On the signed total Roman domination and domatic numbers of graphs

Abstract

A signed total Roman dominating function on a graph G is a function f:V(G)⟶{−1,1,2} such that ∑u∈N(v)f(u)≥1 for every vertex v∈V(G), where N(v) is the neighborhood of v, and every vertex u∈V(G) for which f(u)=−1 is adjacent to at least one vertex w for which f(w)=2. The signed total Roman domination number of a graph G, denoted by γstR(G), equals the minimum weight of a signed total Roman dominating function. A set {f1,f2,…,fd} of distinct signed total Roman dominating functions on G with the property that ∑i=1dfi(v)≤1 for each v∈V(G), is called a signed total Roman dominating family (of functions) on G. The maximum number of functions in a signed total Roman dominating family on G is the signed total Roman domatic number of G, denoted by dstR(G). In this paper we continue the investigation of the signed total Roman domination number, and we initiate the study of the signed total Roman domatic number in graphs. We present sharp bounds for γstR(G) and dstR(G). In addition, we determine the total signed Roman domatic number of some graphs.