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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1373, author = {Waldemar Szumny and Andrzej W\l och and Iwona W\l och}, title = {On (k,l)-kernels in D-join of digraphs}, journal = {Discussiones Mathematicae Graph Theory}, volume = {27}, year = {2007}, pages = {457-470}, zbl = {1142.05040}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1373} }
Waldemar Szumny; Andrzej Włoch; Iwona Włoch. On (k,l)-kernels in D-join of digraphs. Discussiones Mathematicae Graph Theory, Tome 27 (2007) pp. 457-470. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1373/
[000] [1] M. Blidia, P. Duchet, H. Jacob and F. Maffray, Some operations preserving the existence of kernels, Discrete Math. 205 (1999) 211-216, doi: 10.1016/S0012-365X(99)00026-6. | Zbl 0936.05047
[001] [2] R. Diestel, Graph Theory (Springer-Verlag, Heidelberg, New-York, Inc., 2005).
[002] [3] H. Galeana-Sanchez, On the existence of kernels and h-kernels in directed graphs, Discrete Math. 110 (1992) 251-225, doi: 10.1016/0012-365X(92)90713-P. | Zbl 0770.05050
[003] [4] M. Kucharska, On (k,l)-kernels of orientations of special graphs, Ars Combin. 60 (2001) 137-147. | Zbl 1068.05504
[004] [5] M. Kucharska, On (k,l)-kernel perfectness of special classes of digraphs, Discuss. Math. Graph Theory 25 (2005) 103-119, doi: 10.7151/dmgt.1265. | Zbl 1074.05043
[005] [6] M. Kwaśnik and I. Włoch, The total number of generalized stable sets and kernels of graphs, Ars Combin. 55 (2000) 139-146.
[006] [7] A. Włoch and I. Włoch, On (k,l)-kernels in generalized products, Discrete Math. 164 (1997) 295-301, doi: 10.1016/S0012-365X(96)00064-7.
[007] [8] I. Włoch, Generalized Fibonacci polynomial of graphs, Ars Combin. 68 (2003) 49-55. | Zbl 1071.05556