Mixed Roman Domination in Graphs

Let $$G = (V, E)$$G=(V,E) be a simple graph with vertex set V and edge set E. A mixed Roman dominating function (MRDF) of G is a function $$f: V\cup E\rightarrow \{0,1,2\}$$f:V∪E→{0,1,2} satisfying the condition every element $$x\in V\cup E$$x∈V∪E for which $$f(x)= 0$$f(x)=0 is adjacent or incident to at least one element $$y\in V\cup E$$y∈V∪E for which $$f(y) = 2$$f(y)=2. The weight of a MRDF f is $$\omega (f)=\sum _{x\in V\cup E}f(x)$$ω(f)=∑x∈V∪Ef(x). The mixed Roman domination number of G is the minimum weight of a mixed Roman dominating function of G. In this paper, we initiate the study of the mixed Roman domination number and we present bounds for this parameter. We characterize the graphs attaining an upper bound and the graphs having small mixed Roman domination numbers.