$G$-strongly positive scripts and critical configurations of chip firing games on digraphs

TL;DR

This paper introduces G-strongly positive scripts to efficiently recognize critical configurations in chip firing games on digraphs, generalizing previous results and providing new algorithms and combinatorial proofs.

Contribution

It presents G-strongly positive scripts for CFG recognition, an algorithm for minimal scripts, and a combinatorial proof of duality between critical and super-stable configurations.

Findings

G-strongly positive scripts effectively recognize critical configurations.

The algorithm finds the minimal G-strongly positive script, reducing recognition time.

Critical configurations are shown to be non-stable when firing inverse subsets of vertices.

Abstract

We show a collection of scripts, called $G$-strongly positive scripts, which is used to recognize critical configurations of a chip firing game (CFG) on a multi-digraph with a global sink. To decrease the time of the process of recognition caused by the stabilization we present an algorithm to find the minimum G-strongly positive script. From that we prove the non-stability of configurations obtained from a critical configuration by firing inversely any non-empty multi-subset of vertices. This result is a generalization of a very recent one by Aval \emph{et.al} which is applied for CFG on undirected graphs. Last, we give a combinatorial proof for the duality between critical and super-stable configurations.