Game domination subdivision number of a graph

Abstract

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 $\gamma (G^{\prime })$ of $G^{\prime }$ while $\mathcal A$ tries to maximize it. If both $\mathcal A$ and $\mathcal D$ play according to their optimal strategies, $\gamma (G^{\prime })$ is well defined. We call this number the game domination subdivision number of $G$ and denote it by $\gamma _{gs}(G)$. 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.