{
  "id": "2311.07219",
  "version": "v1",
  "published": "2023-11-13T10:29:22.000Z",
  "updated": "2023-11-13T10:29:22.000Z",
  "title": "On Blockers and Transversals of Maximum Independent Sets in Co-Comparability Graphs",
  "authors": [
    "Felicia Lucke",
    "Bernard Ries"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "In this paper, we consider the following two problems: (i) Deletion Blocker($\\alpha$) where we are given an undirected graph $G=(V,E)$ and two integers $k,d\\geq 1$ and ask whether there exists a subset of vertices $S\\subseteq V$ with $|S|\\leq k$ such that $\\alpha(G-S) \\leq \\alpha(G)-d$, that is the independence number of $G$ decreases by at least $d$ after having removed the vertices from $S$; (ii) Transversal($\\alpha$) where we are given an undirected graph $G=(V,E)$ and two integers $k,d\\geq 1$ and ask whether there exists a subset of vertices $S\\subseteq V$ with $|S|\\leq k$ such that for every maximum independent set $I$ we have $|I\\cap S| \\geq d$. We show that both problems are polynomial-time solvable in the class of co-comparability graphs by reducing them to the well-known Vertex Cut problem. Our results generalize a result of [Chang et al., Maximum clique transversals, Lecture Notes in Computer Science 2204, pp. 32-43, WG 2001] and a recent result of [Hoang et al., Assistance and interdiction problems on interval graphs, Discrete Applied Mathematics 340, pp. 153-170, 2023].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-13T10:29:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximum independent set",
      "co-comparability graphs",
      "well-known vertex cut problem",
      "maximum clique transversals",
      "undirected graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}