Summary: We study the half-dependent problem for the king graph $K_{n}$. We give proofs to establish the values $h(K_{n})$ for $n \in {1,2,3,4,5,6}$ and an upper bound for $h(K_{n})$ in general. These proofs are independent of computer assisted results. Also, we introduce a two-player game whose winning strategy is tightly related with the values $h(K_{n})$. This strategy is analyzed here for $n = 3$ and some facts are given for the case $n = 4$. Although the rules of the game are very simple, the winning strategy is highly complex even for $n = 4$.

05D99, 91A05, 91A24

upper bounds, winning strategies, maximal configurations, domination in graphs