On the Existence of (k,l)-Kernels in Infinite Digraphs: A Survey

H. Galeana-Sánchez; C. Hernández-Cruz

Discussiones Mathematicae Graph Theory (2014)

  • Volume: 34, Issue: 3, page 431-466
  • ISSN: 2083-5892

Abstract

top
Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent (if u, v ∈ N, u 6= v, then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) − N then there exists v ∈ N such that d(u, v) ≤ l) set of vertices. A k-kernel is a (k, k −1)-kernel. This work is a survey of results proving sufficient conditions for the existence of (k, l)-kernels in infinite digraphs. Despite all the previous work in this direction was done for (2, 1)-kernels, we present many original results concerning (k, l)-kernels for distinct values of k and l. The original results are sufficient conditions for the existence of (k, l)- kernels in diverse families of infinite digraphs. Among the families that we study are: transitive digraphs, quasi-transitive digraphs, right/left pretransitive digraphs, cyclically k-partite digraphs, κ-strong digraphs, k-transitive digraphs, k-quasi-transitive digraphs

How to cite

top

H. Galeana-Sánchez, and C. Hernández-Cruz. "On the Existence of (k,l)-Kernels in Infinite Digraphs: A Survey." Discussiones Mathematicae Graph Theory 34.3 (2014): 431-466. <http://eudml.org/doc/268081>.

@article{H2014,
abstract = {Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent (if u, v ∈ N, u 6= v, then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) − N then there exists v ∈ N such that d(u, v) ≤ l) set of vertices. A k-kernel is a (k, k −1)-kernel. This work is a survey of results proving sufficient conditions for the existence of (k, l)-kernels in infinite digraphs. Despite all the previous work in this direction was done for (2, 1)-kernels, we present many original results concerning (k, l)-kernels for distinct values of k and l. The original results are sufficient conditions for the existence of (k, l)- kernels in diverse families of infinite digraphs. Among the families that we study are: transitive digraphs, quasi-transitive digraphs, right/left pretransitive digraphs, cyclically k-partite digraphs, κ-strong digraphs, k-transitive digraphs, k-quasi-transitive digraphs},
author = {H. Galeana-Sánchez, C. Hernández-Cruz},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {kernel; k-kernel; (k; l)-kernel; infinite digraph; -kernel; -kernel},
language = {eng},
number = {3},
pages = {431-466},
title = {On the Existence of (k,l)-Kernels in Infinite Digraphs: A Survey},
url = {http://eudml.org/doc/268081},
volume = {34},
year = {2014},
}

TY - JOUR
AU - H. Galeana-Sánchez
AU - C. Hernández-Cruz
TI - On the Existence of (k,l)-Kernels in Infinite Digraphs: A Survey
JO - Discussiones Mathematicae Graph Theory
PY - 2014
VL - 34
IS - 3
SP - 431
EP - 466
AB - Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent (if u, v ∈ N, u 6= v, then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) − N then there exists v ∈ N such that d(u, v) ≤ l) set of vertices. A k-kernel is a (k, k −1)-kernel. This work is a survey of results proving sufficient conditions for the existence of (k, l)-kernels in infinite digraphs. Despite all the previous work in this direction was done for (2, 1)-kernels, we present many original results concerning (k, l)-kernels for distinct values of k and l. The original results are sufficient conditions for the existence of (k, l)- kernels in diverse families of infinite digraphs. Among the families that we study are: transitive digraphs, quasi-transitive digraphs, right/left pretransitive digraphs, cyclically k-partite digraphs, κ-strong digraphs, k-transitive digraphs, k-quasi-transitive digraphs
LA - eng
KW - kernel; k-kernel; (k; l)-kernel; infinite digraph; -kernel; -kernel
UR - http://eudml.org/doc/268081
ER -

References

top
  1. [1] J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications (Springer-Verlag, Berlin Heidelberg New York, 2002). Zbl1001.05002
  2. [2] J. Bang-Jensen and J. Huang, Quasi-transitive digraphs, J. Graph Theory 20 (1995) 141-161. doi:10.1002/jgt.3190200205[Crossref] Zbl0832.05048
  3. [3] C. Berge, Graphs (North-Holland, Amsterdam New York, 1985). 
  4. [4] D. Bród, A. W loch and I. W loch, On the existence of (k, k − 1)-kernels in directed graphs, J. Math. Appl. 28 (2006) 7-12. 
  5. [5] M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, The Strong Perfect Graph Theorem, Ann. of Math. 164 (2006) 51-229. doi:10.4007/annals.2006.164.51[Crossref] Zbl1112.05042
  6. [6] V. Chvátal, On the computational complexity of finding a kernel, Report No. CRM-300, Centre de Recherches Mathematiques, Universite de Montreal, 1973. 
  7. [7] V. Chvátal and L. Lov´asz, Every directed graph has a semi-kernel , Lecture Notes in Math. 411 (1974) 175. doi:10.1007/BFb0066192[Crossref] 
  8. [8] R. Diestel, Graph Theory 3rd Edition (Springer-Verlag, Berlin Heidelberg New York, 2005). 
  9. [9] P. Duchet, Graphes noyau-parfaits, Ann. Discrete Math. 9 (1980) 93-101. doi:10.1016/S0167-5060(08)70041-4[Crossref] 
  10. [10] P. Duchet and H. Meyniel, Kernels in directed graphs: a poison game, Discrete Math. 115 (1993) 273-276. doi:10.1016/0012-365X(93)90496-G[Crossref] Zbl0773.05052
  11. [11] P.L. Erdös and L. Soukup, Quasi-kernels and quasi-sinks in infinite digraphs, Discrete Math. 309 (2009) 3040-3048. doi:10.1016/j.disc.2008.08.006[Crossref][WoS] Zbl1208.05033
  12. [12] A.S. Fraenkel, Combinatorial game theory foundations applied to digraph kernels, Electron. J. Combin. 4 (1997) #R10. Zbl0884.05045
  13. [13] H. Galeana-Sánchez and M. Guevara, Some sufficient conditions for the existence of kernels in infinite digraphs, Discrete Math. 309 (2009) 3680-3693. doi:10.1016/j.disc.2008.01.025[WoS][Crossref] Zbl1225.05110
  14. [14] H. Galeana-Sánchez and C. Hernández-Cruz, Cyclically k-partite digraphs and k- kernels, Discuss. Math. Graph Theory 31 (2011) 63-78. doi:10.7151/dmgt.1530[Crossref] Zbl1284.05114
  15. [15] H. Galeana-Sánchez and C. Hernández-Cruz, k-kernels in generalizations of transi- tive digraphs, Discuss. Math. Graph Theory 31 (2011) 293-312. doi:10.7151/dmgt.1546[Crossref] 
  16. [16] H. Galeana-Sánchez and C. Hernández-Cruz, On the existence of (k, l)-kernels in digraphs with a given circumference, AKCE Int. J. Graphs Combin. (2013), to appear. Zbl1304.05058
  17. [17] H. Galeana-Sánchez and C. Hernández-Cruz, k-kernels in k-transitive and k-quasi- transitive digraphs, Discrete Math. 312 (2012) 2522-2530. doi:10.1016/j.disc.2012.05.005[WoS][Crossref] Zbl1246.05067
  18. [18] H. Galeana-Sánchez and C. Hernández-Cruz, k-kernels in multipartite tournaments, AKCE Int. J. Graphs Combin. 8 (2011) 181-198. 
  19. [19] H. Galeana-Sánchez, C. Hernández-Cruz and M.A. Ju´arez-Camacho, On the exis- tence and number of (k+1)-kings in k-quasi-transitive digraphs, Discrete Math. 313 (2013) 2582-2591. doi:10.1016/j.disc.2013.08.007[WoS][Crossref] Zbl1281.05068
  20. [20] H. Galeana-Sánchez and H.A. Rincón, A sufficient condition for the existence of k-kernels in digraphs, Discuss. Math. Graph Theory 18 (1998) 197-204. doi:10.7151/dmgt.1075[Crossref] Zbl0928.05032
  21. [21] A. Ghouila-Houri, Caractérization des graphes non orient´es dont onpeut orienter les arrˆetes de mani`ere `a obtenir le graphe dune relation dordre, Comptes Rendus de l’Acad´emie des Sciences Paris 254 (1962) 1370-1371. Zbl0105.35503
  22. [22] P. Hell and C. Hernández-Cruz, On the complexity of the 3-kernel problem in some classes of digraphs, Discuss. Math. Graph Theory 34 (2014) 167-201. doi:10.7151/dmgt.1727 [Crossref] 
  23. [23] P. Hell and J. Nešetřil, Graphs and Homomorphisms (Oxford University Press, 2004). doi:10.1093/acprof:oso/9780198528173.001.0001[Crossref] Zbl1062.05139
  24. [24] 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[Crossref] 
  25. [25] M. Kwaśnik, On (k, l)-kernels on graphs and their products, Doctoral Dissertation, Technical University of Wroc law, Wroc law, 1980. 
  26. [26] M. Kwaśnik, The generalizaton of Richardson’s theorem, Discuss. Math. 4 (1981) 11-14. 
  27. [27] M. Kwaśnik, A. W loch and I. W loch, Some remarks about (k, l)-kernels in directed and undirected graphs, Discuss. Math. 13 (1993) 29-37. 
  28. [28] V. Neumann-Lara, Semin´ucleos de una digr´afica, Anales del Instituto de Matem´aticas II (1971). 
  29. [29] M. Richardson, On weakly ordered systems, Bull. Amer. Math. Soc. 52 (1946) 113-116. doi:10.1090/S0002-9904-1946-08518-3[Crossref] Zbl0060.06506
  30. [30] R. Rojas-Monroy and I. Villarreal-Vald´es, Kernels in infinite digraphs, AKCE Int. J. Graphs Combin. 7 (2010) 103-111. 
  31. [31] W. Szumny, A.W loch and I.W loch, On (k, l)-kernels in D-join of digraphs, Discuss. 
  32. Math. Graph Theory 27 (2007) 457-470. doi:10.7151/dmgt.1373[Crossref] 
  33. [32] W. Szumny, A. W loch and I. W loch, On the existence and on the number of (k, l)- kernels in the lexicographic product of graphs, Discrete Math. 308 (2008) 4616-4624. doi:10.1016/j.disc.2007.08.078[Crossref][WoS] 
  34. [33] J. von Neumann, O.Morgenstern, Theory of Games and Economic Behavior (Princeton University Press, Princeton, 1953). Zbl0053.09303

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.