Game domination subdivision number of a graph

The game domination subdivision number of a graph $G$ is defined by the following game. Two players $\mathcal{D}$ and $\mathcal{A}$ , $\mathcal{D}$ playing first, alternately mark or subdivide an edge of $G$ which is not yet marked nor subdivided. The game ends when all the edges of $G$ are marked or subdivided and results in a new graph $G^{\prime }$ . The purpose of $\mathcal{D}$ is to minimize the domination number $\mathit{\gamma }(G^{\prime })$ of $G^{\prime }$ while $\mathcal{A}$ tries to maximize it. If both $\mathcal{A}$ and $\mathcal{D}$ play according to their optimal strategies, $\mathit{\gamma }(G^{\prime })$ is well defined. We call this number the game domination subdivision number of $G$ and denote it by ${\mathit{\gamma }}_{\mathit{gs}}(G)$ . In this paper we initiate the study of the game domination subdivision number of a graph and present sharp bounds on the game domination subdivision number of a tree.