Topology of cut complexes of graphs

Abstract

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 [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 Kn \times K2, using techniques from algebraic topology, discrete Morse theory, and equivariant poset topology.