Chordal and sequentially Cohen-Macaulay clutters

TL;DR

This paper generalizes the concept of chordality from graphs to clutters, showing that chordal clutters have desirable algebraic properties like shellability and linear resolutions, and classifies obstructions to shellability.

Contribution

It introduces a new definition of chordal clutters extending graph theory, and explores their algebraic and geometric properties, including classification of obstructions.

Findings

Independence complex of chordal clutter is shellable.

Circuit ideal of certain complement has a linear resolution.

Classified all obstructions to shellability on 6 vertices.

Abstract

We extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a chordal clutter is shellable, hence sequentially Cohen-Macaulay; and the circuit ideal of a certain complement to such a clutter has a linear resolution. Minimal non-chordal clutters are also closely related to obstructions to shellability, and we give some general families of such obstructions, together with a classification by computation of all obstructions to shellability on 6 vertices.