Journal of the
Korean Mathematical Society JKMS

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 of the technique of linear programming and its duality.

Keywords: $\{k\}$-dominating function, $\{k\}$-domination game, monotonic $\{k\}$-domination game, core, balancedness

MSC numbers: 91A12, 68R01