SIAM Journal on Discrete Mathematics
Society for Industrial and Applied Mathematics
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.
graph pebbling, diameter two, NP-complete, AMS subject classifications, 68Q17, 68R05, 68R10
Cusack, Charles A., Timothy Lewis, Daniel Simpson and Samuel Taggart. "The Complexity of Pebbling in Diameter Two Graphs*." SIAM Journal on Discrete Mathematics 26, no. 3 (2012): 919-928.
∗Received by the editors August 11, 2011; accepted for publication (in revised form) April 6, 2012; published electronically July 10, 2012. This work was supported by the NSF under grant DUE0851293.