{
  "abstract": "We study a new variant of the classical linkage problem for digraphs, which we call {\\bf unilateral linkages.} A digraph $D$ is {\\bf unilaterally $\\bm{k}$-linked} if it satisfies that for every choice of $2k$ distinct vertices $s_1,s_2,\\ldots{},s_k,t_1,t_2,\\ldots{},t_k$ there are vertex disjoint directed paths $P_1,\\ldots{},P_k$ in $D$ such that $P_i$ is either an $(s_i,t_i)$-path or a $(t_i,s_i)$-path for each $i\\in [k]$. This is a relaxation of the property of $D$ being $k$-linked, where we insist that each $P_i$ must go from $s_i$ to $t_i$. Clearly every $k$-linked digraph is also unilaterally $k$-linked. It is easy to see that the unilateral linkage problem is NP-complete for general digraphs and that it is trivial for semicomplete digraphs (digraphs with no pair of non-adjacent vertices). We show how Thomassen's construction of highly connected digraphs which are not 2-linked can be modified to show that there is no connectivity which guarantees that a digraph is unilaterally 2-linked. We show that every 4-strong split digraph and every 5-strong semicomplete multipartite digraph is unilaterally 2-linked. Both bounds are best possible. A digraph is {\\bf co-bipartite} if its vertex set can be covered by two induced semicomplete digraphs. We prove that every 6-strong co-bipartite digraph is unilaterally 2-linked. Here we do not know whether the correct answer is 5. We also prove that every 16-strong digraph with independence number at most 2 is unilaterally 2-linked.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36895",
  "author_total": 4,
  "authors": [
    {
      "name": "Jørgen Bang-Jensen"
    },
    {
      "name": "Xiaoying Chen"
    },
    {
      "name": "Panpan Cheng"
    },
    {
      "name": "Jin Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ctbw51g1vbpwq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36895",
  "fulltext_url": null,
  "id": "ctbw51g1vbpwq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36895",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7224124446",
  "pdf_url": "https://arxiv.org/pdf/2610.36895",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ctbw51g1vbpwq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36895",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36895"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The unilateral 2-linkage problem",
  "title_text": "The unilateral 2-linkage problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/ctbw51g1vbpwq",
  "venue": "arXiv",
  "versions": []
}