The complex of $r$-co-connected subgraphs, chordality and Fröberg's theorem
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| Format: | Preprint |
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2025
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| author | Deshpande, Priyavrat Roy, Amit Sawant, Rutuja |
| author_facet | Deshpande, Priyavrat Roy, Amit Sawant, Rutuja |
| contents | We introduce a new family of pure simplicial complexes, called the $r$-co-connected complex of $G$ with respect to $A$, $Σ_r(A,G)$, where $r\geq 1$ is a natural number, $G$ is a simple graph, and $A$ is a subset of vertices. Interestingly, when $A$ is empty, this complex is precisely the Alexander dual of the $r$-independence complex of $G$. We focus on uncovering the relationship between the topological and combinatorial properties of the complex and the algebraic and homological properties of the Stanley-Reisner ideal of the dual complex. First, we prove that $Σ_r(A,G)$ is vertex decomposable whenever the induced subgraph $G[A]$ is connected and nonempty, yielding a versatile deletion-link calculus for higher independence via Alexander duality. Furthermore, when $A=\emptyset$ and $r \ge 2$, we establish that for several significant classes of graphs - including chordal, co-chordal, cographs, cycles, complements of cycles, and certain grid graphs - the properties of vertex decomposability, shellability, and Cohen-Macaulayness are equivalent and precisely characterized by the co-chordality of the associated clutter $\mathrm{Con}_r(G)$. These results extend Fröberg's theorem to the setting of $r$-connected ideals for these graph classes and motivate a conjecture concerning the linear resolution property of $r$-connected ideals in general. We also construct examples separating shellability from vertex decomposability. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_25710 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The complex of $r$-co-connected subgraphs, chordality and Fröberg's theorem Deshpande, Priyavrat Roy, Amit Sawant, Rutuja Combinatorics Commutative Algebra 05E45, 05C75, 05C69, 05E40 We introduce a new family of pure simplicial complexes, called the $r$-co-connected complex of $G$ with respect to $A$, $Σ_r(A,G)$, where $r\geq 1$ is a natural number, $G$ is a simple graph, and $A$ is a subset of vertices. Interestingly, when $A$ is empty, this complex is precisely the Alexander dual of the $r$-independence complex of $G$. We focus on uncovering the relationship between the topological and combinatorial properties of the complex and the algebraic and homological properties of the Stanley-Reisner ideal of the dual complex. First, we prove that $Σ_r(A,G)$ is vertex decomposable whenever the induced subgraph $G[A]$ is connected and nonempty, yielding a versatile deletion-link calculus for higher independence via Alexander duality. Furthermore, when $A=\emptyset$ and $r \ge 2$, we establish that for several significant classes of graphs - including chordal, co-chordal, cographs, cycles, complements of cycles, and certain grid graphs - the properties of vertex decomposability, shellability, and Cohen-Macaulayness are equivalent and precisely characterized by the co-chordality of the associated clutter $\mathrm{Con}_r(G)$. These results extend Fröberg's theorem to the setting of $r$-connected ideals for these graph classes and motivate a conjecture concerning the linear resolution property of $r$-connected ideals in general. We also construct examples separating shellability from vertex decomposability. |
| title | The complex of $r$-co-connected subgraphs, chordality and Fröberg's theorem |
| topic | Combinatorics Commutative Algebra 05E45, 05C75, 05C69, 05E40 |
| url | https://arxiv.org/abs/2510.25710 |