The Complexity of Pebbling in Diameter Two Graphs

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 from 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 an unsolvable configuration on a diameter two graph can have 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.