{
  "abstract": "Given a simple graph $G = (V, E)$ on $v$ vertices and two distinct vertices $x, y \\in V$, the co-degree $c_{x,y}$ associated to the pair $\\{x, y\\}$ is the number of their common neighbors in the graph $G$. The co-degree sequence of $G$, denoted by $γ(G)$, is the list of all the co-degrees associated to all the possible pairs of distinct vertices, arranged in non-increasing order. In this paper we consider the following problem, which can be viewed as a generalization of a result by Erdős and Gallai as well as of the Erdős, Rényi and Sós' friendship theorem: given an integer $v\\geqslant 2$ and a sequence $γ$ of nonnegative integers arranged in non-increasing order, establish if there exists a simple graph on $v$ vertices having $γ$ as its co-degree sequence and, in case of positive answer, provide such a graph. We provide a full answer to this problem for the class of planar $C_4$-free graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08803",
  "author_total": 4,
  "authors": [
    {
      "name": "Michela Ascolese"
    },
    {
      "name": "Pietro Negrini"
    },
    {
      "name": "Silvia Maria Carla Pagani"
    },
    {
      "name": "Marco Antonio Pellegrini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e5hxz14nzbjkr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08803",
  "fulltext_url": null,
  "id": "e5hxz14nzbjkr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08803",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7212077265",
  "pdf_url": "https://arxiv.org/pdf/2609.08803",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e5hxz14nzbjkr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08803",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08803"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A graph reconstruction problem involving common neighbors",
  "title_text": "A graph reconstruction problem involving common neighbors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/e5hxz14nzbjkr",
  "venue": "arXiv",
  "versions": []
}