A maximizing characteristic for critical configurations of chip-firing games on digraphs

Aval et al. proved in Aval et al. (2016) that starting from a critical configuration of a chip-firing game on an undirected graph, one can never achieve a stable configuration by reverse firing any non-empty subsets of its vertices. In this paper, we generalize the result to digraphs with a global sink where reverse firing subsets of vertices is replaced with reverse firing multi-subsets of vertices. Consequently, a combinatorial proof for the duality between critical configurations and superstable configurations on digraphs is given. Finally, by introducing the concept of energy vector assigned to each configuration, we show that critical and superstable configurations are the unique ones with the greatest and smallest (w.r.t. the containment order), respectively, energy vectors in each of their equivalence classes. � 2018 Elsevier B.V.