Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Chordal graphs, higher independence and vertex decomposable complexes   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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The categories of this paper are: nlin
1
The categories of this paper are: cs
24
The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: adap-org
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: econ