Signed star domatic number of a graph

Let image be a simple graph without isolated vertices with vertex set image and edge set image. A function image is said to be a signed star dominating function on image if image for every vertex image of image, where image. A set image of signed star dominating functions on image with the property that image for each image, is called a signed star dominating family (of functions) on image. The maximum number of functions in a signed star dominating family on image is the signed star domatic number of image, denoted by image.