Pebbling in Planar Graphs
Dr. Charles Cusack
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 is NP-complete, even in graphs that are planar. We also prove that reachability of a vertex can be determined in O(n6 ) time in planar graphs of diameter two.
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