Chordal and sequentially Cohen-Macaulay clutters

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.

20 pages. v2 fixes typos and improves exposition. v3 attributes prior work on shedding faces by Jonsson. v4 has minor updates for publication

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• Chains of modular elements and shellability
• Shedding vertices and Ass-decomposable monomial ideals
• Graded Betti numbers of a hyperedge ideal associated to join of graphs