Roman game domination number of a graph

The Roman game domination number of an undirected graph G is defined by the following game. Players $$\mathcal {A}$$A and $$\mathcal {D}$$D orient the edges of the graph G alternately, with $$\mathcal {D}$$D playing first, until all edges are oriented. Player $$\mathcal {D}$$D (frequently called Dominator) tries to minimize the Roman domination number of the resulting digraph, while player $$\mathcal {A}$$A (Avoider) tries to maximize it. This game gives a unique number depending only on G, if we suppose that both $$\mathcal {A}$$A and $$\mathcal {D}$$D play according to their optimal strategies. This number is called the Roman game domination number of G and is denoted by $$\gamma _{Rg}(G)$$γRg(G). In this paper we initiate the study of the Roman game domination number of a graph and we establish some bounds on $$\gamma _{Rg}(G)$$γRg(G). We also determine the Roman game domination number for some classes of graphs.