Student Author(s)

Mark Powers

Faculty Mentor(s)

Dr. Charles Cusack, Computer Science and Mathematics; Dr. Airat Bekmetjev, Mathematics

Document Type


Event Date



Graph pebbling is a mathematical game played on a connected graph. A configuration places a nonnegative number of pebbles on each vertex. A move between a pair of adjacent vertices removes two pebbles from one vertex and places one pebble on the other. A configuration is solvable if a sequence of pebbling moves can be made to place a pebble on any given vertex. The pebbling number of a graph is the minimum number of pebbles so that any configuration is solvable. A graph satisfies the two-pebbling property if any configuration of more than twice the pebbling number minus the number of vertices with pebbles allows for placing two pebbles on any vertex after applying a sequence of pebbling moves. If a graph does not have the two-pebbling property, it is a Lemke graph. We say that two vertices are doppelgangers if they are adjacent to the same vertices. By adding vertices to a graph that are doppelgangers of existing vertices, the pebbling number does not decrease and does not increases more than the number of added vertices. By adding any number of doppelgangers to previous Lemke graphs in a particular manner, we are able to construct another graph that is also a Lemke graph. Additionally, we have created new algorithms to determine solvabilty. Using this along with four nondeterministic algorithms, we have been able to find all Lemke graphs with up to nine vertices.