A Graph Pebbling Algorithm on Weighted Graphs - Sieben, Nandor (Journal of Graph Algorithms and Applications, 2010)
A pebbling move on a weighted graph removes some pebbles at a vertexand adds one pebble at an adjacent vertex. The number of pebbles removedis the weight of the edge connecting the vertices. A vertex is reachablefrom a pebble distribution if it is possible to move a pebble to thatvertex using pebbling moves. The pebbling number of a weighted graphis the smallest number m needed to guarantee that any vertex isreachable from any pebble distribution of m pebbles. Regular pebblingproblems on unweighted graphs are special cases when the weight onevery edge is 2. A regular pebbling problem often simplifies to apebbling problem on a simpler weighted graph. We present an algorithmto find the pebbling number of weighted graphs. We use this algorithmtogether with graph simplifications to find the regular pebbling numberof all connected graphs with at most nine vertices. <!---->