The signed Roman domatic number of a digraph - Seyed Mahmoud Sheikholeslami; Lutz Volkmann (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2015)
Let D be a finite and simple digraph with vertex set V(D).A signed Roman dominating function on the digraph D isa function f:V (D){-1, 1, 2} such that_uN^-[v]f(u)1 for every vV(D), where N^-[v] consists of v andall inner neighbors of v, and every vertex uV(D) for which f(u)=-1 has an innerneighbor v for which f(v)=2. A set {f_1,f_2,…,f_d} of distinct signedRoman dominating functions on D with the property that _i=1^df_i(v)1 for eachvV(D), is called a signed Roman dominating family (of functions) on D. The maximumnumber of functions in a signed Roman dominating family on D is the signed Roman domaticnumber of D, denoted by d_sR(D). In this paper we initiate the study of signed Romandomatic number in digraphs and we present some sharp bounds for d_sR(D). In addition, wedetermine the signed Roman domatic number of some digraphs. Some of our results are extensionsof well-known properties of the signed Roman domatic number of graphs.