{
  "abstract": "A two-terminal graph is a simple graph equipped with two distinguished vertices, called terminals. Let $T_{n,m}$ be the class consisting of all non-isomorphic two-terminal graphs on $n$ vertices and $m$ edges. Let $G$ be any two-terminal graph in $T_{n,m}$, and let $d$ be any positive integer. For every $ρ\\in [0,1]$, the \\emph{$d$-constrained two-terminal reliability of $G$ at $ρ$}, denoted $R_G^d(ρ)$, is the probability that $G$ has some path of length at most $d$ joining its terminals after each of its edges is independently deleted with probability $ρ$. We say $G$ is a \\emph{$d$-uniformly most reliable two-terminal graph} ($d$-UMRTTG) if for each $H$ in $T_{n,m}$ and every $ρ\\in [0,1]$ it holds that $R_{G}^d(ρ)\\geq R_{H}^{d}(ρ)$. Previous works studied the existence of $d$-UMRTTG in $T_{n,m}$ when $d$ is greater than or equal to $n-1$, or equivalently, when the distance constraint is dropped. In this work, a characterization of all $1$-UMRTTGs and $2$-UMRTTGs is given. Several structural properties of 3-UMRTTGs are established. For every $d\\geq 3$, we prove the existence of a unique $d$-UMRTTG in $T_{n,\\binom{n}{2}-1}$. In contrast, for every $d\\geq 4$ and every $n\\geq 11$, no $d$-UMRTTG exists in $T_{n,m}$ whenever $20\\leq m\\leq 3n-9$ or $3n-5\\leq m\\leq \\binom{n}{2}-2$. Additionally, for each $d\\geq 5$, it is proved that no $d$-UMRTTG exists in $T_{n,m}$ when $n\\geq 11$ and $m=3n-8$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36120",
  "author_total": 1,
  "authors": [
    {
      "name": "Pablo Romero"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8jhhv5yfr8vn9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36120",
  "fulltext_url": null,
  "id": "8jhhv5yfr8vn9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36120",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7224081480",
  "pdf_url": "https://arxiv.org/pdf/2610.36120",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8jhhv5yfr8vn9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36120",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36120"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Existence of uniformly most reliable two-terminal graphs with distance constraints",
  "title_text": "Existence of uniformly most reliable two-terminal graphs with distance constraints",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/8jhhv5yfr8vn9",
  "venue": "arXiv",
  "versions": []
}