On Pebbling Graphs by their Blocks
Abstract
Graph pebbling is a game played on a connected graph G. A player purchases pebbles at a dollar a piece, and hands them to an adversary who distributes them among the vertices of G (called a configuration) and chooses a target vertex r. The player may make a pebbling move by taking two pebbles off of one vertex and moving one pebble to a neighboring vertex. The player wins the game if he can move k pebbles to r. The value of the game (G,k), called the kpebbling number of G, is the minimum cost to the player to guarantee a win. That is, it is the smallest positive integer m of pebbles so that, from every configuration of size m, one can move k pebbles to any target. In this paper, we use the block structure of graphs to investigate pebbling numbers, and we present the exact pebbling number of the graphs whose blocks are complete. We also provide an upper bound for the kpebbling number of diametertwo graphs, which can be the basis for further investigation into the pebbling numbers of graphs with blocks that have diameter at most two.
 Publication:

arXiv eprints
 Pub Date:
 November 2008
 DOI:
 10.48550/arXiv.0811.3238
 arXiv:
 arXiv:0811.3238
 Bibcode:
 2008arXiv0811.3238C
 Keywords:

 Mathematics  Combinatorics;
 91A43;
 05C99
 EPrint:
 20 pages, 7 figures