Chordal graphs, higher independence and vertex decomposable complexes

F. M. Abdelmalek, Priyavrat Deshpande, Shuchita Goyal, A. Roy, Anurag Singh
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引用次数: 2

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 $r \ge 2$.
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弦图,高独立性和顶点可分解复合体
给定一个简单无向图$G$,存在一个简单复形$\mathrm{Ind}(G)$,称为独立复形,其面对应于$G$的独立集。这是一个研究得很好的概念,因为它为交换代数、图论和代数拓扑之间的相互作用提供了肥沃的土壤。许多作者所追求的研究方向之一是确定相关独立复合体为Cohen-Macaulay的图类。例如,已知当$G$是弦图时,复体$\mathrm{Ind}(G)$实际上是顶点可分解的,这是Cohen-Macaulay阶梯中最强的条件。在本文中,我们考虑独立复合体的推广。给定$r\geq 1$,如果诱导子图的连通分量的基数不超过$r$,则顶点集的子集称为$r$独立的。所有$G$的$r$独立子集的集合形成一个简单复合体,称为$r$独立复合体,用$\mathrm{Ind}_r(G)$表示。我们知道,当$G$是弦图时,复$\mathrm{Ind}_r(G)$具有球楔的同伦类型。因此,很自然地要问这些复合体中哪些是可壳化的,甚至是顶点可分解的。利用Woodroofe的弦超图概念,证明了当底层弦图为树时,这些复合体总是可壳的。进一步,利用顶点可分理想的概念,我们证明了对于毛虫图,相关的$r$无关复数对于$r$的所有值都是顶点可分解的。我们还构造了$2r+2$顶点上的弦图,使得它们的$r$无关复合体对于任何$r \ge 2$都不是连续的Cohen-Macaulay。
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