On strong avoiding games
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866916764990832640 |
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| author | Stojaković, Miloš Stratijev, Jelena |
| author_facet | Stojaković, Miloš Stratijev, Jelena |
| contents | Given an increasing graph property $\cal F$, the strong Avoider-Avoider $\cal F$ game is played on the edge set of a complete graph. Two players, Red and Blue, take turns in claiming previously unclaimed edges with Red going first, and the player whose graph possesses $\cal F$ first loses the game. If the property $\cal F$ is "containing a fixed graph $H$", we refer to the game as the $H$ game.
We prove that Blue has a winning strategy in two strong Avoider-Avoider games, $P_4$ game and ${\cal CC}_{>3}$ game, where ${\cal CC}_{>3}$ is the property of having at least one connected component on more than three vertices.
We also study a variant, the strong CAvoider-CAvoider games, with additional requirement that the graph of each of the players must stay connected throughout the game. We prove that Blue has a winning strategy in the strong CAvoider-CAvoider games $S_3$ and $P_4$, as well as in the $Cycle$ game, where the players aim at avoiding all cycles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_07971 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On strong avoiding games Stojaković, Miloš Stratijev, Jelena Computer Science and Game Theory 91A24, 05C57, 91A46 G.2.2 Given an increasing graph property $\cal F$, the strong Avoider-Avoider $\cal F$ game is played on the edge set of a complete graph. Two players, Red and Blue, take turns in claiming previously unclaimed edges with Red going first, and the player whose graph possesses $\cal F$ first loses the game. If the property $\cal F$ is "containing a fixed graph $H$", we refer to the game as the $H$ game. We prove that Blue has a winning strategy in two strong Avoider-Avoider games, $P_4$ game and ${\cal CC}_{>3}$ game, where ${\cal CC}_{>3}$ is the property of having at least one connected component on more than three vertices. We also study a variant, the strong CAvoider-CAvoider games, with additional requirement that the graph of each of the players must stay connected throughout the game. We prove that Blue has a winning strategy in the strong CAvoider-CAvoider games $S_3$ and $P_4$, as well as in the $Cycle$ game, where the players aim at avoiding all cycles. |
| title | On strong avoiding games |
| topic | Computer Science and Game Theory 91A24, 05C57, 91A46 G.2.2 |
| url | https://arxiv.org/abs/2204.07971 |