Bounds on the game isolation number and exact values for paths and cycles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908929043202048 |
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| author | Bujtás, Csilla Dravec, Tanja Henning, Michael A. Klavžar, Sandi |
| author_facet | Bujtás, Csilla Dravec, Tanja Henning, Michael A. Klavžar, Sandi |
| contents | 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 $ι_{\rm g}(G)$ 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 $ι_{\rm g}'(G)$. In this paper, $ι_{\rm g}(C_n)$, $ι_{\rm g}(P_n)$, $ι_{\rm g}'(C_n)$, and $ι_{\rm g}'(P_n)$ are determined for all $n$. It is proved that there are only two graphs that attain equality in the upper bound $ι_{\rm g}(G) \le \frac{1}{2}|V(G)|$, and that there are precisely eleven graphs which attain equality in the upper bound $ι_{\rm g}'(G) \le \frac{1}{2}|V(G)|$. For trees $T$ of order at least three it is proved that $ι_{\rm g}(T) \le \frac{5}{11}|V(T)|$. A new infinite family of graphs $G$ is also constructed for which $ι_{\rm g}(G) = ι_{\rm g}'(G) = \frac{3}{7}|V(G)|$ holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08503 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds on the game isolation number and exact values for paths and cycles Bujtás, Csilla Dravec, Tanja Henning, Michael A. Klavžar, Sandi Combinatorics 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 $ι_{\rm g}(G)$ 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 $ι_{\rm g}'(G)$. In this paper, $ι_{\rm g}(C_n)$, $ι_{\rm g}(P_n)$, $ι_{\rm g}'(C_n)$, and $ι_{\rm g}'(P_n)$ are determined for all $n$. It is proved that there are only two graphs that attain equality in the upper bound $ι_{\rm g}(G) \le \frac{1}{2}|V(G)|$, and that there are precisely eleven graphs which attain equality in the upper bound $ι_{\rm g}'(G) \le \frac{1}{2}|V(G)|$. For trees $T$ of order at least three it is proved that $ι_{\rm g}(T) \le \frac{5}{11}|V(T)|$. A new infinite family of graphs $G$ is also constructed for which $ι_{\rm g}(G) = ι_{\rm g}'(G) = \frac{3}{7}|V(G)|$ holds. |
| title | Bounds on the game isolation number and exact values for paths and cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.08503 |