Shellability of Higher Independence Complexes of Graphs

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.