Signed mixed Roman domination numbers in graphs

Let (G = (V;E)) be a simple graph with vertex set (V) and edge set (E). A signed mixed Roman dominating function (SMRDF) of (G) is a function (f: Vcup Erightarrow {-1,1,2}) satisfying the conditions that (i) (sum _{yin N_m[x]}f(y)ge 1) for each (xin Vcup E), where (N_m[x]) is the set, called mixed closed neighborhood of (x), consists of (x) and the elements of (Vcup E) adjacent or incident to (x) (ii) every element (xin Vcup E) for which (f(x) = -1) is adjacent or incident to at least one element (yin Vcup E) for which (f(y) = 2). The weight of a SMRDF (f) is (omega (f)=sum _{xin Vcup E}f(x)). The signed mixed Roman domination number (gamma _{sR}^*(G)) of (G) is the minimum weight of a SMRDF of (G). In this paper we initiate the study of the signed mixed Roman domination number and we present bounds for this parameter. In particular, we determine this parameter for some classes of graphs.