Acyclic Orientations and Spanning Trees

TL;DR

This paper introduces polytopal complexes related to acyclic orientations, explores their algebraic and combinatorial properties, and applies these findings to stochastic processes, graph enumeration, and bootstrap percolation.

Contribution

It generalizes dualities between tree and permutohedron ideals and provides new combinatorial representations and algebraic tools for acyclic orientations and spanning forests.

Findings

Polytopal complexes resolve combinatorial polynomial ideals related to acyclic orientations.

Rooted spanning forests can be represented as non-crossing trees, revealing structural insights.

Applications include Markov chains on acyclic orientations, expected graph properties, and bootstrap percolation analysis.

Abstract

We introduce polytopal cell complexes associated with partial acyclic orientations of a simple graph, which generalize acyclic orientations. Using the theory of cellular resolutions, two of these polytopal cell complexes are observed to minimally resolve certain special combinatorial polynomial ideals related to acyclic orientations. These ideals are explicitly found to be Alexander dual, which relative to comparable results in the literature, generalizes in a cleaner and more illuminating way the well-known duality between permutohedron and tree ideals. The combinatorics underlying these results naturally leads to a canonical way to represent rooted spanning forests of a labelled simple graph as non-crossing trees, and these representations are observed to carry a plethora of information about generalized tree ideals and acyclic orientations of a graph, and about non-crossing partitions of a totally ordered set. A small sample of the enumerative and structural consequences of collecting and organizing this information are studied in detail. Applications of this combinatorial miscellanea are then introduced and explored, namely: Stochastic processes on state space equal to the set of all acyclic orientations of a simple graph, including irreducible Markov chains, which exhibit stationary distributions ranging from linear extensions-based to uniform; a surprising formula for the expected number of acyclic orientations of a random graph; and a purely algebraic presentation of the main problem in bootstrap percolation, likely making it tractable to explore the set of all percolating sets of a graph with a computer.