Walker-Breaker Games on Gn,pG_{n,p}

The Maker-Breaker connectivity game and Hamilton cycle game belong to the best studied games in positional games theory, including results on biased games, games on random graphs and fast winning strategies. Recently, the Connector-Breaker game variant, in which Connector has to claim edges such that her graph stays connected throughout the game, as well as the Walker-Breaker game variant, in which Walker has to claim her edges according to a walk, have received growing attention. For instance, London and Pluh\'ar studied the threshold bias for the Connector-Breaker connectivity game on a complete graph $K_n$, and showed that there is a big difference between the cases when Maker's bias equals $1$ or $2$. Moreover, a recent result by the first and third author as well as Kirsch shows that the threshold probability $p$ for the $(2:2)$ Connector-Breaker connectivity game on a random graph $G\sim G_{n,p}$ is of order $n^{-2/3+o(1)}$. We extent this result further to Walker-Breaker games and prove that this probability is also enough for Walker to create a Hamilton cycle

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.

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.

On the odd cycle game and connected rules

We study the positional game where two players, Maker and Breaker, alternately select respectively 1 and b previously unclaimed edges of Kn. Maker wins if she succeeds in claiming all edges of some odd cycle in Kn and Breaker wins otherwise. Improving on a result of Bednarska and Pikhurko, we show that Maker wins the odd cycle game if b ≤ (4− √ 6)/5 +o(1)) n. We furthermore introduce “connected rules” and study the odd cycle game under them, both in the Maker-Breaker as well as in the Client-Waiter variant

Spartan Daily, April 1, 1986

Volume 86, Issue 39https://scholarworks.sjsu.edu/spartandaily/7428/thumbnail.jp

Spartan Daily, November 26, 1940

Volume 29, Issue 46https://scholarworks.sjsu.edu/spartandaily/3208/thumbnail.jp

Triangles, Long Paths, and Covered Sets

In chapter 2, we consider a generalization of the well-known Maker-Breaker triangle game for uniform hypergraphs in which Maker tries to build a triangle by choosing one edge in each round and Breaker tries to prevent her from doing so by choosing q edges in each round. The main result is the analysis of a new Breaker strategy using potential functions, introduced by Glazik and Srivastav (2019). Both bounds are of the order Θ(n3/2) so they are asymptotically optimal. The constant for the lower bound is 2-o(1) and for the upper bound it is 3√2. In chapter 3, we describe another Maker-Breaker game, namely the P3-game in which Maker tries to build a path of length 3. First, we show that the methods of chapter 2 are not applicable in this scenario and give an intuition why that might be the case. Then, we give a more simple counting argument to bound the threshold bias. In chapter 4, we consider the longest path problem which is a classic NP-hard problem that arises in many contexts. Our motivation to investigate this problem in a big-data context was the problem of genome-assembly, where a long path in a graph that is constructed of the reads of a genome potentially represents a long contiguous sequence of the genome. We give a semi-streaming algorithm. Our algorithm delivers results competitive to algorithms that do not have a restriction on the amount of memory. In chapter 5, we investigate the b-SetMultiCover problem, a classic combinatorial problem which generalizes the set cover problem. Using an LP-relaxation and analysis with the bounded differences inequality of C. McDiarmid (1989), we show that there is a strong concentration around the expectation