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 at least one pebble on r. A configuration is solvable if every vertex is reachable. We prove that determining reachability of a vertex and solvability of a configuration are NP-complete on planar graphs. We also prove that both reachability and solvability can be determined in O(n6) time on planar graphs with diameter two. Finally, for outerplanar graphs, we present a linear algorithm for determining reachability and a quadratic algorithm for determining solvability. To prove this result, we provide linear algorithms to determine all possible maximal configurations of pebbles that can be placed on the endpoints of a path and on two adjacent vertices in a cycle. © 2014 Elsevier B.V. All rights reserved.
Lewis, T., Cusack, C. A., & Dion, L. (2014). The complexity of pebbling reachability and solvability in planar and outerplanar graphs. Discrete Applied Mathematics, 172, 62–74. https://doi.org/10.1016/j.dam.2014.03.008