Shellability of Higher Independence Complexes of Graphs

TL;DR

This paper studies the shellability of higher independence complexes of graphs, providing conditions, constructions, and classifications for specific graph families, extending known results in combinatorial topology.

Contribution

It introduces new criteria and methods for determining shellability of $r$-independence complexes in structured graphs like block graphs and trees.

Findings

Established sufficient conditions for shellability based on graph parameters.

Developed constructive techniques for generating shellable complexes.

Extended classical independence complex results to higher independence complexes.

Abstract

This paper investigates the shellability of $r$-independence complexes $\mathcal{I}_r(G)$, a generalization of classical independence complexes introduced by Paolini and Salvetti. For a graph $G$, a subset $A \subseteq V(G)$ is $r$-independent if every connected component of the induced subgraph $G[A]$ has at most $r$ vertices. The associated simplicial complex $\mathcal{I}_r(G)$ has been the subject of significant interest due to its connections to combinatorial topology and commutative algebra. We address the classification problem for shellable $r$-independence complexes, focusing on block graphs, trees, and related families. Our main results establish sufficient conditions for shellability based on structural graph parameters such as diameter and forbidden subgraphs. Furthermore, we develop constructive techniques for generating shellable complexes through graph operations, including star-clique attachments, clique whiskering, and clique cycle constructions. These results extend and refine earlier work on classical independence complexes and provide a framework for understanding the topological and algebraic properties of higher independence complexes in structured graph families.