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...

An upper bound for the domination number of the direct product of graphs is proved. It in particular implies that for any graphs G and H, γ(G × H) ≤ 3γ(G)γ(H). Graphs with arbitrarily large domination numbers are constructed for which this bound is attained. Concerning the upper domination number we prove that Γ(G × H) ≥ Γ(G)Γ(H), thus confirming a conjecture from [16]. Finally, for paired-domi...

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...

Let γt(G) and γpr(G) denote the total domination and the paired domination numbers of graph G, respectively, and let G ¤ H denote the Cartesian product of graphs G and H. In this paper, we show that γt(G)γt(H) ≤ 5γt(G ¤ H), which improves the known result γt(G)γt(H) ≤ 6γt(G ¤ H) given by Henning and Rall.