Signed Edge \(k\)-Independence in Graphs

Let \( G = (V, E) \) be a simple graph with vertex set \( V \) and edge set \( E \). If \( k \geq 2 \) is an integer, then the signed edge \( k \)-independence function of \( G \) is a function \( f : E \to \{-1, 1\} \) such that \(\sum_{e’ \in N[e]} f(e’) \leq k – 1\) for each \( e \in E \). The weight of a signed edge \( k \)-independence function \( f \) is \(\omega(f) = \sum_{e \in E} f(e).\) The signed edge \( k \)-independence number \( \alpha_k^s(G) \) of \( G \) is the maximum weight of a signed edge \( k \)-independence function of \( G \). In this paper, we initiate the study of the signed edge \( k \)-independence number and we present bounds for this parameter. In particular, we determine this parameter for some classes of graphs.