Domination Game Research Articles

Cutting lemma and union lemma for the domination game

Two new techniques are introduced into the theory of the domination game. The cutting lemma bounds the game domination number of a partially dominated graph with the game domination number of a suitably modified partially dominated graph. The union lemma bounds the S-game domination number of a disjoint union of paths using appropriate weighting functions. Using these tools a conjecture asserting that the so-called three legged spiders are game domination critical graphs is proved. An extended cutting lemma is also derived and all game domination critical trees on 18, 19, and 20 vertices are listed.

Connected domination game

In this paper we introduce a domination game based on the notion of connected domination. Let G = (V,E) be a connected graph of order at least 2. We define a connected domination game on G as follows: The game is played by two players, Dominator and Staller. The players alternate taking turns choosing a vertex of G (Dominator starts). A move of a player by choosing a vertex v is legal, if (1) the vertex v dominates at least one additional vertex that was not dominated by the set of previously chosen vertices and (2) the set of all chosen vertices induces a connected subgraph of G. The game ends when none of the players has a legal move (i.e., G is dominated). The aim of Dominator is to finish as soon as possible, Staller has an opposite aim. Let D be the set of played vertices obtained at the end of the connected domination game (D is a connected dominating set of G). The connected game domination number of G, denoted cg(G), is the minimum cardinality of D, when both players played optimally on G. We provide an upper bound on cg(G) in terms of the connected domination number. We also give a tight upper bound on this parameter for the class of 2-trees. Next, we investigate the Cartesian product of a complete graph and a tree, and we give exact values of the connected game domination number for such a product, when the tree is a path or a star. We also consider some variants of the game, in particular, a Staller-start game.

Domination game and minimal edge cuts

In this paper a relationship is established between the domination game and minimal edge cuts. It is proved that the game domination number of a connected graph can be bounded above in terms of the size of minimal edge cuts. In particular, if C a minimum edge cut of a connected graph G, then γg(G)≤γg(G∖C)+2κ′(G). Double-Staller graphs are introduced in order to show that this upper bound can be attained for graphs with a bridge. The obtained results are used to extend the family of known traceable graphs whose game domination numbers are at most one-half their order. Along the way two technical lemmas, which seem to be generally applicable for the study of the domination game, are proved.

Domination game on uniform hypergraphs

In this paper we introduce and study the domination game on hypergraphs. This is played on a hypergraph H by two players, namely Dominator and Staller, who alternately select vertices such that each selected vertex enlarges the set of vertices dominated so far. The game is over if all vertices of H are dominated. Dominator aims to finish the game as soon as possible, while Staller aims to delay the end of the game. If each player plays optimally and Dominator starts, the length of the game is the invariant ‘game domination number’ denoted by γg(H). This definition is the generalization of the domination game played on graphs and it is a special case of the transversal game on hypergraphs. After some basic general results, we establish an asymptotically tight upper bound on the game domination number of k-uniform hypergraphs. In the remaining part of the paper we prove that γg(H)≤5n∕9 if H is a 3-uniform hypergraph of order n and does not contain isolated vertices. This also implies the following new result for graphs: If G is an isolate-free graph on n vertices and each of its edges is contained in a triangle, then γg(G)≤5n∕9.

Game domination numbers of a disjoint union of paths and cycles

The domination game is played on a graph G by two players, Dominator and Staller, who alternately chooses a vertex of G in such a way that at least one new vertex is dominated. The game ends when all vertices are dominated. Dominator aims to finish the game in as few moves as possible while Staller aims to finish the game in as many moves as possible. The game domination number γg (G) (respectively γʹg (G)) is the total number of moves both players use in a game which Dominator (respectively Staller) starts and both players use optimal strategies. In this paper we determine the game domination numbers of a disjoint union of paths and cycles.

THE DOMINATION GAME ON SPLIT GRAPHS

We investigate the domination game and the game domination number $\unicode[STIX]{x1D6FE}_{g}$ in the class of split graphs. We prove that $\unicode[STIX]{x1D6FE}_{g}(G)\leq n/2$ for any isolate-free $n$-vertex split graph $G$, thus strengthening the conjectured $3n/5$ general bound and supporting Rall’s $\lceil n/2\rceil$-conjecture. We also characterise split graphs of even order with $\unicode[STIX]{x1D6FE}_{g}(G)=n/2$.

Effect of predomination and vertex removal on the game total domination number of a graph

The game total domination number, γtg, was introduced by Henning et al. in 2015. In this paper we study the effect of vertex predomination on the game total domination number. We prove that γtg(G|v)≥γtg(G)−2 holds for all vertices v of a graph G and present infinite families attaining the equality. To achieve this, some new variations of the total domination game are introduced. The effect of vertex removal is also studied. We show that γtg(G)≤γtg(G−v)+4 and γtg′(G)≤γtg′(G−v)+4.

Eternally dominating large grids

In the m-Eternal Domination game, a team of guard tokens initially occupies a dominating set on a graph G. An attacker then picks a vertex without a guard on it and attacks it. The guards defend against the attack: one of them has to move to the attacked vertex, while each remaining one can choose to move to one of his neighboring vertices. The new guards' placement must again be dominating. This attack-defend procedure continues eternally. The guards win if they can eternally maintain a dominating set against any sequence of attacks, otherwise the attacker wins.The m-eternal domination number for a graph G is the minimum amount of guards such that they win against any attacker strategy in G (all guards move model). We study rectangular grids and provide the first known general upper bound on the m-eternal domination number for these graphs. Our novel strategy implements a square rotation principle and eternally dominates m×n grids by using approximately mn5 guards, which is asymptotically optimal even for ordinary domination.

On domination game stable graphs and domination game edge-critical graphs

Let γg(G) be the game domination number of a graph G. For any vertex v∈V(G), we denote by G|v a partially dominated graph G in which the vertex v is already dominated. In this paper, based on the domination game critical graph, two related definitions to the game domination number are introduced as follows. A graph G is domination game stable (or γg-stable for short) if γg(G)=γg(G|v) for any vertex v∈V(G) and is domination game edge-critical (or γg-edge-critical for short) if γg(G−e)>γg(G) for any edge e∈E(G).A γg-stable (γg-edge-critical resp.) graph G is called k-γg-stable (k-γg-edge-critical resp.) if γg(G)=k. We characterize the k-γg-stable graphs with k∈{1,2} and 3-γg-stable trees and show that the Kneser graphs K(n,2) with n≥5 are γg-stable. Also allγg-stable paths and cycles are completely determined, respectively. Moreover, we prove the existence of k-γg-stable graphs for any positive integer k. In addition, all complete r-partite γg-stable and γg-edge-critical graphs are determined, respectively. The characterization of k-γg-edge-critical graphs is given for k∈{1,2} with an example of a3-γg-edge-critical graph. Two sufficient conditions are presented for non-γg-edge-critical graphs. Finally we propose several open related problems.

Infinite Families of Circular and Möbius Ladders that are Total Domination Game Critical

Let $$\gamma _\mathrm{tg}(G)$$ denote the game total domination number of a graph G, and let G|v mean that a vertex v of G is declared to be already totally dominated. A graph G is total domination game critical if $$\gamma _\mathrm{tg}(G|v) < \gamma _\mathrm{tg}(G)$$ holds for every vertex v in G. If $$\gamma _\mathrm{tg}(G) = k$$ , then G is further called k- $$\gamma _\mathrm{tg}$$ -critical. In this paper, we prove that the circular ladder $$C_{4k} \,\square \,K_2$$ is 4k- $$\gamma _{\mathrm{tg}}$$ -critical and that the Mobius ladder $$\mathrm{ML}_{2k}$$ is 2k- $$\gamma _{\mathrm{tg}}$$ -critical.

Game total domination critical graphs

In the total domination game played on a graph G, players Dominator and Staller alternately select vertices of G, as long as possible, such that each vertex chosen increases the number of vertices totally dominated. Dominator (Staller) wishes to minimize (maximize) the number of vertices selected. The game total domination number, γtg(G), of G is the number of vertices chosen when Dominator starts the game and both players play optimally. If a vertex v of G is declared to be already totally dominated, then we denote this graph by G|v. In this paper the total domination game critical graphs are introduced as the graphs G for which γtg(G|v)<γtg(G) holds for every vertex v in G. If γtg(G)=k, then G is called k-γtg-critical. It is proved that the cycle Cn is γtg-critical if and only if n(mod6)∈{0,1,3} and that the path Pn is γtg-critical if and only if n(mod6)∈{2,4}. 2-γtg-critical and 3-γtg-critical graphs are also characterized as well as 3-γtg-critical joins of graphs.

On the Game Total Domination Number

The total domination game is a two-person competitive optimization game, where the players, Dominator and Staller, alternately select vertices of an isolate-free graph G. Each vertex chosen must strictly increase the number of vertices totally dominated. This process eventually produces a total dominating set of G. Dominator wishes to minimize the number of vertices chosen in the game, while Staller wishes to maximize it. The game total domination number of G, $$\gamma _{\mathrm{tg}}(G)$$ , is the number of vertices chosen when Dominator starts the game and both players play optimally. Recently, Henning, Klavžar, and Rall proved that $$\gamma _{\mathrm{tg}}(G) \le \frac{4}{5}n$$ holds for every graph G which is given on n vertices such that every component of it is of order at least 3; they also conjectured that the sharp upper bound would be $$\frac{3}{4}n$$ . Here, we prove that $$\gamma _{\mathrm{tg}}(G) \le \frac{11}{14}n$$ holds for every G which contains no isolated vertices or isolated edges.

World Domination Games and its Impact on the 21st Century

No other element impacted the historical conditions of the preceding 100 years to such an extent as the war to secure and control the world's reserves of petroleum. Sustainable economic growth after 1873, that discouraged British Empire, arose mechanical economies in Europe. Central Asia remained the object of rivalries and machination by the giant countries of the Europe. World Domination Games started from Pillage Games that lead towards many “Games” such as Great Game, New Great Game, Game Changer and New Game Changer. All prefect countries desire to have a control over the world for the last two centuries. Their efforts turn into numerous clashes and clashes led towards wars. In the twentieth century wars transformed not only their names but also their genetics that has profound impact on the 21st Century. This laid foundation of the emerging new superpowers in every century.

The competitiveness of games in professional sports leagues

A sports game is about competition. The competitiveness of a game is important in terms of fan interest as a non-competitive game will cause fans attending a game to lose interest and be more likely to leave the game early. Fans viewing the game live will find another activity to do. Teams ahead or behind in a non-competitive game may also be more likely to substitute reserve players to reduce risk of injury to key players or gain experience for lesser-used players. Changes in how a team plays in a non-competitive game also impact secondary competitions such as betting or fantasy sports due to player behavior changes in a non-competitive game. In this work we examine the competitiveness of games in six professional sports leagues. We obtain the list of scoring events for each game and map each event into a discrete segment, based on minutes played or outs recorded, for each game. A significant advantage of our approach is that all sports are analyzed in the same manner with the only difference between leagues being the number of segments into which each game is divided. By dividing each segment number by the total number of segments for the league we can compute a common percentage-based timeline to compare the competitiveness of different leagues through the course of a game. A distinguishing feature of our work is that we explore a number of potential measures of competitiveness using our data. These measures include whether a team is leading, whether a team is leading by more than a given number of scores, whether a team is in the lead and wins the game, whether a team is in the lead for good, whether a team already winning goes on to win or stay ahead for good, and the probability that a team with given lead at a given point in the game will win. We analyze the games in each professional sports league using each of these competitiveness metrics and find that MLB games are clearly the least competitive compared with games from each of the other leagues in our study. MLB games have the highest percentage of game segments that are played in less competitive situations. MLS and BPL games tend to be the most competitive in general; largely because about half of game time for these leagues is spent with the game tied. However, if a team does take the lead in one of these leagues then they, along with MLB, have the highest chance that this team will not relinquish the lead. Results for the NFL and NHL show that their games are neither relatively the most or least competitive. We believe variation in the competitiveness of the leagues occurs because of the nature of the sports in terms of how they are played and scored as well as the specific leagues themselves. We go on to find that NFL home teams spend the most amount of game time ahead for good and NHL home teams spend the least. Using this ahead-for-good metric, there is the most amount difference for home and away teams in MLS while there is the least in MLB. Finally we use this same ahead-for-good metric to evaluate the competitiveness performance of all teams across the six leagues in our study. The Denver Broncos of the NFL spent the most game time ahead for good of all professional teams while Chelsea of the BPL spent the least amount of time behind for good. The Golden State Warriors of the NBA and the New England Patriots of the NFL had the most dominant game performance based upon the difference in ahead-for-good versus behind-for-good game time.

The game total domination problem is log-complete in PSPACE

In this paper, we continue the study of the total domination game in graphs introduced in Henning et al. (2015) [10]. A vertex totally dominates another vertex in a graph G if they are neighbors. A total dominating set of G is a set S of vertices of G such that every vertex of G is totally dominated by a vertex in S. The total domination game played on G consists of two players, named Dominator and Staller, who alternately take turns choosing vertices of G such that each chosen vertex totally dominates at least one vertex not totally dominated by the vertices previously chosen. The game ends when the set of vertices chosen becomes a total dominating set in G. Dominator wishes to end the game with a minimum number of vertices chosen, and Staller wishes to end the game with as many vertices chosen as possible. The game total domination number of G is the number of vertices chosen when Dominator starts the game and both players play optimally. In this paper, we show that verifying whether the game total domination number of a graph is bounded by a given integer is log-complete in PSPACE.

Trees with equal total domination and game total domination numbers

In this paper, we continue the study of the total domination game in graphs introduced in Henning et al. (2015), where the players Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a total dominating set S of G in which every vertex is totally dominated by a vertex in S. Dominator wishes to minimize the number of vertices chosen, while Staller wishes to maximize it. The game total domination number, γtg(G), (respectively, Staller-start game total domination number, γtg′(G)) of G is the number of vertices chosen when Dominator (respectively, Staller) starts the game and both players play optimally. For general graphs G, sometimes γtg(G)>γtg′(G). We show that if G is a forest with no isolated vertex, then γtg(G)≤γtg′(G). Using this result, we characterize the trees with equal total domination and game total domination number.

Domination game on paths and cycles

Domination game is a game on a simple graph played by two players, Dominator and Staller, who are alternating in taking turns. In each turn a player chooses a vertex in such a way that at least one new vertex gets dominated by this move. The game ends when all vertices are dominated, and thus no legal move is possible. As the names of the players suggest, Dominator tries to finish the game as fast as possible, while Staller wants to prolong its end as long as she can. By γ g ( γ g ʹ ) we denote the total number of moves in the game when Dominator (resp. Staller) starts, and both players play according to their optimal strategies. In a manuscript from 2012, Kinnersley et al. determined γ g and γ g ʹ for paths and cycles, but have not yet published this very important result. In this paper we give an alternative proof for these formulas. Our approach also explicitly describes optimal strategies for both players.

Evolutionary games on cycles with strong selection.

Evolutionary games on graphs describe how strategic interactions and population structure determine evolutionary success, quantified by the probability that a single mutant takes over a population. Graph structures, compared to the well-mixed case, can act as amplifiers or suppressors of selection by increasing or decreasing the fixation probability of a beneficial mutant. Properties of the associated mean fixation times can be more intricate, especially when selection is strong. The intuition is that fixation of a beneficial mutant happens fast in a dominance game, that fixation takes very long in a coexistence game, and that strong selection eliminates demographic noise. Here we show that these intuitions can be misleading in structured populations. We analyze mean fixation times on the cycle graph under strong frequency-dependent selection for two different microscopic evolutionary update rules (death-birth and birth-death). We establish exact analytical results for fixation times under strong selection and show that there are coexistence games in which fixation occurs in time polynomial in population size. Depending on the underlying game, we observe inherence of demographic noise even under strong selection if the process is driven by random death before selection for birth of an offspring (death-birth update). In contrast, if selection for an offspring occurs before random removal (birth-death update), then strong selection can remove demographic noise almost entirely.

How long can one bluff in the domination game?

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), n ≥ 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.

Humorlessness (Three Monologues and a Hairpiece)

Humorlessness (Three Monologues and a Hairpiece)