Faculty Mentor(s)

Dr. Charles Cusack, Hope College

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Given a simple, connected graph, a pebbling 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 under a configuration if there exists a sequence of pebbling moves that places one pebble on r. A configuration is solvable if every vertex is reachable. We prove tight bounds on the number of vertices with two and three pebbles that can exist in an unsolvable configuration on a diameter two graph in terms of the size of the graph. We also prove that determining reachability of a vertex is NP-complete, even in graphs of diameter two.


This work was supported by the National Science Foundation Research Experience for Undergraduates Program grant No. 0851293.