Lattices generated by Chip Firing Game models: criteria and recognition algorithm

TL;DR

This paper establishes a polynomial-time criterion for recognizing and constructing Chip Firing Game (CFG) generated lattices, advancing understanding of their structure and extending to specific graph classes.

Contribution

It provides the first necessary and sufficient criterion for CFG-generated lattices and offers an efficient algorithm for their recognition and construction.

Findings

Established a polynomial-time recognition criterion for CFG lattices

Developed an algorithm to construct CFGs for given lattices

Extended the recognition approach to CFGs on undirected and acyclic graphs

Abstract

It is well-known that the class of lattices generated by Chip Firing games (CFGs) is strictly included in the class of upper locally distributive lattices (ULD). However a necessary and sufficient criterion for this class is still an open question. In this paper we settle this problem by giving such a criterion. This criterion provides a polynomial-time algorithm for constructing a CFG which generates a given lattice if such a CFG exists. Going further we solve the same problem on two other classes of lattices which are generated by CFGs on the classes of undirected graphs and directed acyclic graphs.