Two Player Graph Pebbling is an extension of graph pebbling. Players Mover and Defender use pebbling moves, the act of removing two pebbles from one vertex and placing one pebble on an adjacent vertex, to win. If a specified vertex has a pebble on it, then Mover wins. If a specified vertex is pebble-free and there are no more valid pebbling moves, then Defender wins. The Two-Player Pebbling Number of a graph G, η(G), is the minimum m such that for every arrangement of m pebbles and for any specified vertex, Mover can win. We specify the winning player for powers of a path.
Isaak, Garth; Prudente, Matthew; and Marcinik, Joseph M. III
"A Pebbling Game on Powers of Paths,"
Communications on Number Theory and Combinatorial Theory: Vol. 4, Article 1.
Available at: https://research.library.kutztown.edu/contact/vol4/iss1/1