Fast strategies in biased Maker--Breaker games

We study the biased $(1:b)$ Maker--Breaker positional games, played on the edge set of the complete graph on $n$ vertices, $K_n$. Given Breaker's bias $b$, possibly depending on $n$, we determine the bounds for the minimal number of moves, depending on $b$, in which Maker can win in each of the two standard graph games, the Perfect Matching game and the Hamilton Cycle game

Maker-Breaker total domination games on cubic graphs

We study Maker-Breaker total domination game played by two players, Dominator and Staller on the connected cubic graphs. Staller (playing the role of Maker) wins if she manages to claim an open neighbourhood of a vertex. Dominator wins otherwise (i.e. if he can claim a total dominating set of a graph). For certain graphs on $n\geq 6$ vertices, we give the characterization on those which are Dominator's win and those which are Staller's win

Pozicione igre na grafovima

\section*{Abstract} We study Maker-Breaker games played on the edges of the complete graph on $n$ vertices, $K_n$, whose family of winning sets \cF consists of all edge sets of subgraphs $G\subseteq K_n$ which possess a predetermined monotone increasing property. Two players, Maker and Breaker, take turns in claiming $a$, respectively $b$, unclaimed edges per move. We are interested in finding the threshold bias b_{\cF}(a) for all values of $a$, so that for every $b$, b\leq b_{\cF}(a), Maker wins the game and for all values of $b$, such that b>b_{\cF}(a), Breaker wins the game. We are particularly interested in cases where both $a$ and $b$ can be greater than $1$. We focus on the \textit{Connectivity game}, where the winning sets are the edge sets of all spanning trees of $K_n$ and on the  \textit{Hamiltonicity game}, where the winning sets are the edge sets of all Hamilton cycles on $K_n$. Next, we consider biased $(1:b)$ Avoider-Enforcer games, also played on the edges of $K_n$. For every constant $k\geq 3$ we analyse the $k$-star game, where Avoider tries to avoid claiming $k$ 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 $K_{2,2}$-game, where Avoider wants to avoid claiming all the edges of some graph isomorphic to $K_{2,2}$ in $K_n$.   Finally, we search for the fast winning strategies for Maker in Perfect matching game and Hamiltonicity game, again played on the edge set of $K_n$. Here, we look at the biased $(1:b)$ games, where Maker's bias is 1, and Breaker's bias is $b, b\ge 1$.\section*{Izvod} Prou\v{c}avamo takozvane Mejker-Brejker (Maker-Breaker) igre koje se igraju na granama kompletnog grafa sa $n$ \v{c}vorova, $K_n$, \v{c}ija familija pobedni\v{c}kih skupova \cF obuhvata sve skupove grana grafa $G\subseteq K_n$ 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 $a$, odnosno $b$, 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 $a$, tako da za svako $b$, b\le b_{\cF}(a), Mejker pobe\dj uje u igri, a za svako $b$, takvo da je b>b_{\cF}(a), Brejker pobe\dj uje. Posebno nas interesuju slu\v{c}ajevi u kojima oba parametra $a$ i $b$ 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 $K_n$, kao i igri Hamiltonove konture, gde su pobedni\v{c}ki skupovi grane svih Hamiltonovih kontura grafa $K_n$. Zatim posmatramo igre tipa Avojder-Enforser (Avoider-Enforcer), sa biasom $(1:b)$, koje se tako\dj e igraju na granama kompletnog grafa sa $n$ \v{c}vorova, $K_n$. Za svaku konstantu $k$, $k\ge 3$ analiziramo igru $k$-zvezde (zvezde sa $k$ krakova), u kojoj \textit{Avojder} poku\v{s}va da izbegne da ima $k$ 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). %$f^{mon}$, $f^-$ and $f^+$. Tako\dj e, posmatramo i monotonu verziju $K_{2,2}$-igre, gde Avojder \v{z}eli da izbegne da graf koji \v{c}ine njegove grane sadr\v{z}i graf izomorfan sa $K_{2,2}$. 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 $K_n$. Ovde posmatramo asimetri\v{c}ne igre gde je bias Mejkera 1, a bias Brejkera $b$, $b\ge 1$

Pozicione igre na grafovima

\section*{Abstract} We study Maker-Breaker games played on the edges of the complete graph on $n$ vertices, $K_n$, whose family of winning sets \cF consists of all edge sets of subgraphs $G\subseteq K_n$ which possess a predetermined monotone increasing property. Two players, Maker and Breaker, take turns in claiming $a$, respectively $b$, unclaimed edges per move. We are interested in finding the threshold bias b_{\cF}(a) for all values of $a$, so that for every $b$, b\leq b_{\cF}(a), Maker wins the game and for all values of $b$, such that b>b_{\cF}(a), Breaker wins the game. We are particularly interested in cases where both $a$ and $b$ can be greater than $1$. We focus on the \textit{Connectivity game}, where the winning sets are the edge sets of all spanning trees of $K_n$ and on the  \textit{Hamiltonicity game}, where the winning sets are the edge sets of all Hamilton cycles on $K_n$. Next, we consider biased $(1:b)$ Avoider-Enforcer games, also played on the edges of $K_n$. For every constant $k\geq 3$ we analyse the $k$-star game, where Avoider tries to avoid claiming $k$ 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 $K_{2,2}$-game, where Avoider wants to avoid claiming all the edges of some graph isomorphic to $K_{2,2}$ in $K_n$.   Finally, we search for the fast winning strategies for Maker in Perfect matching game and Hamiltonicity game, again played on the edge set of $K_n$. Here, we look at the biased $(1:b)$ games, where Maker's bias is 1, and Breaker's bias is $b, b\ge 1$.\section*{Izvod} Prou\v{c}avamo takozvane Mejker-Brejker (Maker-Breaker) igre koje se igraju na granama kompletnog grafa sa $n$ \v{c}vorova, $K_n$, \v{c}ija familija pobedni\v{c}kih skupova \cF obuhvata sve skupove grana grafa $G\subseteq K_n$ 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 $a$, odnosno $b$, 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 $a$, tako da za svako $b$, b\le b_{\cF}(a), Mejker pobe\dj uje u igri, a za svako $b$, takvo da je b>b_{\cF}(a), Brejker pobe\dj uje. Posebno nas interesuju slu\v{c}ajevi u kojima oba parametra $a$ i $b$ 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 $K_n$, kao i igri Hamiltonove konture, gde su pobedni\v{c}ki skupovi grane svih Hamiltonovih kontura grafa $K_n$. Zatim posmatramo igre tipa Avojder-Enforser (Avoider-Enforcer), sa biasom $(1:b)$, koje se tako\dj e igraju na granama kompletnog grafa sa $n$ \v{c}vorova, $K_n$. Za svaku konstantu $k$, $k\ge 3$ analiziramo igru $k$-zvezde (zvezde sa $k$ krakova), u kojoj \textit{Avojder} poku\v{s}va da izbegne da ima $k$ 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). %$f^{mon}$, $f^-$ and $f^+$. Tako\dj e, posmatramo i monotonu verziju $K_{2,2}$-igre, gde Avojder \v{z}eli da izbegne da graf koji \v{c}ine njegove grane sadr\v{z}i graf izomorfan sa $K_{2,2}$. 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 $K_n$. Ovde posmatramo asimetri\v{c}ne igre gde je bias Mejkera 1, a bias Brejkera $b$, $b\ge 1$

Fast Strategies in Waiter-Client Games on KnK_n

Waiter-Client games are played on some hypergraph $(X,\mathcal{F})$, where $\mathcal{F}$ denotes the family of winning sets. For some bias $b$, during each round of such a game Waiter offers to Client $b+1$ elements of $X$, of which Client claims one for himself while the rest go to Waiter. Proceeding like this Waiter wins the game if she forces Client to claim all the elements of any winning set from $\mathcal{F}$. In this paper we study fast strategies for several Waiter-Client games played on the edge set of the complete graph, i.e. $X=E(K_n)$, in which the winning sets are perfect matchings, Hamilton cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.Comment: 38 page

Spanning Structures in Walker--Breaker Games

We study the biased $(2:b)$ Walker--Breaker games, played on the edge set ofthe complete graph on $n$ vertices, $K_n$. These games are a variant of theMaker--Breaker games with the restriction that Walker (playing the role ofMaker) has to choose her edges according to a walk. We look at the two standardgraph games -- the Connectivity game and the Hamilton Cycle game and show thatWalker can win both games even when playing against Breaker whose bias is ofthe order of magnitude $n/ \ln n$