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On k-independent sets and (k, l)-kernels in the corona of graphs

Szumny W., Włoch A.

Journal of Mathematics and Applications

2007

Vol. 29

127-135

A subset S ⊆ V(G) is a k-independent set if no two of its vertices are in distance less than k. In this paper we study fc-independent sets and (k, l)-kernels (i.e. k-independent sets being l-dominating simultaneously) in the corona of graphs. We describe an arbitrary k-independent set of the corona and next we determine the Fibonacci number and the generalized Fibonacci number of the corona of special graphs. We give the necessary and sufficient conditions for the existence of (k, l)-kernels in the corona.

Fibonacci numbers of trees

Startek M., Włoch A., Włoch I.

Journal of Mathematics and Applications

2006

Vol. 28

137-145

In [6] it was presented a graph-representation of the Fibonacci numbers Fn and Lucas numbers Ln. It is interesting to know that they are the totał numbers of independent sets of undirected graphs Pn and Cn, respectively. More general concept of the number of all k-independent sets of graphs Pn and Cn was discussed in [5]. In [6], [7] it was bounded the number of all independent sets of a tree Tn. In this paper we propose the method which estimate the number Fk(Tn) of all k-independent sets of Tn. We also describe graphs G for which the numbers Fk(G) are the generalizations of the Fibonacci numbers.