Domination Game Research Articles

Maker-Breaker domination game played on corona products of graphs

In the Maker-Breaker domination game, Dominator and Staller play on a graph G by taking turns in which each player selects a not yet played vertex of G. Dominator’s goal is to select all the vertices in a dominating set, while Staller aims to prevent this from happening. In this paper, the game is investigated on corona products of graphs. Its outcome is determined as a function of the outcome of the game on the second factor. Staller-Maker-Breaker domination numbers are determined for arbitrary corona products, while Maker-Breaker domination numbers of corona products are bounded from both sides. All the bounds presented are demonstrated to be sharp. Corona products as well as general graphs with small (Staller-)Maker-Breaker domination numbers are described.

Indicated total domination game

Indicated total domination game

Maker-Breaker domination number for Cartesian products of path graphs $P_2$ and $P_n$

We study the Maker-Breaker domination game played by Dominator and Staller on the vertex set of a given graph. Dominator wins when the vertices he has claimed form a dominating set of the graph. Staller wins if she makes it impossible for Dominator to win, or equivalently, she is able to claim some vertex and all its neighbours. Maker-Breaker domination number $\gamma_{MB}(G)$ ($\gamma '_{MB}(G)$) of a graph $G$ is defined to be the minimum number of moves for Dominator to guarantee his winning when he plays first (second). We investigate these two invariants for the Cartesian product of any two graphs. We obtain upper bounds for the Maker-Breaker domination number of the Cartesian product of two arbitrary graphs. Also, we give upper bounds for the Maker-Breaker domination number of the Cartesian product of the complete graph with two vertices and an arbitrary graph. Most importantly, we prove that $\gamma'_{MB}(P_2\square P_n)=n$ for $n\geq 1$, $\gamma_{MB}(P_2\square P_n)$ equals $n$, $n-1$, $n-2$, for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq 13$, respectively. For the disjoint union of $P_2\square P_n$s, we show that $\gamma_{MB}'(\dot\cup_{i=1}^k(P_2\square P_n)_i)=k\cdot n$ ($n\geq 1$), and that $\gamma_{MB}(\dot\cup_{i=1}^k(P_2\square P_n)_i)$ equals $k\cdot n$, $k\cdot n-1$, $k\cdot n-2$ for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq 13$, respectively.

Strategie zarządzania kłopotliwym dziedzictwem kulturowym na Warmii i Mazurach

The article sets out to characterize two main strategies of dealing with the resources of dissonant cultural heritage located in Warmia and Mazury. This type of heritage includes any material or symbolic objects that arouse a sense of dissonance in the social consciousness of representatives of certain social groups. One of the strategies undertaken by regional social actors is described as the strategy of Polonization. The media of memory are filtered so that the story about the past and cultural specificity of the region created by such actors fits the ethnocentrically defined Polishness. The second strategy, called cultural succession, consists in the inclusion of all elements of the existing cultural heritage, regardless of their ethnic or political origin. Those implementing it are striving to create a multi-cultural region. The social actors implementing the above strategies are in conflict because they are playing a game of symbolic domination.

Progress towards the 1/2-Conjecture for the domination game

The domination game is played on a graph G by two players, Dominator and Staller, who alternate in selecting vertices until each vertex in the graph G is contained in the closed neighbourhood of the set of selected vertices. Dominator’s aim is to reach this state in as few moves as possible, whereas Staller wants the game to last as long as possible. In this paper, we prove that if G has n vertices and minimum degree at least 2, then Dominator has a strategy to finish the domination game on G within 10n/17+1/17 moves, thus making progress towards a conjecture by Bujtás, Iršič and Klavžar.

The Maker–Maker domination game in forests

We study the Maker–Maker version of the domination game introduced in 2018 by Duchêne 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.

Fast winning strategies for Staller in the Maker–Breaker domination game

The Maker–Breaker domination game is played on a graph G by two players, called Dominator and Staller, who alternately choose a vertex that has not been played so far. Dominator wins the game if his moves form a dominating set. Staller wins if she plays all vertices from a closed neighborhood of a vertex v∈V(G). Dominator’s fast winning strategies were studied earlier. In this work, we concentrate on the cases when Staller has a winning strategy in the game. We introduce the invariant γSMB′(G) (resp., γSMB(G)) which is the smallest integer k such that, under any strategy of Dominator, Staller can win the game by playing at most k vertices, if Staller (resp., Dominator) plays first on the graph G.We prove some basic properties of γSMB(G) and γSMB′(G) and study the parameters’ changes under some operators as taking the disjoint union of graphs or deleting a cut vertex. We show that the inequality δ(G)+1≤γSMB′(G)≤γSMB(G) always holds and that for every three integers r,s,t with 2≤r≤s≤t, there exists a graph G such that δ(G)+1=r, γSMB′(G)=s, and γSMB(G)=t. We prove exact formulas for γSMB′(G) where G is a path or it is a tadpole graph which is obtained from the disjoint union of a cycle and a path by adding one edge between them.

Maker-Breaker domination game on trees when Staller wins

In the Maker-Breaker domination game played on a graph $G$, 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 $\gamma_{\rm SMB}(G)$ (resp., $\gamma_{\rm SMB}'(G)$) denotes the minimum number of moves Staller needs to win. For every positive integer $k$, trees $T$ with $\gamma_{\rm SMB}'(T)=k$ are characterized and a general upper bound on $\gamma_{\rm SMB}'$ is proved. Let $S = S(n_1,\dots, n_\ell)$ be the subdivided star obtained from the star with $\ell$ edges by subdividing its edges $n_1-1, \ldots, n_\ell-1$ times, respectively. Then $\gamma_{\rm SMB}'(S)$ is determined in all the cases except when $\ell\ge 4$ and each $n_i$ is even. The simplest formula is obtained when there are at least two odd $n_i$s. If $n_1$ and $n_2$ are the two smallest such numbers, then $\gamma_{\rm SMB}'(S(n_1,\dots, n_\ell))=\lceil \log_2(n_1+n_2+1)\rceil$. For caterpillars, exact formulas for $\gamma_{\rm SMB}$ and for $\gamma_{\rm SMB}'$ are established.

4-Total domination game critical graphs

The total domination game is played on a simple graph [Formula: see text] with no isolated vertices by two players, Dominator and Staller, who alternate choosing a vertex in [Formula: see text]. Each chosen vertex totally dominates its neighbors. In this game, each chosen vertex must totally dominates at least one new vertex not totally dominated before. The game ends when all vertices in [Formula: see text] are totally dominated. Dominator’s goal is to finish the game as soon as possible, and Staller’s goal is to prolong it as much as possible. The game total domination number is the number of chosen vertices when both players play optimally, denoted by [Formula: see text] when Dominator starts the game and denoted by [Formula: see text] when Staller starts the game. If a vertex [Formula: see text] in [Formula: see text] is declared to be already totally dominated, then we denote this graph by [Formula: see text]. A total domination game critical graph is a graph [Formula: see text] for which [Formula: see text] holds for every vertex [Formula: see text] in [Formula: see text]. If [Formula: see text], then [Formula: see text] is called [Formula: see text]-[Formula: see text]-critical. In this work, we characterize some 4-[Formula: see text]-critical graphs.

Aspiration-driven co-evolution of cooperation with individual behavioral diversity.

In evolutionary game, aspiration-driven updates and imitation updates are the two dominant game models, and individual behavior patterns are mainly categorized into two types: node player and link player. In more recent studies, the mixture strategy of different types of players has been proven to improve cooperation substantially. Motivated by such a co-evolution mechanism, we combine aspiration dynamics with individual behavioral diversity, where self-assessed aspirations are used to update imitation strategies. In this study, the node players and the link players are capable to transform into each other autonomously, which introduces new features to cooperation in a diverse population as well. In addition, by driving all the players to form specific behavior patterns, the proposed mechanism achieves a survival environment optimization of the cooperators. As expected, the interaction between node players and link players allows the cooperator to avoid the invasion of the defector. Based on the experimental evaluation, the proposed work has demonstrated that the co-evolution mechanism has facilitated the emergence of cooperation by featuring mutual transformation between different players. We hope to inspire a new way of thinking for a promising solution to social dilemmas.

President Putin: A holy mad man or a brave but trapped chicken in Ukraine?

To choose an effective strategy to stop Putin and Russia’s aggressive ambitions in Ukraine and in wider Eastern Europe, Western political and military leaders need to understand his modus operandi and strategic logic. This article focuses on the benefits of game theory in explaining the behavior of the warring parties in Ukraine. It argues that, from an initial “mad man” or a game of chicken strategy chosen by Putin, he has been forced to gradually shift toward a less promising dollar auction model. If the mad man strategy was meant to deter opposing powers from testing Putin’s limits, now the dominant game has locked Putin into a self-driven game with ever higher bids and complications for any exit. For an effective response from the West, the key elements are the Ukrainian will to fight but also the brinkmanship of the West in choosing the right moment to force the Kremlin out of the game.

Mycielskian of graphs with small game domination number

Domination game is a game played on a finite, undirected graph G, between two players Dominator and Staller. During the game, the players alternately choose vertices of G such that each chosen vertex dominates at least one new vertex that is not dominated by previously chosen vertices. The aim of Dominator is to finish the game as early as possible while that of Staller is to delay the process as much as possible. The game domination number γg(G) is the total number of moves in the game when Dominator starts and both players play optimally. Similarly the staller start game domination number γg′(G) is the total number of moves in the game when Staller starts and both players play optimally. Here the domination game in the Mycielskian of a graph is studied. We establish bounds for the game domination number of Mycielskian of a graph in terms of its domination number and also in terms of its game domination number. The Mycielskian of a graph with small game domination number are characterised.

A call to action: Standardizing white-tailed deer harvest data in the Midwestern United States and implications for quantitative analysis and disease management

Recreational hunting has been the dominant game management and conservation mechanism in the United States for the past century. However, there are numerous modern-day issues that reduce the viability and efficacy of hunting-based management, such as fewer hunters, overabundant wildlife populations, limited access, and emerging infectious diseases in wildlife. Quantifying the drivers of recreational harvest by hunters could inform potential management actions to address these issues, but this is seldom comprehensively accomplished because data collection practices limit some analytical applications (e.g., differing spatial scales of harvest regulations and harvest data). Additionally, managing large-scale issues, such as infectious diseases, requires collaborations across management agencies, which is challenging or impossible if data are not standardized. Here we discuss modern issues with the prevailing wildlife management framework in the United States from an analytical point of view with a case study of white-tailed deer (Odocoileus virginianus) in the Midwest. We have four aims: (1) describe the interrelated processes that comprise hunting and suggest improvements to current data collections systems, (2) summarize data collection systems employed by state wildlife management agencies in the Midwestern United States and discuss potential for large-scale data standardization, (3) assess how aims 1 and 2 influence managing infectious diseases in hunted wildlife, and (4) suggest actionable steps to help guide data collection standards and management practices. To achieve these goals, Wisconsin Department of Natural Resources disseminated a questionnaire to state wildlife agencies (Illinois, Indiana, Iowa, Kentucky, Michigan, Minnesota, Missouri, Ohio, Wisconsin), and we report and compare their harvest management structures, data collection practices, and responses to chronic wasting disease. We hope our “call to action” encourages re-evaluation, coordination, and improvement of harvest and management data collection practices with the goal of improving the analytical potential of these data. A deeper understanding of the strengths and deficiencies of our current management systems in relation to harvest and management data collection methods could benefit the future development of comprehensive and collaborative management and research initiatives (e.g., adaptive management) for wildlife and their diseases.

Maker-Breaker total domination game on cubic graphs

We study Maker--Breaker total domination game played by two players, Dominator and Staller, on the connected cubic graphs. Staller (playing the role of Maker) wins if she manages to claim an open neighbourhood of a vertex. Dominator wins otherwise (i.e.\ if he can claim a total dominating set of a graph). For certain graphs on $n\geq 6$ vertices, we give the characterization on those which are Dominator's win and those which are Staller's win.

Game Strategy Selection of Multiple Participants in Cloud Manufacturing Alliance

Introducing the cloud platform operator (CPO) to manage all kinds of manufacturing services on the cloud service platform, the game strategy selection of the multiple participants in the cloud manufacturing alliance is one of current research hotspots of the cloud manufacturing mode. Based on the game theory, this article elaborates the multi-participants business collaboration process and proposes the game strategy selection framework of the multiple participants in the cloud manufacturing alliance. From the perspectives of the manufacturing resource demander (MRD), the manufacturing resource provider (MRP), and the CPO, the game types are analyzed, the benefit functions are established, the existences of game equilibria are proved, and their corresponding constraint conditions are obtained. Integrating the dominant game strategies of the multiple participants, the dominant combination strategies of the above three game equilibria are proposed and the intersection conditions of Nash equilibrium from the mixed perspective are obtained. A set of experimental data from the practical case is given to prove that the solution space of Nash equilibrium from the mixed perspective is not empty. This research provides the theoretical basis for the landing application of the cloud manufacturing mode and the technical support for the development of the cloud manufacturing service platform. It has the important academic value and the broad application prospects.

Predominating a Vertex in the Connected Domination Game

The connected domination game is played just as the domination game, with an additional requirement that at each stage of the game the vertices played induce a connected subgraph. The number of moves in a D-game (an S-game, resp.) on a graph G when both players play optimally is denoted by \(\gamma _\mathrm{cg}(G)\) (\(\gamma _\mathrm{cg}'(G)\), resp.). Connected Game Continuation Principle is established as a substitute for the classical Continuation Principle which does not hold for the connected domination game. Let G|x denote the graph G together with a declaration that the vertex x is already dominated. The first main result asserts that if G is a graph with \(\gamma _\mathrm{cg}(G) \ge 3\) and \(x \in V(G)\), then \(\gamma _\mathrm{cg}(G|x) \le 2 \gamma _\mathrm{cg}(G) - 3\) and the bound is sharp. The second main theorem states that if G is a graph with \(n(G) \ge 2\) and \(x \in V(G)\), then \(\gamma _\mathrm{cg}(G|x) \ge \left\lceil \frac{1}{2} \gamma _\mathrm{cg}(G) \right\rceil\) and the bound is sharp. Graphs G and their vertices x for which \(\gamma _\mathrm{cg}'(G|x) = \infty\) holds are also characterized.

Connected domination game: Predomination, Staller-start game, and lexicographic products

The connected domination game was introduced in 2019 by Borowiecki, Fiedorowicz and Sidorowicz as another variation of the domination game. We answer a problem from their paper regarding the relation between the number of moves in a game where Dominator/Staller starts the game. Additionally, we study the relation to the diameter and present graphs with small game connected domination number. We also determine the value on the lexicographic products, and consider the effect of predomination of a vertex.

Association between offensive and defensive playing style variables and ranking position in a national football league

ABSTRACT The aim of this study was to determine the association of playing style and efficacy variables with football success in a professional football league. Match statistics were obtained from 23 football teams competing in the 2017–2018 and 2018–2019 seasons of the Spanish national league(LaLiga). Offensive and defensive playing style and efficacy variables were calculated. Pearson’s correlation coefficient tests and principal component analysis (PCA) were applied to establish the influence of these variables on the number of points obtained at the end of the season and on the ranking position. In isolation, the efficacy of completion showed the highest association with ranking points and position. A two-dimension PCA explained 77.8% of the variance in the ranking position. In dimension-1 (58.5%), game initiative and attack building, and in dimension-2 (19.3%), efficacy of defensive containment and a lower rate of long passes were within the variables that explained more variance in the ranking position. Success in football, measured by ranking position at the end of the Spanish national league, was associated with several playing style and efficacy variables. Overall, a dominant game style with high efficacy to finish attacking plays, and an offensive game initiative, are most associated with successful football.

The domination game played on diameter 2 graphs

Let $$\gamma _g(G)$$ be the game domination number of a graph G. It is proved that if $$\mathrm{diam}(G) = 2$$ , then $$\gamma _g(G) \le \left\lceil \frac{n(G)}{2} \right\rceil - \left\lfloor \frac{n(G)}{11}\right\rfloor $$ . The bound is attained: if $$\mathrm{diam}(G) = 2$$ and $$n(G) \le 10$$ , then $$\gamma _g(G) = \left\lceil \frac{n(G)}{2} \right\rceil $$ if and only if G is one of seven sporadic graphs with $$n(G)\le 6$$ or the Petersen graph, and there are exactly ten graphs of diameter 2 and order 11 that attain the bound.

Eternal Domination: D-Dimensional Cartesian and Strong Grids and Everything in Between

In the eternal domination game played on graphs, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices on their turn. The goal is to determine the eternal domination number \(\gamma ^{\infty }_{all}\) of a graph, which is the minimum number of guards required to defend against an infinite sequence of attacks. This paper first continues the study of the eternal domination game on strong grids \(P_n\boxtimes P_m\). Cartesian grids \(P_n \square P_m\) have been vastly studied with tight bounds existing for small grids such as \(k\times n\) grids for \(k\in \{2,3,4,5\}\). It was recently proven that \(\gamma ^{\infty }_{all}(P_n \square P_m)=\gamma (P_n \square P_m)+O(n+m)\) where \(\gamma (P_n \square P_m)\) is the domination number of \(P_n \square P_m\) which lower bounds the eternal domination number [Lamprou et al. Eternally dominating large grids. Theoretical Computer Science, 794:27–46, 2019]. We prove that, for all \(n,m\in \mathbb {N^*}\) such that \(m\ge n\), \(\lfloor \frac{n}{3} \rfloor \lfloor \frac{m}{3} \rfloor +\Omega (n+m)=\gamma _{all}^{\infty } (P_{n}\boxtimes P_{m})=\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil + O(m\sqrt{n})\) (note that \(\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil\) is the domination number of \(P_n\boxtimes P_m\)). We then generalise our technique to prove that \(\gamma _{all}^{\infty }(G)=\gamma (G)+o(\gamma (G))\) for all graphs \(G\in {\mathcal {F}}\), where \({\mathcal {F}}\) is a large family of D-dimensional grids which are supergraphs of the D-dimensional Cartesian grid and subgraphs of the D-dimensional strong grid. In particular, \({\mathcal {F}}\) includes both the D-dimensional Cartesian grid and the D-dimensional strong grid.