{
  "abstract": "Given a finite planar point set $X=\\{x_1,\\ldots,x_n\\}$, let $a_1<a_2< \\cdots < a_s$ denote the $s \\leq {n \\choose 2}$ different (Euclidean) distances between points in $X$. The aim is to retain as many of the points as possible such that certain imposed inter-pair distances are avoided. For example, we show that: (A) Given a set $X$ of $n$ points in the plane, an independent set of size $0.2293 n$ in the unit distance graph determined by $X$ can be found in $O(n^2)$ time. For any fixed $k \\in \\mathbf{N}$ there exists a constant $c_k>0$ such that for any set $X$ of $n$ points in the plane, and set $D$ of size $k$, there exists a subset $X' \\subseteq X$ of size $|X'| \\geq c_k |X|$ in which no pair of points is at some distance in $D$ (we say that $X'$ avoids distances in $D$). Such a set can be found efficiently. (B) (i) There is a subset of $X$ of size $n/2$ that avoids the maximum distance; this bound cannot be improved. (ii) There is a subset of $X$ of size $8n/47$ that avoids both the minimum and maximum distance; on the other hand, there exists $X$ such that at most $5n/16$ points can be retained with this property. (iii) There is a subset of $X$ of size $n/7$ whose diameter is at most half the original diameter; on the other hand, there exists $X$ such that at most $n/6$ points can be retained with this property. (C) (i) For any $\\varepsilon>0$, a $(1-\\varepsilon)$-approximation for \\textsc{MIS} in the minimum distance graph of $n$ points in the plane can be found in $2^{O(1/\\varepsilon)} n^{O(1)}$ time. (ii) Given a set $X$ of $n$ points in the plane, a maximum independent set in the graph of diameters can be computed in $O(n^2)$ time. The techniques used to handle the planar case can be extended to $\\mathbf{R}^d$ for any fixed dimension $d$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24803",
  "author_total": 2,
  "authors": [
    {
      "name": "Adrian Dumitrescu"
    },
    {
      "name": "Zsolt Lángi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2543gw27w57wk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24803",
  "fulltext_url": null,
  "id": "2543gw27w57wk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24803",
  "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",
      "cs.CG"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1704|Computational Geometry and Mesh Generation"
    ]
  ],
  "openalex": "W7223032010",
  "pdf_url": "https://arxiv.org/pdf/2610.24803",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2543gw27w57wk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24803",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24803"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Avoiding Distances",
  "title_text": "Avoiding Distances",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/2543gw27w57wk",
  "venue": "arXiv",
  "versions": []
}