Graph pebbling can be extended to a two-player game on a graph G, called Two-Player Graph Pebbling, with players Mover and Defender. The players each use pebbling moves, the act of removing two pebbles from one vertex and placing one of the pebbles on an adjacent vertex, to win. Mover wins if they can place a pebble on a specified vertex. Defender wins if the specified vertex is pebble-free and there are no more pebbling moves on the vertices of G. 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 paths, cycles, and the join of certain graphs.
Isaak, Garth; Prudente, Matthew; Potylycki, Andrea; Fagley, William; and Marcinik, Joseph
"On Two-Player Pebbling,"
Communications on Number Theory and Combinatorial Theory: Vol. 3, Article 3.
Available at: https://research.library.kutztown.edu/contact/vol3/iss1/3