On the Number of \( k \)-Dominating Independent Sets in Planar Graphs

A set \(J_k\) of graph vertices is said to be \(k\) -dominating independent ( \(k \geq 1\) ) if its vertices are pairwise adjacent and every vertex not in \(J_k\) is adjacent to at least \(k\) vertices in \(J_k\) . In the present paper, we obtain new upper bounds for the number of \(k\) -dominating independent sets for \(k \geq 2\) in some planar graph classes.