Research Specifications

A signed double Roman dominating function (SDRDF) on a graph G is a function f : V ! f􀀀1; 1; 2; 3g such that Σ u2N[u] f(u)  1 for every v 2 V (G), and every vertex u 2 V (G) for which f(u) = 􀀀1, is adjacent to at least two neighbors assigned 2 under f or one neighbor w with f(w) = 3, and if f(u) = 1, then vertex u must have at least one neighbor w with f(w)  2. The weight of a signed double Roman dominating function f is the value, f(V (G)) = Σ u2V (G) f(u). The signed double Roman domination number sdR(G) of G is the minimum weight of a SDRDF of G. A set ff1; f2; :::; fdg of distinct signed double Roman dominating functions on G with the property that Σd i=1 fi(v)  2 for each v 2 V (G), is called a signed double Roman dominating family (of functions) on G. The maximum number of functions in a signed double Roman dominating family on G is the signed double Roman domatic number of G, denoted by dsdR(G). In this work we initiate the study of the signed double Roman domatic number in graphs and we present some sharp bounds for dsdR(G). In addition, we determine the Roman domatic number of some graphs. signed double Roman dominating function, signed double Roman domination number, signed double Roman domatic number