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For a graphG = ( V , E )with vertex setV = V ( G )and edge setE = E ( G )    , a Roman{ 3 }    -dominating function (R    { 3 }    -DF) is a functionf : V ( G ) → { 0 , 1 , 2 , 3 }having the property that∑  u ∈  N G   ( v )    f  ( u )  ≥ 3    , iff ( v ) = 0    , and∑  u ∈  N G   ( v )    f  ( u )  ≥ 2    , iff ( v ) = 1for any vertexv ∈ V ( G )    . The weight of a Roman{ 3 }    -dominating function f is the sumf  ( V )  =  ∑  v ∈ V ( G )   f  ( v )and the minimum weight of a Roman{ 3 }    -dominating function on G is the Roman{ 3 }    -domination number of G, denoted byγ  { R 3 }    ( G ). Let G be a graph with no isolated vertices. The total Roman{ 3 }    -dominating function on G is an R    { 3 }    -DF f on G with the additional property that every vertexv ∈ Vwithf ( v ) ≠ 0has a neighbor w withf ( w ) ≠ 0    . The minimum weight of a total Roman{ 3 }    -dominating function on G, is called the total Roman{ 3 }    -domination number denoted byγ  t { R 3 }    ( G ). We initiate the study of total Roman{ 3 }    -domination and show its relationship to other domination parameters. We present an upper bound on the total Roman{ 3 }    -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{ 3 }    -domination for bipartite graphs.

Total Roman {3}-domination in Graphs