29 research outputs found

    Fast winning strategies in Avoider-Enforcer games

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    In numerous positional games the identity of the winner is easily determined. In this case one of the more interesting questions is not {\em who} wins but rather {\em how fast} can one win. These type of problems were studied earlier for Maker-Breaker games; here we initiate their study for unbiased Avoider-Enforcer games played on the edge set of the complete graph KnK_n on nn vertices. For several games that are known to be an Enforcer's win, we estimate quite precisely the minimum number of moves Enforcer has to play in order to win. We consider the non-planarity game, the connectivity game and the non-bipartite game

    Positional Games

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    Positional games are a branch of combinatorics, researching a variety of two-player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex, to purely abstract games played on graphs and hypergraphs. It is closely connected to many other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic combinatorics, and to computer science. We survey the basic notions of the field, its approaches and tools, as well as numerous recent advances, standing open problems and promising research directions.Comment: Submitted to Proceedings of the ICM 201

    Pozicione igre na grafovima

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    \section*{Abstract} We study Maker-Breaker games played on the edges of the complete graph on nn vertices, KnK_n, whose family of winning sets \cF consists of all edge sets of subgraphs GKnG\subseteq K_n which possess a predetermined monotone increasing property. Two players, Maker and Breaker, take turns in claiming aa, respectively bb, unclaimed edges per move. We are interested in finding the threshold bias b_{\cF}(a) for all values of aa, so that for every bb, b\leq b_{\cF}(a), Maker wins the game and for all values of bb, such that b>b_{\cF}(a), Breaker wins the game. We are particularly interested in cases where both aa and bb can be greater than 11. We focus on the \textit{Connectivity game}, where the winning sets are the edge sets of all spanning trees of KnK_n and on the  \textit{Hamiltonicity game}, where the winning sets are the edge sets of all Hamilton cycles on KnK_n. Next, we consider biased (1:b)(1:b) Avoider-Enforcer games, also played on the edges of KnK_n. For every constant k3k\geq 3 we analyse the kk-star game, where Avoider tries to avoid claiming kk edges incident to the same vertex. We analyse both versions of Avoider-Enforcer games, the strict and the monotone, and for each provide explicit winning strategies for both players. Consequentially, we establish bounds on the threshold biases f^{mon}_\cF, f^-_\cF and f^+_\cF, where \cF is the hypergraph of the game (the family of target sets). We also study the monotone version of K2,2K_{2,2}-game, where Avoider wants to avoid claiming all the edges of some graph isomorphic to K2,2K_{2,2} in KnK_n.   Finally, we search for the fast winning strategies for Maker in Perfect matching game and Hamiltonicity game, again played on the edge set of KnK_n. Here, we look at the biased (1:b)(1:b) games, where Maker's bias is 1, and Breaker's bias is b,b1b, b\ge 1.\section*{Izvod} Prou\v{c}avamo takozvane Mejker-Brejker (Maker-Breaker) igre koje se igraju na granama kompletnog grafa sa nn \v{c}vorova, KnK_n, \v{c}ija familija pobedni\v{c}kih skupova \cF obuhvata sve skupove grana grafa GKnG\subseteq K_n koji imaju neku monotono rastu\'{c}u osobinu. Dva igra\v{c}a, \textit{Mejker} (\textit{Pravi\v{s}a}) i \textit{Brejker} (\textit{Kva\-ri\-\v{s}a}) se smenjuju u odabiru aa, odnosno bb, slobodnih grana po potezu. Interesuje nas da prona\dj emo grani\v{c}ni bias b_{\cF}(a) za sve vrednosti pa\-ra\-me\-tra aa, tako da za svako bb, b\le b_{\cF}(a), Mejker pobe\dj uje u igri, a za svako bb, takvo da je b>b_{\cF}(a), Brejker pobe\dj uje. Posebno nas interesuju slu\v{c}ajevi u kojima oba parametra aa i bb mogu imati vrednost ve\'cu od 1. Na\v{s}a pa\v{z}nja je posve\'{c}ena igri povezanosti, gde su pobedni\v{c}ki skupovi  grane svih pokrivaju\'cih stabala grafa KnK_n, kao i igri Hamiltonove konture, gde su pobedni\v{c}ki skupovi grane svih Hamiltonovih kontura grafa KnK_n. Zatim posmatramo igre tipa Avojder-Enforser (Avoider-Enforcer), sa biasom (1:b)(1:b), koje se tako\dj e igraju na granama kompletnog grafa sa nn \v{c}vorova, KnK_n. Za svaku konstantu kk, k3k\ge 3 analiziramo igru kk-zvezde (zvezde sa kk krakova), u kojoj \textit{Avojder} poku\v{s}va da izbegne da ima kk svojih grana incidentnih sa istim \v{c}vorom. Posmatramo obe verzije ove igre, striktnu i monotonu, i za svaku dajemo eksplicitnu pobedni\v{c}ku strategiju za oba igra\v{c}a. Kao rezultat, dobijamo gornje i donje ograni\v{c}enje za grani\v{c}ne biase f^{mon}_\cF, f^-_\cF i f^+_\cF, gde \cF predstavlja hipergraf igre (familija ciljnih skupova). %fmonf^{mon}, ff^- and f+f^+. Tako\dj e, posmatramo i monotonu verziju K2,2K_{2,2}-igre, gde Avojder \v{z}eli da izbegne da graf koji \v{c}ine njegove grane sadr\v{z}i graf izomorfan sa K2,2K_{2,2}. Kona\v{c}no, \v{z}elimo da prona\dj emo strategije za brzu pobedu Mejkera u igrama savr\v{s}enog me\v{c}inga i Hamiltonove konture, koje se tako\dj e igraju na granama kompletnog grafa KnK_n. Ovde posmatramo asimetri\v{c}ne igre gde je bias Mejkera 1, a bias Brejkera bb, b1b\ge 1

    Mini-Workshop: Positional Games

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    Positional games is one of rapidly developing subjects of modern combinatorics, researching two player perfect information games of combinatorial nature, ranging from recreational games like Tic-Tac-Toe to purely abstract games played on graphs and hypergraphs. Though defined usually in game theoretic terms, the subject has a distinct combinatorial flavor and boasts strong mutual connections with discrete probability, Ramsey theory and randomized algorithms. This mini-workshop was dedicated to summarizing the recent progress in the subject, to indicating possible directions of future developments, and to fostering collaboration between researchers working in various, sometimes apparently distinct directions

    Avoidance Games Are PSPACE-Complete

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    Avoidance games are games in which two players claim vertices of a hypergraph and try to avoid some structures. These games are studied since the introduction of the game of SIM in 1968, but only few complexity results are known on them. In 2001, Slany proved some partial results on Avoider-Avoider games complexity, and in 2017 Bonnet et al. proved that short Avoider-Enforcer games are Co-W[1]-hard. More recently, in 2022, Miltzow and Stojakovi\'c proved that these games are NP-hard. As these games corresponds to the mis\`ere version of the well-known Maker-Breaker games, introduced in 1963 and proven PSPACE-complete in 1978, one could expect these games to be PSPACE-complete too, but the question remained open since then. We prove here that both Avoider-Avoider and Avoider-Enforcer conventions are PSPACE-complete, and as a consequence of it that some particular Avoider-Enforcer games also are

    Pozicione igre na grafovima

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    \section*{Abstract} We study Maker-Breaker games played on the edges of the complete graph on nn vertices, KnK_n, whose family of winning sets \cF consists of all edge sets of subgraphs GKnG\subseteq K_n which possess a predetermined monotone increasing property. Two players, Maker and Breaker, take turns in claiming aa, respectively bb, unclaimed edges per move. We are interested in finding the threshold bias b_{\cF}(a) for all values of aa, so that for every bb, b\leq b_{\cF}(a), Maker wins the game and for all values of bb, such that b>b_{\cF}(a), Breaker wins the game. We are particularly interested in cases where both aa and bb can be greater than 11. We focus on the \textit{Connectivity game}, where the winning sets are the edge sets of all spanning trees of KnK_n and on the  \textit{Hamiltonicity game}, where the winning sets are the edge sets of all Hamilton cycles on KnK_n. Next, we consider biased (1:b)(1:b) Avoider-Enforcer games, also played on the edges of KnK_n. For every constant k3k\geq 3 we analyse the kk-star game, where Avoider tries to avoid claiming kk edges incident to the same vertex. We analyse both versions of Avoider-Enforcer games, the strict and the monotone, and for each provide explicit winning strategies for both players. Consequentially, we establish bounds on the threshold biases f^{mon}_\cF, f^-_\cF and f^+_\cF, where \cF is the hypergraph of the game (the family of target sets). We also study the monotone version of K2,2K_{2,2}-game, where Avoider wants to avoid claiming all the edges of some graph isomorphic to K2,2K_{2,2} in KnK_n.   Finally, we search for the fast winning strategies for Maker in Perfect matching game and Hamiltonicity game, again played on the edge set of KnK_n. Here, we look at the biased (1:b)(1:b) games, where Maker's bias is 1, and Breaker's bias is b,b1b, b\ge 1.\section*{Izvod} Prou\v{c}avamo takozvane Mejker-Brejker (Maker-Breaker) igre koje se igraju na granama kompletnog grafa sa nn \v{c}vorova, KnK_n, \v{c}ija familija pobedni\v{c}kih skupova \cF obuhvata sve skupove grana grafa GKnG\subseteq K_n koji imaju neku monotono rastu\'{c}u osobinu. Dva igra\v{c}a, \textit{Mejker} (\textit{Pravi\v{s}a}) i \textit{Brejker} (\textit{Kva\-ri\-\v{s}a}) se smenjuju u odabiru aa, odnosno bb, slobodnih grana po potezu. Interesuje nas da prona\dj emo grani\v{c}ni bias b_{\cF}(a) za sve vrednosti pa\-ra\-me\-tra aa, tako da za svako bb, b\le b_{\cF}(a), Mejker pobe\dj uje u igri, a za svako bb, takvo da je b>b_{\cF}(a), Brejker pobe\dj uje. Posebno nas interesuju slu\v{c}ajevi u kojima oba parametra aa i bb mogu imati vrednost ve\'cu od 1. Na\v{s}a pa\v{z}nja je posve\'{c}ena igri povezanosti, gde su pobedni\v{c}ki skupovi  grane svih pokrivaju\'cih stabala grafa KnK_n, kao i igri Hamiltonove konture, gde su pobedni\v{c}ki skupovi grane svih Hamiltonovih kontura grafa KnK_n. Zatim posmatramo igre tipa Avojder-Enforser (Avoider-Enforcer), sa biasom (1:b)(1:b), koje se tako\dj e igraju na granama kompletnog grafa sa nn \v{c}vorova, KnK_n. Za svaku konstantu kk, k3k\ge 3 analiziramo igru kk-zvezde (zvezde sa kk krakova), u kojoj \textit{Avojder} poku\v{s}va da izbegne da ima kk svojih grana incidentnih sa istim \v{c}vorom. Posmatramo obe verzije ove igre, striktnu i monotonu, i za svaku dajemo eksplicitnu pobedni\v{c}ku strategiju za oba igra\v{c}a. Kao rezultat, dobijamo gornje i donje ograni\v{c}enje za grani\v{c}ne biase f^{mon}_\cF, f^-_\cF i f^+_\cF, gde \cF predstavlja hipergraf igre (familija ciljnih skupova). %fmonf^{mon}, ff^- and f+f^+. Tako\dj e, posmatramo i monotonu verziju K2,2K_{2,2}-igre, gde Avojder \v{z}eli da izbegne da graf koji \v{c}ine njegove grane sadr\v{z}i graf izomorfan sa K2,2K_{2,2}. Kona\v{c}no, \v{z}elimo da prona\dj emo strategije za brzu pobedu Mejkera u igrama savr\v{s}enog me\v{c}inga i Hamiltonove konture, koje se tako\dj e igraju na granama kompletnog grafa KnK_n. Ovde posmatramo asimetri\v{c}ne igre gde je bias Mejkera 1, a bias Brejkera bb, b1b\ge 1

    Pozicione igre na grafovima

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    \section*{Abstract} We study Maker-Breaker games played on the edges of the complete graph on nn vertices, KnK_n, whose family of winning sets \cF consists of all edge sets of subgraphs GKnG\subseteq K_n which possess a predetermined monotone increasing property. Two players, Maker and Breaker, take turns in claiming aa, respectively bb, unclaimed edges per move. We are interested in finding the threshold bias b_{\cF}(a) for all values of aa, so that for every bb, b\leq b_{\cF}(a), Maker wins the game and for all values of bb, such that b>b_{\cF}(a), Breaker wins the game. We are particularly interested in cases where both aa and bb can be greater than 11. We focus on the \textit{Connectivity game}, where the winning sets are the edge sets of all spanning trees of KnK_n and on the  \textit{Hamiltonicity game}, where the winning sets are the edge sets of all Hamilton cycles on KnK_n. Next, we consider biased (1:b)(1:b) Avoider-Enforcer games, also played on the edges of KnK_n. For every constant k3k\geq 3 we analyse the kk-star game, where Avoider tries to avoid claiming kk edges incident to the same vertex. We analyse both versions of Avoider-Enforcer games, the strict and the monotone, and for each provide explicit winning strategies for both players. Consequentially, we establish bounds on the threshold biases f^{mon}_\cF, f^-_\cF and f^+_\cF, where \cF is the hypergraph of the game (the family of target sets). We also study the monotone version of K2,2K_{2,2}-game, where Avoider wants to avoid claiming all the edges of some graph isomorphic to K2,2K_{2,2} in KnK_n.   Finally, we search for the fast winning strategies for Maker in Perfect matching game and Hamiltonicity game, again played on the edge set of KnK_n. Here, we look at the biased (1:b)(1:b) games, where Maker's bias is 1, and Breaker's bias is b,b1b, b\ge 1.\section*{Izvod} Prou\v{c}avamo takozvane Mejker-Brejker (Maker-Breaker) igre koje se igraju na granama kompletnog grafa sa nn \v{c}vorova, KnK_n, \v{c}ija familija pobedni\v{c}kih skupova \cF obuhvata sve skupove grana grafa GKnG\subseteq K_n koji imaju neku monotono rastu\'{c}u osobinu. Dva igra\v{c}a, \textit{Mejker} (\textit{Pravi\v{s}a}) i \textit{Brejker} (\textit{Kva\-ri\-\v{s}a}) se smenjuju u odabiru aa, odnosno bb, slobodnih grana po potezu. Interesuje nas da prona\dj emo grani\v{c}ni bias b_{\cF}(a) za sve vrednosti pa\-ra\-me\-tra aa, tako da za svako bb, b\le b_{\cF}(a), Mejker pobe\dj uje u igri, a za svako bb, takvo da je b>b_{\cF}(a), Brejker pobe\dj uje. Posebno nas interesuju slu\v{c}ajevi u kojima oba parametra aa i bb mogu imati vrednost ve\'cu od 1. Na\v{s}a pa\v{z}nja je posve\'{c}ena igri povezanosti, gde su pobedni\v{c}ki skupovi  grane svih pokrivaju\'cih stabala grafa KnK_n, kao i igri Hamiltonove konture, gde su pobedni\v{c}ki skupovi grane svih Hamiltonovih kontura grafa KnK_n. Zatim posmatramo igre tipa Avojder-Enforser (Avoider-Enforcer), sa biasom (1:b)(1:b), koje se tako\dj e igraju na granama kompletnog grafa sa nn \v{c}vorova, KnK_n. Za svaku konstantu kk, k3k\ge 3 analiziramo igru kk-zvezde (zvezde sa kk krakova), u kojoj \textit{Avojder} poku\v{s}va da izbegne da ima kk svojih grana incidentnih sa istim \v{c}vorom. Posmatramo obe verzije ove igre, striktnu i monotonu, i za svaku dajemo eksplicitnu pobedni\v{c}ku strategiju za oba igra\v{c}a. Kao rezultat, dobijamo gornje i donje ograni\v{c}enje za grani\v{c}ne biase f^{mon}_\cF, f^-_\cF i f^+_\cF, gde \cF predstavlja hipergraf igre (familija ciljnih skupova). %fmonf^{mon}, ff^- and f+f^+. Tako\dj e, posmatramo i monotonu verziju K2,2K_{2,2}-igre, gde Avojder \v{z}eli da izbegne da graf koji \v{c}ine njegove grane sadr\v{z}i graf izomorfan sa K2,2K_{2,2}. Kona\v{c}no, \v{z}elimo da prona\dj emo strategije za brzu pobedu Mejkera u igrama savr\v{s}enog me\v{c}inga i Hamiltonove konture, koje se tako\dj e igraju na granama kompletnog grafa KnK_n. Ovde posmatramo asimetri\v{c}ne igre gde je bias Mejkera 1, a bias Brejkera bb, b1b\ge 1

    Fast winning strategies in positional games

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    Abstract For the unbiased Maker-Breaker game, played on the hypergraph H, let τ M (H) be the smallest integer t such that Maker can win the game within t moves (if the game is a Breaker's win then set τ M (H) = ∞). Similarly, for the unbiased Avoider-Enforcer game played on H, let τ E (H) be the smallest integer t such that Enforcer can win the game within t moves (if the game is an Avoider's win then set τ E (H) = ∞). In this paper, we investigate τ M and τ E and determine their value for various positional games
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