{
  "annotation_id": "872b40d2-0488-4782-b3f0-e2dcfabacbf4",
  "date_created": "2026-02-17T05:53:15.946000Z",
  "date_modified": "2026-02-17T05:53:15.946000Z",
  "file_hash": "54f3b1a1acb9f1c0ee6b6404d435ab1740ad80d9c9fc39556184630969919744",
  "private": false,
  "record": {
    "abstract": "The notion of mutual visibility in graphs arises from constraining shortest paths by forbidding internal vertices from belonging to a specified subset. Mutual-visibility sets, originally introduced as a tool for studying information flow and structural restrictions in complex networks, have since gained increasing attention due to their theoretical significance and diverse applications. In this paper, a complete characterization of mutual-visibility sets in trees is presented. It is shown that a subset $S$ is a mutual-visibility set of $T$ if and only if it coincides with the set of leaves of the Steiner subtree $T\\langle S\\rangle$. As a consequence, the mutual-visibility number of a tree is equal to the number of its leaves. For trees containing branch vertices, the notion of legs is introduced and an explicit formula for the number of maximal mutual-visibility sets is derived in terms of the corresponding leg lengths. It is proved that every tree is absolute-clear. It is further established that the mutual-visibility number is preserved under the line graph operation for trees with at least two edges, that is, $\\mu(L(T))=\\mu(T)$.",
    "arxiv_id": "2601.08270",
    "authors": [
      "Tonny K B",
      "Shikhi M"
    ],
    "categories": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "On the Mutual-Visibility of Tree Graphs",
    "url": "https://arxiv.org/abs/2601.08270",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "5f76d93d-6fac-46aa-ae04-291368443439",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}