Topology of Cut Complexes II

We continue the study of the k-cut complex 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 ₖ (G₁+G₂) of the disjoint union of two graphs G₁ and G₂, and for the homology representation of ₖ (Kₘ+Kₙ). 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.