Signed domatic number of a graph

Let G be a finite and simple graph with the vertex set image, and let image be a two-valued function. If image for each image, where image is the closed neighborhood of image, then f is a signed dominating function on G. A set image of signed dominating functions on G with the property that image for each image, is called a signed dominating family (of functions) on G. The maximum number of functions in a signed dominating family on G is the signed domatic number on G, denoted by image.