Abstract
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.
Highlights
In this paper we define and study the new problem of connected domination game on graphs
In the present paper we introduce a new domination game based on the notion of connected domination
We define a connected domination game on G as follows: The game is played by two players, Dominator and Staller
Summary
In this paper we define and study the new problem of connected domination game on graphs. We define a connected domination game on G as follows: The game is played by two players, Dominator and Staller. Let D be the set of played vertices at the end of the connected domination game. The connected game domination number of G, denoted γcg(G), is defined as the minimum cardinality of D, when both players played optimally on G. We give examples of graphs for which the upper bound on the connected game domination number given in Theorem 1 is tight. We study the game in which Staller has the first move and the game in which Staller can skip some moves
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