Acta Mathematica Vietnamica

In this paper, we introduce a variation of the chip-firing game on a directed acyclic graph $G = (V, E)$. Starting from a given chip configuration, we can fire a vertex $v$ by sending one chip along one of its outgoing edges to the corresponding neighbors if $v$ has at least one chip. Our main result is to give the collection of energies to show the partial order structure of the configuration space of the game. After that, we consider the case when support graph has only one source, we give the characterization of its reachable configurations and of its fixed points.

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