{
  "abstract": "A $u\\hbox{-}v$ separator in $G$ is a set $S\\subseteq V(G)\\setminus\\{u,v\\}$ such that there is no $u\\hbox{-}v$ path in $G\\setminus S$. A minimal separator in $G$ is a set $S$ such that for some $u,v\\in V(G)\\setminus S$, $S$ is a $u\\hbox{-}v$ separator in $G$ but no proper subset of $S$ is. A set $S$ is bisimplicial if it is the union of at most two cliques. Recognizing graphs in which every minimal separator is the union of at most $k$ cliques is polynomial-time solvable for $k=1$ and NP-hard for every fixed $k\\geq 3$. We settle the remaining case $k=2$ by giving a polynomial-time algorithm to decide whether every minimal separator in a graph is bisimplicial. A set $S$ is triad-free if its independence number $α(S)$ is at most two. Given a graph $G$ and two vertices $u,v$, the problem of finding a $u\\hbox{-}v$ separator $S$ with $α(S)\\leq k$ had the same complexity gap: it was known to be polynomial-time solvable for $k=1$ and NP-hard for every fixed $k\\geq 3$. We resolve the remaining case $k=2$ by giving a polynomial-time algorithm that finds a triad-free $u\\hbox{-}v$ separator or determines that none exists.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19093",
  "author_total": 2,
  "authors": [
    {
      "name": "Maria Chudnovsky"
    },
    {
      "name": "Ilya Maier"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fagq031m7gpza.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19093",
  "fulltext_url": null,
  "id": "fagq031m7gpza",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19093",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7222364792",
  "pdf_url": "https://arxiv.org/pdf/2610.19093",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fagq031m7gpza.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19093",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19093"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Finding non-bisimplicial and triad-free minimal separators",
  "title_text": "Finding non-bisimplicial and triad-free minimal separators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/fagq031m7gpza",
  "venue": "arXiv",
  "versions": []
}