Line digraphs can be obtained by sequences of state splittings, a particular kind of operation widely used in symbolic dynamics [12]. Properties of line digraphs inherited from the source have been studied, for instance in [7] Harminc showed that the cardinalities of the sets of kernels and solutions (kernel's dual definition) of a digraph and its line digraph coincide. We extend this for (k,l)-kernels in the context of state splittings and also look at (k,l)-semikernels, k-Grundy functions and their duals.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1367,
author = {Hortensia Galeana-S\'anchez and Ricardo G\'omez},
title = {(k,l)-kernels, (k,l)-semikernels, k-Grundy functions and duality for state splittings},
journal = {Discussiones Mathematicae Graph Theory},
volume = {27},
year = {2007},
pages = {359-371},
zbl = {1133.05039},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1367}
}
Hortensia Galeana-Sánchez; Ricardo Gómez. (k,l)-kernels, (k,l)-semikernels, k-Grundy functions and duality for state splittings. Discussiones Mathematicae Graph Theory, Tome 27 (2007) pp. 359-371. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1367/
[000] [1] C. Berge, Graphs (North-Holland, Amsterdam, 1985).
[001] [2] M. Boyle and R. Wagoner, Positive algebraic K-theory and shifts of finite type. Modern dynamical systems and applications (Cambridge University Press, 2004) 45-66. | Zbl 1148.37303
[002] [3] H. Galeana-Sánchez, On the existence of (k,l)-kernels in digraphs, Discrete Math. 85 (1990) 99-102, doi: 10.1016/0012-365X(90)90167-G. | Zbl 0729.05020
[003] [4] H. Galeana-Sánchez and Xueliang Li, Semikernels and (k,l)-kernels in digraphs, SIAM J. Disc. Math. 11 (1998) 340-346. | Zbl 0907.05025
[004] [5] H. Galeana-Sánchez, L. Pastrana Ramí rez and H.A. Rincón Mejí a, Semikernels, quasikernels and Grundy functions in the line digraph, SIAM J. Disc. Math. 4 (1991) 80-83. Discrete Math. 59 (1986) 257-265.
[005] [6] R. Gómez, Positive K-theory for finitary isomorphisms of Markov chains, Ergodic Theory and Dynam. Systems. 23 (2003) 1485-1504, doi: 10.1017/S0143385702001700. | Zbl 1060.37007
[006] [7] M. Harminc, Solutions and kernels of a directed graph, Math. Slovaca 32 (1982) 263-267. | Zbl 0491.05029
[007] [8] B. Kitchens, Symbolic dynamics. One-sided, two-sided and countable state Markov shifts (Springer-Verlag, 1998). | Zbl 0892.58020
[008] [9] M. Kwaśnik, On the (k,l)-kernels, Graph Theory ( agów, 1981), 114-121, Lecture Notes in Math., 1018 (Springer, Berlin, 1983).
[009] [10] M. Kwaśnik, A. Włoch and I. Włoch, Some remarks about (k,l)-kernels in directed and undirected graphs, Discuss. Math. 13 (1993) 29-37.
[010] [11] M. Kucharska and M. Kwaśnik, On (k,l)-kernels of special superdigraphs of Pₘ and Cₘ, Discuss. Math. Graph Theory 21 (2001) 95-109, doi: 10.7151/dmgt.1135.
[011] [12] D. Lind and B. Marcus, An introduction to symbolic dynamics and coding (Cambridge University Press, 1995), doi: 10.1017/CBO9780511626302. | Zbl 1106.37301
[012] [13] V. Neumann-Lara, Seminuclei of a digraph, (Spanish) An. Inst. Mat. Univ. Nac. Autónoma México 11 (1971) 55-62. | Zbl 0286.05113