ROMAN GAME DOMINATION SUBDIVISION NUMBER OF A GRAPH

A roman dominating function on a graph g = (v;e) is a function f : v ??! f0; 1; 2gsatisfying the condition that every vertex v for which f(v) = 0 is adjacent to at least one vertex u forwhich f(u) = 2. the weight of a roman dominating function is the value w(f) = pv2v f(v). theroman domination number of a graph g, denoted by r(g), equals the minimum weight of a romandominating function on g. the roman game domination subdivision number of a graph g is denedby the following game. two players d and a, d playing rst, alternately mark or subdivide an edgeof g which is not yet marked nor subdivided. the game ends when all the edges of g are marked orsubdivided and results in a new graph g0. the purpose of d is to minimize the roman dominationnumber r(g0) of g0 while a tries to maximize it. if both a and d play according to their optimalstrategies, r(g0) is well defined. we call this number the roman game domination subdivision numberof g and denote it by rgs(g). in this paper we initiate the study of the roman game dominationsubdivision number of a graph and present sharp bounds on the roman game domination subdivisionnumber of a tree.