Total Cut Complexes of Graphs

TL;DR

This paper introduces the total k-cut complex of a graph, exploring its topological and combinatorial properties across various graph families using algebraic topology and discrete Morse theory.

Contribution

It defines the total k-cut complex inspired by prior work and analyzes its properties for different graph classes, extending understanding of graph complexes.

Findings

Homotopy types characterized for chordal graphs and cycles.

Combinatorial properties studied for bipartite, prism, and grid graphs.

Techniques from algebraic topology and discrete Morse theory applied.

Abstract

Inspired by work of Fr\"oberg (1990), and Eagon and Reiner (1998), we define the \emph{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 $K_n \times K_2$, and grid graphs, using techniques from algebraic topology and discrete Morse theory.