roman game domination subdivision number of a graph

a {em roman dominating function} on a graph $g = (v ,e)$ is a function $f : vlongrightarrow {0, 1, 2}$ satisfying the condition that every vertex $v$ for which $f (v) = 0$ is adjacent to at least one vertex $u$ for which $f (u) = 2$. the {em weight} of a roman dominating function is the value $w(f)=sum_{vin v}f(v)$. the roman domination number of a graph $g$, denoted by $gamma_r(g)$, equals the minimum weight of a roman dominating function on g. the roman 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 $g'$. the purpose of $mathcal d$ is to minimize the roman dominating number $gamma_r(g')$ of $g'$ while $mathcal a$ tries to maximize it. if both $mathcal a$ and $mathcal d$ play according to their optimal strategies, $gamma_r(g')$ is well defined. we call this number the {em roman game domination subdivision number} of $g$ and denote it by $gamma_{rgs}(g)$. in this paper we initiate the study of the roman game domination subdivision number of a graph and present sharp bounds on the roman game domination subdivision number of a tree.