Abstract. We continue the study of the [Formula: see text]-cut complex [Formula: see text] of a graph [Formula: see text] initiated in the paper of Bayer et al. [ SIAM J. Discrete Math., 38 (2024), pp. 1630–1675]. We give explicit formulas for the [Formula: see text]- and [Formula: see text]-polynomials of the cut complex [Formula: see text] of the disjoint union of two graphs [Formula: see text] and [Formula: see text], and for the homology representation of [Formula: see text]. 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.

Abstract. We define the [Formula: see text]- cut complex of a graph [Formula: see text] with vertex set [Formula: see text] to be the simplicial complex whose facets are the complements of sets of size [Formula: see text] in [Formula: see text] inducing disconnected subgraphs of [Formula: see text]. This generalizes the Alexander dual of a graph complex studied by Fröberg [ Topics in Algebra, Part 2, PWN, Warsaw, 1990, pp. 57–70] and Eagon and Reiner [ J. Pure Appl. Algebra, 130 (1998), pp. 265–275]. We describe the effect of various graph operations on the cut complex and study its shellability, homotopy type, and homology for various families of graphs, including trees, cycles, complete multipartite graphs, and the prism [Formula: see text], using techniques from algebraic topology, discrete Morse theory, and equivariant poset topology.