The domination game played on unions of graphs

Abstract In a graph G, a vertex is said to dominate itself and its neighbors. The Domination game is a two player game played on a finite graph. Players alternate turns in choosing a vertex that dominates at least one new vertex. The game ends when no move is possible, that is when the set of chosen vertices forms a dominating set of the graph. One player (Dominator) aims to minimize the size of this set while the other (Staller) tries to maximize it. The game domination number, denoted by γg, is the number of moves when both players play optimally and Dominator starts. The Staller-start game domination number γ g is defined similarly when Staller starts. It is known that the difference between these two values is at most one We first describe a family of graphs that we call no-minus graphs, for which no player gets advantage in passing a move. While it is known that forests are no-minus, we prove that tri-split graphs and dually chordal graphs also are no-minus. Then, we show that the domination game parameters of the union of two no-minus graphs can take only two values according to the domination game parameters of the initial graphs. In comparison, we also show that in the general case, up to four values may be possible

Dominating sequences in grid-like and toroidal graphs

A longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the strong product. In most of the cases exact Grundy domination numbers are determined for products of paths and/or cycles.Comment: 17 pages 3 figure

Igre, porojene iz grafovske dominacije

V delu preučujemo igre na grafih, ki temeljijo na dominaciji. Največ pozornosti posvetimo dominacijski igri, v kateri igralca dominator in zavlačevalka izmenično izbirata vozlišča končnega grafa, dokler izbrana vozlišča ne tvorijo dominacijske množice. Kot je jasno že iz imen igralcev, je dominatorjev cilj čim hitreje zaključiti igro, medtem ko zavlačevalka stremi k čim daljši igri. Igralno dominacijsko število grafa je invarianta, ki nam pove, koliko potez je potrebnih, ko oba igrata optimalno. Potem ko v prvem poglavju predstavimo zgodovino grafovske dominacije ter prve rezultate v povezavi z dominacijsko igro, se v drugem poglavju ukvarjamo z dominacijsko igro na disjunktni uniji grafov, v tretjem pa z igralnim dominacijskim številom na enostavnih družinah grafov. Četrto poglavje posvetimo realizacijam parov igralnega dominacijskega števila z visoko povezanimi družinami grafov, medtem ko v petem skonstruiramo neskončne razrede grafov, ki imajo igralno dominacijsko število (domnevno) maksimalno možno. V šestem poglavju rešimo klasični problem grafovskih invariant, in sicer, kako se invarianta poljubnega grafa spremeni, če mu odvzamemo eno povezavo ali eno vozlišče. V zadnjem poglavju nas zanimajo kombinatorne igre. Podrobneje si pogledamo kombinatorno različico dominacijske igre dom, za katero določimo Sprague-Grundyjeve vrednosti nekaterih enostavnih družin grafov.In the thesis, we study games on graphs that are based on domination. Our main focus will be the domination game that is played by two players, Dominator and Staller, who are alternating in choosing vertices of a finite graph. The game ends when the set of chosen vertices forms a domination set. Dominator\u27s goal is to finish the game in as few moves as possible, while Staller wants to delay the end of the game as long as she can. The total number of moves in the game, when both players are playing optimally, is called the game domination number. In the first chapter, we present the historical background of the domination theory in graphs, and introduce the domination game along with its first results. In Chapter 2, we study the domination game on disjoint unions of graphs, while Chapter 3 is used to present exact formulas for the game domination number of some simple classes of graphs. In the fourth chapter, we find highly connected families that realize all possible pairs of game domination numbers. In Chapter 5, we construct infinite families of 3/5-graphs and 3/5-trees, while Chapter 6 is used to solve a classical problem of graph invariants regarding edge and vertex removal. In the last chapter, we first present an overview of the similar combinatorial games, and then steer our attention towards the combinatorial game dom that has similar rules as the domination game. We compute Sprague-Grundy values for some simple families of graphs

Kritični grafi za dominacijsko igro

The domination game is played on a graph ▫$G$▫ by two players who alternately take turns by choosing a vertex such that in each turn at least one previously undominated vertex is dominated. The game is over when each vertex becomes dominated. One of the players, namely Dominator, wants to finish the game as soon as possible, while the other one wants to delay the end. The number of turns when Dominator starts the game on ▫$G$▫ and both players play optimally is the graph invariant ▫$gamma_g(G)$▫, named the game domination number. Here we study the ▫$gamma_g$▫-critical graphs which are critical with respect to vertex predomination. Besides proving some general properties, we characterize ▫$gamma_g$▫-critical graphs with ▫$gamma_g =2$▫ and with ▫$gamma_g =3$▫, moreover for each ▫$n$▫ we identify the (infinite) class of all ▫$gamma_g$▫-critical ones among the ▫$n$▫th powers ▫$C_N^n$▫ of cycles. Along the way we determine ▫$gamma_g(C_N^n)$▫ for all ▫$n$▫ and ▫$N$▫. Results of a computer search for ▫$gamma_g$▫-critical trees are presented and several problems and research directions are also listed.Dominacijsko igro na grafu ▫$G$▫ igrata dva igralca, ki izmenično izbirata vozlišča grafa tako, da je po vsaki potezi dominirano vsaj eno novo vozlišče. Igra se zaključi, ko so vsa vozlišča dominirana. Eden od igralcev - Dominator - želi igro končati čim hitreje, medtem ko Zavlačevalka želi igro končati čim kasneje. Število potez v igri, ki jo začne Dominator, in ko oba igralca igrata optimalno, imenujemo igralno dominacijsko število in označimo z ▫$gamma_g(G)$▫. V članku raziskujemo ▫$gamma_g$▫-kritične grafe, ki so vpeljani kot grafi kritični glede na predhodno dominacijo vozlišča. Poleg nekaj splošnih rezultatov karakteriziramo ▫$gamma_g$▫-kritične grafe z ▫$gamma_g =2$▫ in z ▫$gamma_g =3$▫. Za vsak $n$ tudi identificiramo neskončen razred grafov vseh ▫$gamma_g$▫-kritičnih grafov izmed ▫$n$▫tih potenc ▫$C_N^n$▫ ciklov. Pri tem določimo ▫$gamma_g(C_N^n)$▫ za vse ▫$n$▫ in ▫$N$▫. Predstavljeni so rezultati računalniškega pregledovanja ▫$gamma_g$▫-kritičnih dreves. Navedenih je tudi več problemov in možnih nadaljnjih raziskovalnih smeri

Domination Game Critical Graphs

The domination game is played on a graph G by two players who alternately take turns by choosing a vertex such that in each turn at least one previously undominated vertex is dominated. The game is over when each vertex becomes dominated. One of the players, namely Dominator, wants to finish the game as soon as possible, while the other one wants to delay the end. The number of turns when Dominator starts the game on G and both players play optimally is the graph invariant γg(G), named the game domination number. Here we study the γg-critical graphs which are critical with respect to vertex predomination. Besides proving some general properties, we characterize γg-critical graphs with γg = 2 and with γg = 3, moreover for each n we identify the (infinite) class of all γg-critical ones among the nth powers CnN of cycles. Along the way we determine γg(CnN) for all n and N. Results of a computer search for γg-critical trees are presented and several problems and research directions are also listed

Kako dolgo se lahko pretvarjamo v dominacijski igri?

The domination game is played on an arbitrary graph ▫$G$▫ by two players, Dominator and Staller. The game is called Game 1 when Dominator starts it, and Game 2 otherwise. In this paper bluff graphs are introduced as the graphs in which every vertex is an optimal start vertex in Game 1 as well as in Game 2. It is proved that every minus graph (a graph in which Game 2 finishes faster than Game 1) is a bluff graph. A non-trivial infinite family of minus (and hence bluff) graphs is established. Minus graphs with game domination number equal to 3 are characterized. Double bluff graphs are also introduced and it is proved that Kneser graphs ▫$K(n,2)$▫, za ▫$n ge 6$▫, are double bluff. The domination game is also studied on generalized Petersen graphs and on Hamming graphs. Several generalized Petersen graphs that are bluff graphs but not vertex-transitive are found. It is proved that Hamming graphs are not double bluff.Dominacijsko igro na grafu ▫$G$▫ igrata dva igralca, Dominator in Zavlačevalka. Ko igro začenja Dominator, ji rečemo Igra 1, sicer pa Igra 2. V članku vpeljemo grafe pretvarjanja kot tiste grafe, v katerih je vsako vozlišče optimalno začetno vozlišče za Igro 1 in tudi za Igro 2. V tem članku je dokazano, da je vsak minus graf (to je graf, v katerem se Igra 2 konča hitreje kot Igra 1) tudi graf pretvarjanja. Predstavimo netrivialno neskončno družine minus grafov (in s tem tudi grafov pretvarjanja). Minus grafi z igralnim dominantnim številom enakim 3 so okarakterizirani. Vpeljemo tudi grafe dvojnega pretvarjanja in dokažemo, da so med njimi Kneserjevi grafi, ▫$K(n,2)$▫, za ▫$n ge 6$▫. Dominacijsko igro obravnavamo tudi v posplošenih Petersenovih grafih in Hammingovih grafih. Odkrijemo več posplošenih Petersenovih grafov, ki so grafi pretvarjanja, niso pa vozliščno tranzitivni. Dokažemo, da Hammingov grafi niso grafi dvojnega pretvarjanja