Topology of Cut Complexes II

TL;DR

This paper advances the understanding of the topology of k-cut complexes in graphs by providing explicit formulas for their combinatorial invariants and analyzing specific graph classes using topological and combinatorial tools.

Contribution

It offers explicit formulas for f- and h-polynomials of cut complexes of disjoint unions and studies their homology representations, extending previous work on graph cut complexes.

Findings

Formulas for f- and h-polynomials of cut complexes of disjoint unions.

Homology representations of cut complexes of complete graphs.

Topological analysis of cut complexes of squared paths and grid graphs.

Abstract

We continue the study of the $k$-cut complex $\Delta_k(G)$ of a graph $G$ initiated in the paper of Bayer, Denker, Jeli\'c Milutinovi\'c, Rowlands, Sundaram and Xue [Topology of cut complexes of graphs, SIAM J. on Discrete Math. 38(2): 1630--1675 (2024)]. We give explicit formulas for the $f$- and $h$-polynomials of the cut complex $\Delta_k(G_1+G_2) $ of the disjoint union of two graphs $G_1$ and $G_2$, and for the homology representation of $\Delta_k(K_m+K_n)$. We also study the cut complex of the squared path and the grid graph. Our techniques include tools from combinatorial topology, discrete Morse theory and equivariant poset topology.