  We define the $k$-cut complex of a graph $G$ with vertex set $V(G)$ to be the
simplicial complex whose facets are the complements of sets of size $k$ in
$V(G)$ inducing disconnected subgraphs of $G$. This generalizes the Alexander
dual of a graph complex studied by Fr\"oberg (1990), and Eagon and Reiner
(1998). 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
$K_n \times K_2$, using techniques from algebraic topology, discrete Morse
theory and equivariant poset topology.
