{
  "id": "2105.02312",
  "version": "v1",
  "published": "2021-05-05T20:21:42.000Z",
  "updated": "2021-05-05T20:21:42.000Z",
  "title": "Lower Bound and Exact Values for the Boundary Independence Broadcast Number of a Tree",
  "authors": [
    "C. M. Mynhardt",
    "L. Neilson"
  ],
  "comment": "15 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:2104.02266",
  "categories": [
    "math.CO"
  ],
  "abstract": "A broadcast on a nontrivial connected graph G is a function f from V(G) to the set {0,1,...,diam(G)} such that f(v) is at most the eccentricity of v for all vertices v of G. The weight of f is the sum of the function values over V(G). A vertex u hears f from v if f(v) is positive and u is within distance f(v) from v. A broadcast f is boundary independent if any vertex that hears f from two or more vertices is at distance f(v) from each such vertex v. The maximum weight of a boundary independent broadcast on G is denoted by {\\alpha}_{bn}(G). We prove a sharp lower bound on {\\alpha}_{bn}(T) for a tree T. Combined with a previously determined upper bound, this gives exact values of {\\alpha}_{bn}(T) for some classes of trees T. We also determine {\\alpha}_{bn}(T) for trees with exactly two branch vertices and use this result to demonstrate the existence of trees for which {\\alpha}_{bn} lies strictly between the lower and upper bounds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-05T20:21:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69"
    ],
    "keywords": [
      "boundary independence broadcast number",
      "exact values",
      "upper bound",
      "boundary independent broadcast",
      "sharp lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}