{
  "id": "2001.05448",
  "version": "v1",
  "published": "2020-01-15T17:42:52.000Z",
  "updated": "2020-01-15T17:42:52.000Z",
  "title": "Higher Independence Complexes of graphs and their homotopy types",
  "authors": [
    "Priyavrat Deshpande",
    "Anurag Singh"
  ],
  "categories": [
    "math.AT",
    "math.CO"
  ],
  "abstract": "For $r\\geq 1$, the $r$-independence complex of a graph $G$ is a simplicial complex whose faces are subset $I \\subseteq V(G)$ such that each component of the induced subgraph $G[I]$ has at most $r$ vertices. In this article, we determine the homotopy type of $r$-independence complexes of certain families of graphs including complete $s$-partite graphs, fully whiskered graphs, cycle graphs and perfect $m$-ary trees. In each case, these complexes are either homotopic to a wedge of equi-dimensional spheres or are contractible. We also give a closed form formula for their homotopy types.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-15T17:42:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "homotopy type",
      "higher independence complexes",
      "simplicial complex",
      "partite graphs",
      "cycle graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}