871 research outputs found
Maker-Breaker domination number
The Maker-Breaker domination game is played on a graph by Dominator and
Staller. The players alternatively select a vertex of that was not yet
chosen in the course of the game. Dominator wins if at some point the vertices
he has chosen form a dominating set. Staller wins if Dominator cannot form a
dominating set. In this paper we introduce the Maker-Breaker domination number
of as the minimum number of moves of Dominator to
win the game provided that he has a winning strategy and is the first to play.
If Staller plays first, then the corresponding invariant is denoted
. Comparing the two invariants it turns out that they
behave much differently than the related game domination numbers. The invariant
is also compared with the domination number. Using the
Erd\H{o}s-Selfridge Criterion a large class of graphs is found for which
holds. Residual graphs are introduced and
used to bound/determine and .
Using residual graphs, and are
determined for an arbitrary tree. The invariants are also obtained for cycles
and bounded for union of graphs. A list of open problems and directions for
further investigations is given.Comment: 20 pages, 5 figure
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Mini-Workshop: Positional Games
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
Maker-Breaker domination game on trees when Staller wins
In the Maker-Breaker domination game played on a graph , Dominator's goal
is to select a dominating set and Staller's goal is to claim a closed
neighborhood of some vertex. We study the cases when Staller can win the game.
If Dominator (resp., Staller) starts the game, then
(resp., ) denotes the minimum number of moves Staller
needs to win. For every positive integer , trees with are characterized. Applying hypergraphs, a general upper bound on
is proved. Let be the subdivided
star obtained from the star with edges by subdividing its edges times, respectively. Then is
determined in all the cases except when and each is even. The
simplest formula is obtained when there are are at least two odd s. If
and are the two smallest such numbers, then . For caterpillars,
exact formulas for and for are
established
Avoidance Games Are PSPACE-Complete
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
How fast can Dominator win in the Maker--Breaker domination game?
We study the Maker--Breaker domination games played by two players, Dominator
and Staller. We give a structural characterization for graphs with
Maker--Breaker domination number equal to the domination number. Specifically,
we show how fast Dominator can win in the game on , for
The Maker-Maker domination game in forests
We study the Maker-Maker version of the domination game introduced in 2018 by
Duch\^ene et al. Given a graph, two players alternately claim vertices. The
first player to claim a dominating set of the graph wins. As the Maker-Breaker
version, this game is PSPACE-complete on split and bipartite graphs. Our main
result is a linear time algorithm to solve this game in forests. We also give a
characterization of the cycles where the first player has a winning strategy
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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