Vertex decomposable graphs and obstructions to shellability

TL;DR

This paper explores the relationship between vertex decomposability and shellability in graphs, establishing new conditions under which graphs are shellable and characterizing obstructions to shellability in flag complexes.

Contribution

It introduces new results linking vertex decomposability with shellability in 5-chordal graphs and characterizes obstructions to shellability in flag complexes.

Findings

5-chordal graphs with no chordless 4-cycles are shellable and Cohen-Macaulay

Characterization of obstructions to shellability in flag complexes

Vertex decomposability preserves shellability in certain graph constructions

Abstract

Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5-chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability.