling number of a graph, γ(G), is the minimum number of pebbles such that, given any configuration of γ(G) pebbles on the vertices of G, pebbling moves can be used to place one pebble on each vertex of G. We define the root vertex of a graph and fix an initial configuration of pebbles on G where we place all pebbles on the root vertex of G. We define the root cover pebbling number, R(G), of a graph to be the minimum number of pebbles needed so that, if R(G) pebbles are placed on the root vertex, pebbling moves can be used to place one pebble on each vertex. We obtain formulas for root cover pebbling numbers of two types of graphs. We use these formulas to compare the cover pebbling number with the root cover pebbling number of paths, stars and fuses. We also determine ways to minimize the root cover pebbling number of a graph