| Title: | Bounds on the game isolation number and exact values for paths and cycles |
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| Authors: | ID Bujtás, Csilla (Author)
ID Dravec, Tanja (Author)
ID Henning, Michael A. (Author)
ID Klavžar, Sandi (Author) |
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| Files: |  PDF - Presentation file, download (416,35 KB)
MD5: 2ACD69B2C51D44023D5F9D8BEE74C84A
 URL - Source URL, visit https://dmtcs.episciences.org/18488
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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: | The isolation game is played on a graph $G$ by two players who take turns playing a vertex such that if $X$ is the set of already played vertices, then a vertex can be selected only if it dominates a vertex from a nontrivial component of $G \setminus N_G[X]$, where $N_G[X]$ is the set of vertices in $X$ or adjacent to a vertex in $X$. Dominator wishes to finish the game with the minimum number of played vertices, while Staller has the opposite goal. The game isolation number $\iota_{\rm g}(G)$ is the number of moves in the Dominator-start game where both players play optimally. If Staller starts the game the invariant is denoted by $\iota_{\rm g}'(G)$. In this paper, $\iota_{\rm g}(C_n)$, $\iota_{\rm g}(P_n)$, $\iota_{\rm g}'(C_n)$, and $\iota_{\rm g}'(P_n)$ are determined for all $n$. It is proved that there are only two graphs that attain equality in the upper bound $\iota_{\rm g}(G) \le \frac{1}{2}|V(G)|$, and that there are precisely eleven graphs which attain equality in the upper bound $\iota_{\rm g}'(G) \le \frac{1}{2}|V(G)|$. For trees $T$ of order at least three it is proved that $\iota_{\rm g}(T) \le \frac{5}{11}|V(T)|$. A new infinite family of graphs $G$ is also constructed for which $\iota_{\rm g}(G) = \iota_{\rm g}'(G) = \frac{3}{7}|V(G)|$ holds. |
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| Keywords: | isolating set, isolation game, paths and cycles, trees |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Year of publishing: | 2026 |
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| Number of pages: | 22 str. |
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| Numbering: | Vol. 28, no. 2 article no. 33 |
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| PID: | 20.500.12556/DiRROS-30029  |
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| UDC: | 519.17 |
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| ISSN on article: | 1365-8050 |
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| DOI: | 10.46298/dmtcs.16132 |
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| COBISS.SI-ID: | 281382915 |
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| Note: |
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| Publication date in DiRROS: | 12.06.2026 |
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| Views: | 82 |
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| Downloads: | 67 |
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