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 $$\gamma _{gs}(G)$$ γ 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.