Total $2$-cut complexes of powers of cycle graphs and Cartesian products of certain graphs

For a positive integer $k$, the \emph{ total $k$-cut complex} of a graph $G$, denoted as ${\mathrm{\Delta }}_k^t(G)$, is the simplicial complex whose facets are $\sigma \subseteq V(G)$ such that $\mathrm{\mid }\sigma \mathrm{\mid }=\mathrm{\mid }V(G)\mathrm{\mid }-k$ and the induced subgraph $G[V(G)\setminus \sigma ]$ does not contain any edge. These complexes were introduced by Bayer et al.\ in \cite{Bayer2024TotalCutcomplex} in connection with commutative algebra. In the same paper, they studied the homotopy types of these complexes for various families of graphs, including cycle graphs $C_n$, squared cycle graphs $C_n^2$, and Cartesian products of complete graphs and path graphs $K_m\mathrm{\square }{P}_2$ and $K_2\mathrm{\square }{P}_n$. In this article, we extend the work of Bayer et al.\ for these families of graphs. We focus on the complexes ${\mathrm{\Delta }}_2^t(G)$ and determine the homotopy types of these complexes for three classes of graphs: (i) $p$-th powers of cycle graphs $C_n^p$ (ii) $K_m\mathrm{\square }{P}_n$ and (iii) $K_m\mathrm{\square }{C}_n$. Using discrete Morse theory, we show that these complexes are homotopy equivalent to wedges of spheres. We also give the number and dimension of spheres appearing in the homotopy type. Our result on powers of cycle graphs $C_n^p$ proves a conjecture of Shen et al.\ about the homotopy type of the complexes ${\mathrm{\Delta }}_2^t(C_n^p)$.