Signed star kk-subdomination numbers in graphs

Let GG be a simple graph without isolated vertex 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 kk-subdominating function of GG if ∑e∈E(v)f(e)≥1∑e∈E(v)f(e)≥1 for at least kk vertices vv of GG, where E(v)={uv∈E(G)∣u∈N(v)}E(v)={uv∈E(G)∣u∈N(v)}. The value min∑e∈E(G)f(e)min∑e∈E(G)f(e), taking over all signed star kk-subdominating function ff of GG is called the signed star kk-subdomination number of GG and denoted by γSSk(G). In this paper we give some bounds on the signed star kk-subdomination number of graphs.