In this paper, we consider cooperative games arising from integer domination problem on graphs. We introduce two games, {k}-domination game and its monotonic relaxed game, and focus on their cores. We first give characterizations of the cores and the relationship between them. Furthermore, a common necessary and sufficient condition for the balancedness of both games is obtained by making use o...

The domination game, played on a graph G, was introduced in [3]. Vertices are chosen, one at a time, by two players Dominator and Staller. Each chosen vertex must enlarge the set of vertices of G dominated to that point in the game. Both players use an optimal strategy–Dominator plays so as to end the game as quickly as possible, Staller plays in such a way that the game lasts as many steps as ...

The r-domination search game on graphs is a game-theoretical approach to several graph and hypergraph parameters including treewidth and hypertree width. The task is to identify the minimum number of cops sufficient to catch the visible and very fast robber. To catch the robber, at least one of the cops should be within at most r edges from him. We give a constant factor approximation algorithm...

17 The domination game is played on a graph G by two players who alternately take 18 turns by choosing a vertex such that in each turn at least one previously undominated 19 vertex is dominated. The game is over when each vertex becomes dominated. One 20 of the players, namely Dominator, wants to finish the game as soon as possible, while 21 the other one wants to delay the end. The number of t...

In the article m dimensional lexicographic noncooperative games   ) ,..., ( 1 m   are defined for the players’ } ,..., 2 , 1 { n N  for which there exists a characteristic function T m v v v v ) ,..., , ( 2 1  . Some main features are proved of v function in a lexicographic case. A lexicographic cooperative game is called a couple   v N , , where v is a real vectorfunction on N subsets ...

In the domination game studied here, Dominator and Staller alternately choose a vertex of a graph G and take it into a set D. The number of vertices dominated by the set D must increase in each single turn and the game ends when D becomes a dominating set of G. Dominator aims to minimize whilst Staller aims to maximize the number of turns (or equivalently, the size of the dominating set D obtai...

In the domination game, introduced by Brešar, Klavžar, and Rall in 2010, Dominator and Staller alternately select a vertex of a graph G. A move is legal if the selected vertex v dominates at least one new vertex – that is, if we have a u ∈ N [v] for which no vertex from N [u] was chosen up to this point of the game. The game ends when no more legal moves can be made, and its length equals the n...

For an arbitrary countable ordinal α, we construct a game in which weak and strong domination coincide, and in which α rounds of elimination of dominated strategies are required to solve a game.

The recently introduced total domination game is studied. This game is played on a graph G by two players, named Dominator and Staller. They 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. Dominator’s goal is to totally dominate the graph as fast as possible, and Staller wis...