{
  "id": "2412.15950",
  "version": "v1",
  "published": "2024-12-20T14:45:37.000Z",
  "updated": "2024-12-20T14:45:37.000Z",
  "title": "Maximal independent sets in graphs with given matching number",
  "authors": [
    "Yongtang Shi",
    "Jianhua Tu",
    "Ziyuan Wang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In a graph $G$, a maximal independent set $I$ is a set of vertices such that no two vertices are adjacent, and the addition of any vertex $v$ not in $I$ would result in a set that is no longer independent. An induced triangle matching in a graph is a set of vertex disjoint triangles forming an induced subgraph, with its size being the number of these triangles. Palmer and Patk\\'{o}s [J. Graph Theory 104 (2023) 440--445] have made a significant contribution by establishing an upper bound on the number of maximal independent sets for graphs of given order that do not contain an induced triangle matching of size $t+1$. Building on their work, this paper extends the analysis to determine the maximum number of maximal independent sets in general graphs, connected graphs, triangle-free graphs, and connected triangle-free graphs with given matching number. Additionally, we characterize the corresponding extremal graphs that achieve this maximum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-20T14:45:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "05C30",
      "05C70"
    ],
    "keywords": [
      "maximal independent set",
      "matching number",
      "induced triangle matching",
      "corresponding extremal graphs",
      "graph theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}