Chordal graphs, higher independence and vertex decomposable complexes
Fred M. Abdelmalek, Priyavrat Deshpande, Shuchita Goyal, Amit Roy, Anurag Singh
TL;DR
The paper investigates higher independence complexes ${\rm Ind}_r(G)$, extending the classical ${\rm Ind}(G)$ by allowing components of size up to $r$. It establishes shellability for ${\rm Ind}_r(G)$ when $G$ is a tree by viewing ${\rm Ind}_r(G)$ as the independence complex of a chordal hypergraph ${\rm Con}_r(G)$, and proves vertex decomposability for ${\rm Ind}_r(CG)$ when $CG$ is a caterpillar via Alexander duals and vertex-splittable ideals. It also shows that even within chordal graphs there are examples where ${\rm Ind}_r(G)$ is not sequentially Cohen–Macaulay for $r\ge2$, highlighting limitations of these strong properties and motivating a conjecture that all tree-based $${\rm Ind}_r(G)$$ are vertex decomposable. The work advances understanding of how graph structure governs topological and algebraic properties of higher independence complexes and uses tools from hypergraph theory and commutative algebra to derive new results and counterexamples.
Abstract
Given a simple undirected graph $G$ there is a simplicial complex $\mathrm{Ind}(G)$, called the independence complex, whose faces correspond to the independent sets of $G$. This is a well studied concept because it provides a fertile ground for interactions between commutative algebra, graph theory and algebraic topology. One of the line of research pursued by many authors is to determine the graph classes for which the associated independence complex is Cohen-Macaulay. For example, it is known that when $G$ is a chordal graph the complex $\mathrm{Ind}(G)$ is in fact vertex decomposable, the strongest condition in the Cohen-Macaulay ladder. In this article we consider a generalization of independence complex. Given $r\geq 1$, a subset of the vertex set is called $r$-independent if the connected components of the induced subgraph have cardinality at most $r$. The collection of all $r$-independent subsets of $G$ form a simplicial complex called the $r$-independence complex and is denoted by $\mathrm{Ind}_r(G)$. It is known that when $G$ is a chordal graph the complex $\mathrm{Ind}_r(G)$ has the homotopy type of a wedge of spheres. Hence it is natural to ask which of these complexes are shellable or even vertex decomposable. We prove, using Woodroofe's chordal hypergraph notion, that these complexes are always shellable when the underlying chordal graph is a tree. Further, using the notion of vertex splittable ideals we show that for caterpillar graphs the associated $r$-independence complex is vertex decomposable for all values of $r$. We also construct chordal graphs on $2r+2$ vertices such that their $r$-independence complexes are not sequentially Cohen-Macaulay for any $r \ge 2$.