For a graph with vertex set and edge set , a Roman -dominating function (R -DF) is a function having the property that , if , and , if for any vertex . The weight of a Roman -dominating function f is the sum and the minimum weight of a Roman -dominating function on G is the Roman -domination number of G, denoted by . Let G be a graph with no isolated vertices. The total Roman -dominating function on G is an R -DF f on G with the additional property that every vertex with has a neighbor w with . The minimum weight of a total Roman -dominating function on G, is called the total Roman -domination number denoted by . We initiate the study of total Roman -domination and show its relationship to other domination parameters. We present an upper bound on the total Roman -domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. Finally, we investigate the complexity of total Roman -domination for bipartite graphs.
This is an open access article distributed under the Creative Commons Attribution License
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited