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arXiv · 2606.21190

Isolation subdivision number of a graph

Magda Dettlaff·Magdalena Lemańska·Merce Mora·Radosław Ziemann·Paweł Żyliński
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Abstract

For a graph $G=(V,E),$ a set $S \subseteq V$ is called an isolating set of $G$ if the set $V-N[S]$ is independent. The minimum cardinality of an isolating set in $G$ is the isolation number of $G$, denoted by $\iota(G).$ Here we introduce the isolation subdivision number of a graph $G$, denoted by ${\rm sd}_\iota(G)$, as the minimum number of edges of $G$ that must be subdivided, where each edge can be subdivided at most once, in order to obtain a graph with isolation number greater than $\iota(G).$ We show that the new parameter is well defined for any non-trivial graph different from a star and that it can be arbitrarily large. We present the values of this parameter for some elementary classes of graphs and establish some basic properties. We show also that $1\leq {\rm sd}_\iota(T)\leq 4$ for any tree $T$ different from a star and characterize all trees $T$ with ${\rm sd}_\iota(T)=1.$

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Magda Dettlaff, Magdalena Lemańska, Merce Mora, Radosław Ziemann, Paweł Żyliński. 2026-06-19. Isolation subdivision number of a graph. https://arxiv.org/abs/2606.21190

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