Let G be a simple graph without isolated vertices with vertex set
V(G) and edge set E(G) and let k be a positive integer. A function f:E(G)⟶{−1,1} is said to be a signed star k-dominating function on G if
∑e∈E(v)​f(e)≥k for every vertex v of G, where
E(v)={uv∈E(G)∣u∈N(v)}. A set {f1​,f2​,…,fd​} of
signed star k-dominating functions on G with the property that
∑i=1d​fi​(e)≤1 for each e∈E(G), is called a signed
star k-dominating family (of functions) on G. The maximum number of
functions in a signed star k-dominating family on G is the signed
star k-domatic number of G, denoted by dkSS​(G)