Roman Subdivision Domination in Graphs.

The subdivision graph S G( ) of a graph G is the graph whose vertex set is the union of the set of vertices and the set of edges of G in which each edge uv is subdivided at once as uw and wv . A Roman dominating function on a subdivision graph S G H ( ) = is a function f V H : 0,1,2 ( ) →{ } satisfying the condition that every vertex u for which f u( ) = 0 is adjacent to at least one vertex v for which f v( ) = 2. The weight of a Roman dominating function is the value ( ( )) ( ) v V H( ) f V H f v ∈ = ? . The minimum weight of a Roman dominating function on a subdivision graph H is called the Roman subdivision domination number of G and is denoted byγ RS (G). In this paper, we study the Roman domination in subdivision graph S G( ) and obtain some results on γ RS (G) in terms of vertices, blocks and other different parameters of the graph G , but not the members of S G( ) . Further we develop its relationship with other different domination parameters of G . Subject classification number: 05C69, 05C70.