The complex of $r$-co-connected subgraphs, chordality and Fröberg's theorem
Priyavrat Deshpande, Amit Roy, Rutuja Sawant
TL;DR
This work develops a unifying framework linking topological combinatorics and commutative algebra through the $r$-co-connected complex $\Sigma_r(A,G)$. By establishing vertex decomposability of $\Sigma_r(A,G)$ when $A\neq\emptyset$ and $G[A]$ is connected, it enables a deletion–link calculus that propagates to the Alexander dual $\Sigma_r(G)=(\mathrm{Ind}_r(G))^{\∨}$. For several graph classes with $r\ge 2$, it extends Fröberg’s theorem by showing that vertex decomposability, shellability, and Cohen–Macaulayness of $\Sigma_r(G)$ are equivalent to the co-chordality of the $r$-connected clutter $\mathrm{Con}_r(G)$, thereby characterizing when the associated $r$-connected ideals have linear resolutions. The paper also establishes a conjecture linking CM-ness to co-chordality and provides counterexamples illustrating nuances between VD and shellability. Collectively, these results advance a Fröberg-type dictionary for higher-degree, $r$-connected ideals and motivate further exploration of the algebraic-topological structure of higher independence complexes.
Abstract
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.