| Title: | Isolation game on graphs |
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| Authors: | ID Brešar, Boštjan (Author)
ID Dravec, Tanja (Author)
ID Johnston, Daniel P. (Author)
ID Kuenzel, Kirsti (Author)
ID Rall, Douglas F. (Author) |
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| Files: |  PDF - Presentation file, download (290,47 KB)
MD5: 23E1043F7BD1E0BF0D9C00E499EC664E
 URL - Source URL, visit https://link.springer.com/article/10.1007/s00373-026-03030-y
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | Given a graph $G$ and a family of graphs $\cal F$, an $\cal F$-isolating set, as introduced by Caro and Hansberg, is any set $S\subset V(G)$ such that $G - N[S]$ contains no member of $\cal F$ as a subgraph. In this paper, we introduce a game in which two players with opposite goals are together building an $\cal F$-isolating set in $G$. Following the domination games, Dominator (Staller) wants that the resulting $\cal F$-isolating set obtained at the end of the game, is as small (as big) as possible, which leads to the graph invariant called the game $\cal F$-isolation number, denoted $\iota_{\rm g}(G,\cal F)$. We prove that the Continuation Principle holds in the $\cal F$-isolation game, and that the difference between the game $\cal F$-isolation numbers when either Dominator or Staller starts the game is at most $1$. Considering two arbitrary families of graphs $\cal F$ and $\cal F'$, we find relations between them that ensure $\iota_{\rm g}(G,{\mathcal{F}}') \leq \iota_{\rm g}(G,{\mathcal{F}})$ for any graph $G$. A special focus is given on the isolation game, which takes place when ${\cal F}=\{K_2\}$. We prove that $\iota_{\rm g}(G,\{K_2\})\le |V(G)|/2$ for any graph $G$, and conjecture that $\lceil 3|V(G)|/7\rceil$ is the actual (sharp) upper bound. We prove that the isolation game on a forest when Dominator has the first move never lasts longer than the one in which Staller starts the game. Finally, we prove good lower and upper bounds on the game isolation numbers of paths $P_n$, which lead to the exact values $\iota_{\rm g}(P_n,\{K_2\})=\left\lfloor\frac{2n+2}{5}\right\rfloor$ when $n \equiv i \pmod 5$ and $i \in \{1,2,3\}$. |
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| Keywords: | isolation number, graph games, domination games, continuation principle, forest |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.04.2026 |
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| Year of publishing: | 2026 |
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| Number of pages: | 13 str. |
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| Numbering: | Vol. 42, iss. 2, article no. 30 |
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| PID: | 20.500.12556/DiRROS-27965  |
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| UDC: | 519.17 |
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| ISSN on article: | 0911-0119 |
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| DOI: | 10.1007/s00373-026-03030-y |
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| COBISS.SI-ID: | 270312963 |
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| Note: |
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| Publication date in DiRROS: | 03.03.2026 |
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| Views: | 38 |
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| Downloads: | 15 |
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| Metadata: |    |
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