Two-Player Graph Pebbling

Abstract

Given a graph G with pebbles on the vertices, we define a pebbling move as removing two pebbles from a vertex u, placing one pebble on a neighbor v, and discarding the other pebble, like a toll. The pebbling number n(G) is the least number of pebbles needed so that every arrangement of n(G) pebbles can place a pebble on any vertex through a sequence of pebbling moves. We introduce a new variation on graph pebbling called two-player pebbling. In this, players called the mover and the defender alternate moves, with the stipulation that the defender cannot reverse the previous move. The mover wins only if they can place a pebble on a specified vertex and the defender wins if the mover cannot. We define n(G), analogously, as the minimum number of pebbles such that given every configuration of the n(G) pebbles and every specified vertex r, the mover has a winning strategy. First, we will investigate upper bounds for n(G) on various classes of graphs and find a certain structure for which the defender has a winning strategy, no matter how many pebbles are in a configuration. Then, we characterize winning configurations for both players on a special class of diameter 2 graphs. Finally, we show winning configurations for the mover on paths using a recursive argument