Graph pebbling algorithms and Lemke graphs

Abstract

Given a simple, connected graph, a pebbling configuration (or just configuration) is a function from its vertex set to the nonnegative integers. A pebbling move between adjacent vertices removes two pebbles from one vertex and adds one pebble to the other. A vertex r is said to be reachable from a configuration if there exists a sequence of pebbling moves that places at least one pebble on r. A configuration is solvable if every vertex is reachable. The pebbling numberπ(G) of a graph G is the minimum integer such that every configuration of size π(G) on G is solvable. A graph G is said to satisfy the two-pebbling property if for any configuration with more than 2π(G)−q pebbles, where q is the number of vertices with pebbles, two pebbles can be moved to any vertex of G. A Lemke graph is a graph that does not satisfy the two-pebbling property. In this paper we present a new algorithm to determine if a vertex is reachable with a given configuration and if a configuration on a graph is solvable. We also discuss straightforward algorithms to compute the pebbling number and to determine whether or not a graph has the two-pebbling property. Finally, we use these algorithms to determine all Lemke graphs on at most 9 vertices, finding many previously unknown Lemke graphs.