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

TL;DR

This paper generalizes a known property of critical configurations in chip-firing games from undirected graphs to directed graphs with sinks, establishing a duality and energy-based characterization of these configurations.

Contribution

It extends the critical configuration property to digraphs with global sinks and introduces an energy vector concept to characterize critical and superstable configurations.

Findings

Critical configurations cannot be stabilized by reverse firing in digraphs.

A duality between critical and superstable configurations is established.

Energy vectors uniquely identify critical and superstable configurations within their classes.

Abstract

Aval et al. proved 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.