Game domination subdivision number of a graph

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