On the existence and on the number of [formula omitted]-kernels in the lexicographic product of graphs

Abstract

In [G. Hopkins, W. Staton, Some identities arising from the Fibonacci numbers of certain graphs, Fibonacci Quart. 22 (1984) 225–228.] and [I. Włoch, Generalized Fibonacci polynomial of graphs, Ars Combinatoria 68 (2003) 49–55] the total number of k -independent sets in the generalized lexicographic product of graphs was given. In this paper we study ( k , l ) -kernels (i.e. k -independent sets being l -dominating, simultaneously) in this product and we generalize some results from [A. Włoch, I. Włoch, The total number of maximal k -independent sets in the generalized lexicographic product of graphs, Ars Combinatoria 75 (2005) 163–170]. We give the necessary and sufficient conditions for the existence of ( k , l ) -kernels in it. Moreover, we construct formulas which calculate the number of all ( k , l ) -kernels, k -independent sets and l -dominating sets in the lexicographic product of graphs for all parameters k , l . The result concerning the total number of independent sets generalizes the Fibonacci polynomial of graphs. Also for special graphs we give some recurrence formulas.