123 research outputs found
Building spanning trees quickly in Maker-Breaker games
For a tree T on n vertices, we study the Maker-Breaker game, played on the
edge set of the complete graph on n vertices, which Maker wins as soon as the
graph she builds contains a copy of T. We prove that if T has bounded maximum
degree, then Maker can win this game within n+1 moves. Moreover, we prove that
Maker can build almost every tree on n vertices in n-1 moves and provide
non-trivial examples of families of trees which Maker cannot build in n-1
moves
Generating random graphs in biased Maker-Breaker games
We present a general approach connecting biased Maker-Breaker games and
problems about local resilience in random graphs. We utilize this approach to
prove new results and also to derive some known results about biased
Maker-Breaker games. In particular, we show that for
, Maker can build a pancyclic graph (that is, a graph
that contains cycles of every possible length) while playing a game on
. As another application, we show that for , playing a game on , Maker can build a graph which
contains copies of all spanning trees having maximum degree with
a bare path of linear length (a bare path in a tree is a path with all
interior vertices of degree exactly two in )
Fast Strategies in Waiter-Client Games on
Waiter-Client games are played on some hypergraph , where
denotes the family of winning sets. For some bias , during
each round of such a game Waiter offers to Client elements of , 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 . In this paper we study fast strategies
for several Waiter-Client games played on the edge set of the complete graph,
i.e. , in which the winning sets are perfect matchings, Hamilton
cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.Comment: 38 page
Fast strategies in biased Maker--Breaker games
We study the biased Maker--Breaker positional games, played on the
edge set of the complete graph on vertices, . Given Breaker's bias
, possibly depending on , we determine the bounds for the minimal number
of moves, depending on , in which Maker can win in each of the two standard
graph games, the Perfect Matching game and the Hamilton Cycle game
Pozicione igre na grafovima
\section*{Abstract} We study Maker-Breaker games played on the edges of the complete graph on vertices, , whose family of winning sets \cF consists of all edge sets of subgraphs which possess a predetermined monotone increasing property. Two players, Maker and Breaker, take turns in claiming , respectively , unclaimed edges per move. We are interested in finding the threshold bias b_{\cF}(a) for all values of , so that for every , b\leq b_{\cF}(a), Maker wins the game and for all values of , such that b>b_{\cF}(a), Breaker wins the game. We are particularly interested in cases where both and can be greater than . We focus on the \textit{Connectivity game}, where the winning sets are the edge sets of all spanning trees of and on the \textit{Hamiltonicity game}, where the winning sets are the edge sets of all Hamilton cycles on . Next, we consider biased Avoider-Enforcer games, also played on the edges of . For every constant we analyse the -star game, where Avoider tries to avoid claiming 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 -game, where Avoider wants to avoid claiming all the edges of some graph isomorphic to in . Finally, we search for the fast winning strategies for Maker in Perfect matching game and Hamiltonicity game, again played on the edge set of . Here, we look at the biased games, where Maker's bias is 1, and Breaker's bias is .\section*{Izvod} Prou\v{c}avamo takozvane Mejker-Brejker (Maker-Breaker) igre koje se igraju na granama kompletnog grafa sa \v{c}vorova, , \v{c}ija familija pobedni\v{c}kih skupova \cF obuhvata sve skupove grana grafa 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 , odnosno , 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 , tako da za svako , b\le b_{\cF}(a), Mejker pobe\dj uje u igri, a za svako , takvo da je b>b_{\cF}(a), Brejker pobe\dj uje. Posebno nas interesuju slu\v{c}ajevi u kojima oba parametra i 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 , kao i igri Hamiltonove konture, gde su pobedni\v{c}ki skupovi grane svih Hamiltonovih kontura grafa . Zatim posmatramo igre tipa Avojder-Enforser (Avoider-Enforcer), sa biasom , koje se tako\dj e igraju na granama kompletnog grafa sa \v{c}vorova, . Za svaku konstantu , analiziramo igru -zvezde (zvezde sa krakova), u kojoj \textit{Avojder} poku\v{s}va da izbegne da ima 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). %, and . Tako\dj e, posmatramo i monotonu verziju -igre, gde Avojder \v{z}eli da izbegne da graf koji \v{c}ine njegove grane sadr\v{z}i graf izomorfan sa . 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 . Ovde posmatramo asimetri\v{c}ne igre gde je bias Mejkera 1, a bias Brejkera ,
How Long Can a Graph be Kept Planar?
The graph (non-)planarity game is played on the complete graph between an Enforcer and an Avoider, each of whom take one edge per round. The game ends when the edges chosen by Avoider form a non-planar subgraph. We show that Avoider can play for turns, improving the previous bound of
Mejker–Brejker igre na grafovima
The topic of this thesis are different variants of Maker–Breaker positional game, where two players Maker and Breaker alternatively take turns in claiming unclaimed edges/vertices of a given graph. We consider Walker–Breaker game, played on the edge set of the graph Kn. Walker, playing the role of Maker is restricted to claim her edges according to a walk, while Breaker can claim any unclaimed edge per move. The focus is on two standard games - the Connectivity game, where Walker has the goal to build a spanning tree on Kn, and the Hamilton Cycle game, where Walker has the goal to build a Hamilton cycle on Kn. We show that Walker with bias 2 can win both games even when playing against Breaker whose bias b is of the order of magnitude n= ln n. Next, we consider (1 : 1) WalkerMaker–WalkerBreaker game on E(Kn),where both Maker and Breaker are walkers and we are interested in seeing how fast WalkerMaker can build spanning tree and Hamilton cycle. Finally, we study Maker–Breaker total domination game played on the vertex set of a given graph. Two players, Dominator and Staller, alternately take turns in claiming unclaimed vertices of the graph. Staller is Maker and wins if she can claim an open neighbourhood of a vertex. Dominator is Breaker and wins if he manages to claim a total dominating set of a graph. For certain connected cubic graphs on n ≥ 6 vertices, we give the characterization of those graphs which are Dominator’s win and those which are Staller’s win.Tema istrazivanja ove disertacije su igre tipa Mejker– Brejker u kojima uˇcestvuju dva igraˇca, Mejker i Brejker, koji naizmjeniˇcno uzimaju slobodne grane/ˇcvorove datog grafa. Bavimo se Voker–Brejker igrama koje se igraju na skupu grana grafa Kn. Voker, u ulozi Mejkera, jeograniˇcen da uzima svoje grane kao da se ˇseta kroz graf, dok Brejker moˇze da uzme bilo koju slobodnu granu grafa. Fokus je na dvije standardne igre - igri povezanosti, gdje Voker ima za cilj da napravi pokrivaju´ce stablo grafa Kn i igri Hamiltonove konture, gdje Voker ima za cilj da napravi Hamiltonovu konturu. Brejker pobjeduje ako sprijeˇci Vokera u ostvarenju njegovog cilja. Pokaza´cemo da Voker sa biasom 2 moˇze da pobijedi u obje igre ˇcak i ako igra protiv Brejkera ˇciji je bias b reda n= ln n. Potom razmatramo (1 : 1) VokerMejker–VokerBrejker igre na Kn, gdje oba igraˇca, i Mejker i Brejker, moraju da biraju grane koje su dio ˇsetnje u njihovom grafu s ciljem odredivanja brze pobjedniˇce strategije VokerMejkera u igri povezanosti i igri Hamiltonove konture. Konaˇcno, istraˇzujemo Mejker–Brejker igre totalne dominacije koje se igraju na skupu ˇcvorova datog grafa. Dva igraˇca, Dom inator i Stoler naizmjeniˇcno uzimaju slobodne ˇcvorove datog grafa. Stoler je Mejker i pobjeduje ako uspije da uzme sve susjede nekog ˇcvora. Dominator je Brejker i pobjeduje ako ˇcvorovi koje uzme dok kraja igre formiraju skup totalne dominacije. Za odredene klase povezanih kubnih grafova reda n ≥ 6, dajemo karakterizaciju onih grafova na kojima Dominator pobjeduje i onih na kojima Stoler pobjeduje.
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