Total Cut Complexes of Graphs

Inspired by work of Fröberg (1990), and Eagon and Reiner (1998), we define the total k-cut complex of a graph G to be the simplicial complex whose facets are the complements of independent sets of size k in G. We study the homotopy types and combinatorial properties of total cut complexes for various families of graphs, including chordal graphs, cycles, bipartite graphs, the prism \(K_n \times K_2\) $K_n\times K_2$ , and grid graphs, using techniques from algebraic topology and discrete Morse theory.