Iwona Włoch

On kernels by monochromatic paths in the corona of digraphs

In this paper we derive necessary and sufficient conditions for the existence of kernels by monochromatic paths in the corona of digraphs. Using these results, we are able to prove the main result of this paper which provides necessary and sufficient conditions for the corona of digraphs to be monochromatic kernel-perfect. Moreover we calculate the total numbers of kernels by monochromatic paths, independent by monochromatic paths sets and dominating by monochromatic paths sets in this digraphs product.

On the existence and on the number of (k,l)-kernels in the lexicographic product of graphs

In [G. Hopkins, W. Staton, Some identities arising from the Fibonacci numbers of certain graphs, Fibonacci Quart. 22 (1984) 225-228.] and [I. Wloch, 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. Wloch, I. Wloch, 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.

On ( k,l )-kernels in generalized products

Abstract We prove some theorems, which describe the parameters k , l of ( k , l )-kernels in the generalized Cartesian product with respect to digraphs and in the generalized lexicographical product with respect to digraphs and graphs.

Generalized sequences and k-independent sets in graphs

In this paper we give a generalization of known sequences and then we give their graph representations. We generalize Fibonacci numbers, Lucas numbers, Pell numbers, Tribonacci numbers and we prove that they are equal to the total number of k-independent sets in special graphs.

Fibonacci numbers and Lucas numbers in graphs

A subset S@?V(G) is independent if no two vertices of S are adjacent in G. In this paper we study the number of independent sets in graphs with two elementary cycles. In particular we determine the smallest number and the largest number of these sets among n-vertex graphs with two elementary cycles. The extremal values of the number of independent sets are described using Fibonacci numbers and Lucas numbers.

Note: Trees with extremal numbers of maximal independent sets including the set of leaves

A subset S of vertices of a graph G is independent if no two vertices in S are adjacent. In this paper we study maximal (with respect to set inclusion) independent sets in trees including the set of leaves. In particular the smallest and the largest number of these sets among n-vertex trees are determined characterizing corresponding trees. We define a local augmentation of trees that preserves the number of maximal independent sets including the set of leaves.

On (k,l)-kernels in D-join of digraphs

In [5] the necessary and sufficient conditions for the existence of (k, l)-kernels in a D-join of digraphs were given if the digraph D is without circuits of length less than k. In this paper we generalize these results for an arbitrary digraph D. Moreover, we give the total number of (k, l)-kernels, k-independent sets and l-dominating sets in a D-join of digraphs.

On a new type of distance Fibonacci numbers

In this paper, we define a new kind of Fibonacci numbers generalized in the distance sense. This generalization is related to distance Fibonacci numbers and distance Lucas numbers, introduced quite recently. We also study distinct properties of these numbers for negative integers. Their representations and interpretations in graphs are also studied.

On Types of Distance Fibonacci Numbers Generated by Number Decompositions

We introduce new types of distance Fibonacci numbers which are closely related with number decompositions. Using special decompositions of the number we give a sequence of identities for them. Moreover, we give matrix generators for distance Fibonacci numbers and their direct formulas.

On

We define in this paper new distance generalizations of the Pell numbers and the companion Pell numbers. We give a graph interpretation of these numbers with respect to a special 3-edge colouring of the graph.