The Total Number of Generalized Stable Sets and Kernels of Graphs

In [8] a graph representation of the Fibonacci numbers \(F_n\) and Lucas numbers \(F_y^*\) was presented. It is interesting to know that they are the total numbers of all stable sets of undirected graphs \(P_n\) and \(C_n\), respectively. In this paper we discuss a more general concept of stable sets and kernels of graphs. Our aim is to determine the total numbers of all \(k\)-stable sets and \((k, k-1)\)-kernels of graphs \(P_n\) and \(C_n\). The results are given by the second-order linear recurrence relations containing generalized Fibonacci and Lucas numbers. Recent problems were investigated in [9], [10].