The odd 2t-pebbling property of graphs

The t-pebbling number, ft(G), of a connected graph G, is the smallest positive integer such that from every placement of ft(G) pebbles, t pebbles can be moved to a specified target vertex by a sequence of pebbling moves, each move taking two pebbles off a vertex and placing one on an adjacent vertex. We say a graph G satisfies the odd 2t?pebbling property if, for any arrangement of pebbles with at least 2ft (G) ? r + 1 pebbles, where r is the number of vertices with an odd number of pebbles in the arrangement, it is possible to put 2t pebbles on any target vertex using pebbling moves. We study the odd 2t-pebbling property of graphs.