Advertisement

Efficient closed domination in digraph products

  • Iztok PeterinEmail author
  • Ismael G. Yero
Article
  • 14 Downloads

Abstract

A digraph D is an efficient closed domination digraph if there exists a subset S of V(D) for which the closed out-neighborhoods centered in vertices of S form a partition of V(D). In this work we deal with efficient closed domination digraphs among several product of digraphs. We completely describe the efficient closed domination digraphs among lexicographic and strong products of digraphs. We characterize those direct products of digraphs that are efficient closed domination digraphs, where factors are either two cycles or two paths. Among Cartesian product of digraphs, we describe all such efficient closed domination digraphs such that they are a Cartesian product digraph either with a cycle or with a star.

Keywords

Efficient closed domination Digraphs Products of digraphs 

Mathematics Subject Classification

05C69 05C76 

Notes

References

  1. Abay-Asmerom G, Hammack RH, Taylor DT (2008) Total perfect codes in tensor products of graphs. Ars Combin 88:129–134MathSciNetzbMATHGoogle Scholar
  2. Abay-Asmerom G, Hammack RH, Taylor DT (2009) Perfect \(r\)-codes in strong products of graphs. Bull Inst Combin Appl 55:66–72MathSciNetzbMATHGoogle Scholar
  3. Bange DW, Barkauskas AE, Slater PJ (1978) Disjoint dominating sets in trees. Sandia Laboratories Report, SAND 78-1087JGoogle Scholar
  4. Barkauskas AE, Host LH (1993) Finding efficient dominating sets in oriented graphs. Congr Numer 98:27–32MathSciNetzbMATHGoogle Scholar
  5. Biggs N (1973) Perfect codes in graphs. J Combin Theory Ser B 15:289–296MathSciNetCrossRefGoogle Scholar
  6. Chelvam TT (2012) Efficient open domination in Cayley graphs. Appl Math Lett 25:1560–1564MathSciNetCrossRefGoogle Scholar
  7. Cockayne EJ, Hartnell BL, Hedetniemi ST, Laskar R (1993) Perfect domination in graphs. J Comb Inf Syst Sci 18:136–148MathSciNetzbMATHGoogle Scholar
  8. Cowen R, Hechler SH, Kennedy JW, Steinberg A (2007) Odd neighborhood transversals on grid graphs. Discrete Math 307:2200–2208MathSciNetCrossRefGoogle Scholar
  9. Dejter IJ (2008) Perfect domination in regular grid graphs. Australas J Combin 42:99–114MathSciNetzbMATHGoogle Scholar
  10. Dorbec P, Gravier S, Klavžar S, Špacapan S (2006) Some results on total domination in direct products of graphs. Discuss Math Graph Theory 26:103–112MathSciNetCrossRefGoogle Scholar
  11. Gavlas H, Schultz K (2002) Efficient open domination. Electron Notes Discrete Math 11:681–691MathSciNetCrossRefGoogle Scholar
  12. Gravier S (2002) Total domination number of grid graphs. Discrete Appl Math 121:119–128MathSciNetCrossRefGoogle Scholar
  13. Hammack R, Imrich W, Klavžar S (2011) Handbook of product graphs, 2nd edn. CRC Press, Boca RatonzbMATHGoogle Scholar
  14. Huang J, Xu J-M (2008) The bondage numbers and efficient dominations of vertex-transitive graphs. Discret Math 308:571–582MathSciNetCrossRefGoogle Scholar
  15. Jerebic J, Klavžar S, Špacapan S (2005) Characterizing \(r\)-perfect codes in direct products of two and three cycles. Inf Process Lett 94:1–6MathSciNetCrossRefGoogle Scholar
  16. Jha PK (2014) Tight-optimal circulants vis-à-vis twisted tori. Discrete Appl Math 175:24–34MathSciNetCrossRefGoogle Scholar
  17. Klavžar S, Špacapan S, Žerovnik J (2006) An almost complete description of perfect codes in direct products of cycles. Adv Appl Math 37:2–18MathSciNetCrossRefGoogle Scholar
  18. Klavžar S, Peterin I, Yero IG (2017) Graphs that are simultaneously efficient open domination and efficient closed domination graphs. Discrete Appl Math 217:613–621MathSciNetCrossRefGoogle Scholar
  19. Klostermeyer WF, Goldwasser JL (2006) Total perfect codes in grid graphs. Bull Inst Combin Appl 46:61–68MathSciNetzbMATHGoogle Scholar
  20. Knor M, Potočnik P (2012) Efficient domination in cubic vertex-transitive graphs. Eur J Combin 33:1755–1764MathSciNetCrossRefGoogle Scholar
  21. Kraner Šumenjak T, Peterin I, Rall DF, Tepeh A (2016) Partitioning the vertex set of G to make GDH an efficient open domination graph. Discrete Math Theoret Comput Sci 18:#10Google Scholar
  22. Kuziak D, Peterin I, Yero IG (2014) Efficient open domination in graph products. Discrete Math Theoret Comput Sci 16:105–120MathSciNetzbMATHGoogle Scholar
  23. Martinez C, Beivide R, Gabidulin E (2007) Perfect codes for metrics induced by circulant graphs. IEEE Trans Inf Theory 53:3042–3052MathSciNetCrossRefGoogle Scholar
  24. McAndrew MH (1963) On the product of directed graphs. Proc Am Math Soc 14:600–606MathSciNetCrossRefGoogle Scholar
  25. Mollard M (2011) On perfect codes in Cartesian products of graphs. Eur J Combin 32:398–403MathSciNetCrossRefGoogle Scholar
  26. Niepel Ĺ, Černý A (2009) Efficient domination in directed tori and the Vizing’s conjecture for directed graphs. Ars Combin 91:411–422MathSciNetzbMATHGoogle Scholar
  27. Schaudt O (2012) Efficient total domination in digraphs. J Discrete Algorithms 15:32–42MathSciNetCrossRefGoogle Scholar
  28. Schwenk AJ, Yue BQ (2005) Efficient dominating sets in labeled rooted oriented trees. Discrete Math 305(13):276–298MathSciNetCrossRefGoogle Scholar
  29. Shiau AC, Shiau T-H, Wang Y-L (2017) Efficient absorbants in generalized de Bruijn digraphs. Discrete Optim 25:77–85MathSciNetCrossRefGoogle Scholar
  30. Smart CB, Slater PJ (1995) Complexity results for closed neighborhood order parameters. Congr Numer 112:83–96MathSciNetzbMATHGoogle Scholar
  31. Sohn MY, Chen X-G, Hu F-T (2018) On efficiently total dominatable digraphs. Bull Malays Math Sci Soc 41:1749–1758MathSciNetCrossRefGoogle Scholar
  32. Taylor DT (2009) Perfect \(r\)-codes in lexicographic products of graphs. Ars Combin 93:215–223MathSciNetzbMATHGoogle Scholar
  33. Wang Y-L, Wu KH, Kloks T (2013) On perfect absorbants in generalized de Bruijn digraphs. In: Proceedings of the frontiers in algorithmics and algorithmic aspects in information and management, third joint international conference, FAW-AAIM, pp 303–314Google Scholar
  34. Wu KH, Wang Y-L, Kloks T (2017) On efficient absorbant conjecture in generalized de Bruijn digraphs. Int J Comput Math 94:922–932MathSciNetCrossRefGoogle Scholar
  35. Žerovnik J (2008) Perfect codes in direct products of cycles: a complete characterization. Adv Appl Math 41:197–205MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Electrical Engineering and Computer ScienceUniversity of MariborMariborSlovenia
  2. 2.Departamento de Matemáticas, Escuela Politécnica Superior de AlgecirasUniversidad de CádizAlgecirasSpain

Personalised recommendations