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Graphs and Combinatorics

, Volume 30, Issue 4, pp 909–932 | Cite as

Distinguishing-Transversal in Hypergraphs and Identifying Open Codes in Cubic Graphs

  • Michael A. Henning
  • Anders Yeo
Original Paper

Abstract

The open neighborhood N(v) of a vertex v in a graph G is the set of vertices adjacent to v in G. A graph is twin-free (or open identifiable) if every two distinct vertices have distinct open neighborhoods. A separating open code in G is a set C of vertices such that \({N(u) \cap C \neq N(v) \cap C}\) for all distinct vertices u and v in G. An open dominating set, or total dominating set, in G is a set C of vertices such that \({N(u) \cap C \ne N(v) \cap C}\) for all vertices v in G. An identifying open code of G is a set C that is both a separating open code and an open dominating set. A graph has an identifying open code if and only if it is twin-free. If G is twin-free, we denote by \({\gamma^{\rm IOC}(G)}\) the minimum cardinality of an identifying open code in G. A hypergraph H is identifiable if every two edges in H are distinct. A distinguishing-transversal T in an identifiable hypergraph H is a subset T of vertices in H that has a nonempty intersection with every edge of H (that is, T is a transversal in H) such that T distinguishes the edges, that is, \({e \cap T \neq f \cap T}\) for every two distinct edges e and f in H. The distinguishing-transversal number \({\tau_D(H)}\) of H is the minimum size of a distinguishing-transversal in H. We show that if H is a 3-uniform identifiable hypergraph of order n and size m with maximum degree at most 3, then \({20\tau_D(H) \leq 12n + 3m}\) . Using this result, we then show that if G is a twin-free cubic graph on n vertices, then \({\gamma^{\rm IOC}(G) \leq 3n/4}\) . This bound is achieved, for example, by the hypercube.

Keywords

Distinguishing transversals Hypergraphs Identifying opencodes Total domination 

Mathematics Subject Classification

05C69 

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References

  1. 1.
    Chvátal V., McDiarmid C.: Small transversals in hypergraphs. Combinatorica 12, 19–26 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    De Bontridder K.M.J., Halldórsson B.V., Halldórsson M.M., Hurkens C.A.J, Lenstra J.K, Ravi R., Stougie L.: Approximation algorithms for the test cover problem. Math. Program. Ser. B 98, 477–491 (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Foucaud F., Guerrini E., Kovšea M., Naserasra R., Parreaub A., Valicova P.: Extremal graphs for the identifying code problem. Europ. J. Combin. 32, 628–638 (2011)CrossRefzbMATHGoogle Scholar
  4. 4.
    Haynes, T.W., Hedetniemi, S.T., Slater P.J.: Fundamentals of Domination in Graphs, Marcel Dekker, Inc. New York 1998Google Scholar
  5. 5.
    Haynes T.W., Knisley D.J., Seier E., Zou Y.: A quantitative analysis of secondary RNA structure using domination based parameters on trees. BMC Bioinform. 7, 108 (2006)CrossRefGoogle Scholar
  6. 6.
    Henning M.A.: Recent results on total domination in graphs: A survey. Discret. Math. 309, 32–63 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Henning M.A., Yeo A.: Hypergraphs with large transversal number and with edge sizes at least three. J. Graph Theory 59, 326–348 (2008)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Henning M.A., Yeo A.: Identifying vertex covers in graphs. Electronic J. Combin. 19(4), #P32 (2012)MathSciNetGoogle Scholar
  9. 9.
    Honkala I., Karpovsky M.G., Litsyn S.: On the identification of vertices and edges using cycles. Lecture Note Comput. Sci. 2227, 308–314 (2001)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Honkala I., Karpovsky M.G., Litsyn S.: Cycles identifying vertices and edges in binary hypercubes and 2-dimensional tori. Discret. Appl. Math. 129, 409–419 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Honkala I., Laihonen T., Ranto S.: On strongly identifying codes. Discret. Math. 254, 191–205 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Karpovsky M.G., Chakrabarty K., Levitin L.B.: On a new class of codes for identifying vertices in graphs. IEEE Trans. Inform. Theory 44, 599–611 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Lai F.C., Chang G.J.: An upper bound for the transversal numbers of 4-uniform hypergraphs. J. Combin. Theory Ser. B 50, 129–133 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Laifenfeld, M.. Trachtenberg, A., Cohen, R., Starobinski, D.: Joint monitoring and routing in wireless sensor networks using robust identifying codes. In: Proceedings of IEEE Broadnets , pp. 197–06, 2007Google Scholar
  15. 15.
  16. 16.
    Moncel, J.: Codes Identifants dans les graphes. PhD thesis, Université Joseph Fourier Grenoble I, France, 2005 - available online at http://tel.archives-ouvertes.fr/tel-00010293/en/
  17. 17.
    Moret M.M.E., Shapiro H.D.: On minimizing a set of tests. SIAM J. Sci. Stat. Comput. 6, 983–1003 (1985)CrossRefGoogle Scholar
  18. 18.
    Ray, S., Ungrangsi, R., De Pellegrini, F., Trachtenberg, A., Starobinski, D.: Robust location detection in emergency sensor networks. Proceedings of IEEE INFOCOM 2003, pp. 1044–1053, 2003Google Scholar
  19. 19.
    Seo S.J., Slater P.J.: Open neighborhood locating-dominating sets. Australas. J. Combin. 46, 109–120 (2010)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Seo S.J., Slater P.J.: Open neighborhood locating-dominating in trees. Discret. Appl. Math. 159, 484–489 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Thomassé S., Yeo A.: Total domination of graphs and small transversals of hypergraphs. Combinatorica 27, 473–487 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Tuza Z.s.: Covering all cliques of a graph. Discret. Math. 86, 117–126 (1990)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of JohannesburgJohannesburgSouth Africa

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