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  "path": "/abs/2602.22980v1",
  "publishedAt": "2026-02-27T01:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Karl Bartolo",
    "Peter Borg",
    "Magda Dettlaff",
    "Magdalena Lemańska",
    "Paweł Żyliński"
  ],
  "textContent": "**Authors:** Karl Bartolo, Peter Borg, Magda Dettlaff, Magdalena Lemańska, Paweł Żyliński\n\nThis paper introduces the notion of $(ι,q)$-critical graphs. The isolation number of a graph $G$, denoted by $ι(G)$ and also known as the vertex-edge domination number, is the minimum number of vertices in a set $D$ such that the subgraph induced by the vertices not in the closed neighbourhood of $D$ has no edges. A graph $G$ is $(ι,q)$-critical, $q \\ge 1$, if the subdivision of any $q$ edges in $G$ gives a graph with isolation number greater than $ι(G)$ and there exists a set of $q-1$ edges such that subdividing them gives a graph with isolation number equal to $ι(G)$. We prove that for each integer $q \\ge 1$ there exists a $(ι,q)$-critical graph, while for a given graph $G$, the admissible values of $q$ satisfy $1 \\le q \\le |E(G)| - 1$. In addition, we provide a general characterisation of $(ι,1)$-critical graphs as well as a constructive characterisation of $(ι,1)$-critical trees.",
  "title": "Isolation critical graphs under multiple edge subdivision"
}