{
  "id": "2111.03298",
  "version": "v1",
  "published": "2021-11-05T07:21:50.000Z",
  "updated": "2021-11-05T07:21:50.000Z",
  "title": "Relating the total domination number and the annihilation number for quasi-trees and some composite graphs",
  "authors": [
    "Hongbo Hua",
    "Xinying Hua",
    "Sandi Klavžar",
    "Kexiang Xu"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The total domination number $\\gamma_{t}(G)$ of a graph $G$ is the cardinality of a smallest set $D\\subseteq V(G)$ such that each vertex of $G$ has a neighbor in $D$. The annihilation number $a(G)$ of $G$ is the largest integer $k$ such that there exist $k$ different vertices in $G$ with the degree sum at most $m(G)$. It is conjectured that $\\gamma_{t}(G)\\leq a(G)+1$ holds for every nontrivial connected graph $G$. The conjecture has been proved for graphs with minimum degree at least $3$, trees, certain tree-like graphs, block graphs, and cactus graphs. In the main result of this paper it is proved that the conjecture holds for quasi-trees. The conjecture is verified also for some graph constructions including bijection graphs, Mycielskians, and the newly introduced universally-identifying graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-11-05T07:21:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "total domination number",
      "annihilation number",
      "composite graphs",
      "quasi-trees",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}