{
  "id": "2306.17076",
  "version": "v1",
  "published": "2023-06-29T16:27:53.000Z",
  "updated": "2023-06-29T16:27:53.000Z",
  "title": "A combinatorial characterization of $S_2$ binomial edge ideals",
  "authors": [
    "Davide Bolognini",
    "Antonio Macchia",
    "Giancarlo Rinaldo",
    "Francesco Strazzanti"
  ],
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "Several algebraic properties of a binomial edge ideal $J_G$ can be interpreted in terms of combinatorial properties of its associated graph $G$. In particular, the so-called cut-point sets of a graph $G$, special sets of vertices that disconnect $G$ in a minimal way, play an important role since they are in bijection with the minimal prime ideals of $J_G$. In this paper we establish the first graph-theoretical characterization of binomial edge ideals $J_G$ satisfying Serre's condition $(S_2)$ by proving that this is equivalent to having $G$ accessible, which means that $J_G$ is unmixed and the cut-point sets of $G$ form an accessible set system. The proof relies on the combinatorial structure of the Stanley-Reisner simplicial complex of a multigraded generic initial ideal of $J_G$, whose facets can be described in terms of cut-point sets. Another key step in the proof consists in proving the equivalence between accessibility and strong accessibility for the collection of cut sets of $G$ with $J_G$ unmixed. This result, interesting on its own, provides the first relevant class of set systems for which the previous two notions are equivalent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-29T16:27:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "binomial edge ideal",
      "combinatorial characterization",
      "cut-point sets",
      "set system",
      "first relevant class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}