The ACO Seminar (2022–2023)

In this talk, we introduce two new families of simplicial complexes defined from graphs. Given a graph $G$, we define cut complexes $\Delta_k(G)$ and total cut complexes $\Delta^t_k(G)$. These complexes are formed from the complements of disconnected sets (for cut complexes) and independent sets (for total cut complexes) in graphs. A famous result by Fröberg and reinterpreted by Eagon and Reiner states that the $2-$ (total) cut complex of a graph $G$ is vertex decomposable if and only if $G$ is chordal. A motivation for this work is to generalize Fröberg’s theorem by extending Eagon and Reiner’s construction.

We will discuss several results about what graph properties and operations can tell us about the topology of $\Delta_k(G)$ and $\Delta^t_k(G)$, such as their shellability and the homotopy type. In particular, we prove that the total cut complex $\Delta^t_k(G)$ is vertex decomposable for any chordal graph $G$.