Signed star domatic number of a graph

Let GG be a simple graph without isolated vertices with vertex set V(G)V(G) and edge set E(G)E(G). A function f:E(G)⟶{−1,1}f:E(G)⟶{−1,1} is said to be a signed star dominating function on GG if ∑e∈E(v)f(e)≥1∑e∈E(v)f(e)≥1 for every vertex vv of GG, where E(v)={uv∈E(G)∣u∈N(v)}E(v)={uv∈E(G)∣u∈N(v)}. A set {f1,f2,…,fd}{f1,f2,…,fd} of signed star dominating functions on GG with the property that ∑i=1dfi(e)≤1 for each e∈E(G)e∈E(G), is called a signed star dominating family (of functions) on GG. The maximum number of functions in a signed star dominating family on GG is the signed star domatic number of GG, denoted by dSS(G)dSS(G).In this paper we study the properties of the signed star domatic number dSS(G)dSS(G). In particular, we determine the signed domatic number of some classes of graphs.