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  "slug": "median-radius-obstructions",
  "title": "Superexponential strict-improvement distances in median graphs",
  "shortTitle": "When a better graph center is far away",
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  "oneLine": "Integer separation forces very long strictly improving moves, even in median graphs.",
  "abstract": "This unrefereed candidate realizes minimum integer separation norm exactly as the distance to a strict radius improvement on a grid, then transfers the construction to ordinary eccentricity without increasing cube dimension. An explicit Fibonacci family disproves dimension-linear locality. An attributed Alon–Vũ theorem gives a superexponential lower bound in every sufficiently large dimension, with a matching leading logarithmic exponent for corner profiles. This is not an algorithmic running-time lower bound.",
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    "median graphs",
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    "integer separation"
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    "Exact Fibonacci first-improvement distance 2F_d − 1 in every odd dimension d ≥ 3.",
    "Dimension-preserving transfer from integer separation to ordinary graph eccentricity.",
    "All-dimensions superexponential lower bound using Alon–Vũ.",
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    "Determine upper locality bounds for ordinary eccentricity on all finite median graphs.",
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    "Find smaller graphs realizing large strict-improvement distances."
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    {
      "citation": "Ducoffe (2026), Radius functions in median graphs: source Open Problem 2, p. 506.",
      "url": "https://ems.press/content/serial-article-files/53603",
      "doi": null
    },
    {
      "citation": "Alon and Vũ (1997), Anti-Hadamard matrices, coin weighing, threshold gates, and indecomposable hypergraphs: external large-weight theorem.",
      "url": "https://doi.org/10.1006/jcta.1997.2780",
      "doi": "10.1006/jcta.1997.2780"
    },
    {
      "citation": "Bergé, Ducoffe and Habib (2025), Quasilinear-time eccentricities computation, and more, on median graphs.",
      "url": "https://arxiv.org/abs/2410.10235",
      "doi": "10.48550/arXiv.2410.10235"
    },
    {
      "citation": "Chalopin et al. (2025), On G^p-unimodality of radius functions in graphs.",
      "url": "https://arxiv.org/abs/2503.15011",
      "doi": "10.48550/arXiv.2503.15011"
    }
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