Signed star k-domatic number of a graph

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 $f:E(G)longrightarrow {-1, 1}$ is said to be a signed star $k$-dominating function on $G$ if $sum_{ein E(v)}f(e)ge k$ for every vertex $v$ of $G$, where $E(v)={uvin E(G)mid uin N(v)}$. A set ${f_1,f_2,ldots,f_d}$ of signed star $k$-dominating functions on $G$ with the property that $sum_{i=1}^df_i(e)le 1$ for each $ein E(G)$, 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 $d_{kSS}(G)$.