Abstract
Consider a distribution of pebbles on the vertices of a graph G . A pebbling move consists of the removal of two pebbles from a vertex and then placing one pebble at an adjacent vertex. The optimal pebbling number of G , denoted f opt ( G ) , is the least number of pebbles, such that for some distribution of f opt ( G ) pebbles, a pebble can be moved to any vertex of G . We give sharp lower and upper bounds for f opt ( G ) for G of diameter d . For graphs of diameter two (respectively, three) we characterize the classes of graphs having f opt ( G ) equal to a value between 2 and 4 (respectively, between 3 and 8).
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