Total k-rainbow domination subdivision number in graphs

A total k-rainbow dominating function (TkRDF) of G is a function f from the vertex set V (G) to the set of all subsets of the set {1, . . . , k} such that (i) for any vertex v ∈ V (G) with f(v) = ∅ the condition ⋃ u∈N(v) f(u) = {1, . . . , k} is fulfilled, where N(v) is the open neighborhood of v, and (ii) the subgraph of G induced by {v ∈ V (G) | f(v) 6= ∅} has no isolated vertex. The total k-rainbow domination number, γtrk(G), is the minimum weight of a TkRDF on G. The total k-rainbow domination subdivision number sdγtrk(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total k-rainbow domination number. In this paper, we initiate the study of total k-rainbow domination subdivision number in graphs and we present sharp bounds for sdγtrk(G). In addition, we determine the total 2-rainbow domination subdivision number of complete bipartite graphs and show that the total 2-rainbow domination subdivision number can be arbitrary large.