Finite record sets of chip-firing games

A finite graph with an assignment of non-negative integers to vertices gives chip-firing games.Chip-firing games determine languages (sets of words) called the record sets of legal games.Björner, Lovász and Shor found several properties that are satisfied by record sets.In this paper, we will find two more properties of record sets.Under the assumption that the record set is finite and the game fires only two vertices, these properties characterize the record sets of graphs.