Signed total Roman domination and domatic numbers in graphs

A signed total Roman dominating function (STRDF) on a graph G is a function f:V(G)⟶{−1,1,2} satisfying (i) ∑x∈NG(u)f(x)≥1 for each vertex u∈V(G) and its neighborhood NG(u) in G and, (ii) every vertex u∈V(G) with f(u)=−1, there exists a vertex v∈NG(u) with f(v)=2. The minimum number ∑u∈V(G)f(u) among all STRDFs f on G is denoted by γstR(G). A set {f1,…,fd} of distinct STRDFs on G is called a signed total Roman dominating family on G if ∑i=1dfi(u)≤1 for each u∈V(G). We use dstR(G) to denote the maximum number of functions among all signed total Roman dominating families on G. Our purpose in this paper is to examine the effects on γstR(G) when G is modified by removing or subdividing an edge. In addition, we determine the number dstR(G) for the case that G is a complete graph or bipartite graph.