A note on independence complexes of chordal graphs and dismantling

TL;DR

This paper characterizes the topological properties of independence complexes of chordal graphs, showing they are contractible if dismantlable and homotopy equivalent to a sphere if their core is a cross-polytopal sphere, using tree models.

Contribution

It establishes a precise topological characterization of independence complexes of chordal graphs based on dismantlability and core structure, linking graph properties to topological types.

Findings

Independence complex is contractible iff dismantlable.

Homotopy type is a sphere iff core is a cross-polytopal sphere.

Uses tree models of chordal graphs for proofs.

Abstract

We show that the independence complex of a chordal graph is contractible if and only if this complex is dismantlable (strong collapsible) and it is homotopy equivalent to a sphere if and only if its core is a cross-polytopal sphere. The proof uses the properties of tree models of chordal graphs.