Signed Roman (Total) Domination Numbers of Complete Bipartite Graphs and Wheels

A signed (res.signed total) Roman dominating function,SRDF (res.STRDF) for short,of a graph G =(V,E) is a function f:V → {-1,1,2} satisfying the conditions that (i) ∑v∈Nv] f(v) ≥ 1 (res.∑ v∈N(v) f(v) ≥ 1) for any v ∈ V,where Nv] is the closed neighborhood and N(v) is the neighborhood of v,and (ii) every vertex v for which f(v) =-1 is adjacent to a vertex u for which f(u) =2.The weight of a SRDF (res.STRDF) is the sum of its function values over all vertices.The signed (res.signed total) Roman domination number of G is the minimum weight among all signed (res.signed total) Roman dominating functions of G.In this paper,we compute the exact values of the signed (res.signed total) Roman domination numbers of complete bipartite graphs and wheels.