On Graph Identification Problems and the Special Case of Identifying Vertices Using Paths

  • Florent Foucaud
  • Matjaž Kovše
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7643)


In this paper, we introduce the identifying path cover problem: an identifying path cover of a graph G is a set \(\mathcal P\) of paths such that each vertex belongs to a path of \(\mathcal P\), and for each pair u,v of vertices, there is a path of \(\mathcal P\) which includes exactly one of u,v. This problem is related to a large variety of identification problems. We investigate the identifying path cover problem in some families of graphs. In particular, we derive the optimal size of an identifying path cover for paths, cycles, hypercubes and topologically irreducible trees and give an upper bound for all trees. We give lower and upper bounds on the minimum size of an identifying path cover for general graphs, and discuss their tightness. In particular, we show that any connected graph G has an identifying path cover of size at most \(\left\lceil\frac{2(|V(G)|-1)}{3}\right\rceil\). We also study the computational complexity of the associated optimization problem, in particular we show that when the length of the paths is asked to be of a fixed value, the problem is APX-complete.


Test cover Identification Paths Approximation 


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  1. 1.
    Auger, D., Charon, I., Hudry, O., Lobstein, A.: Watching systems in graphs: an extension of identifying codes. Discrete Appl. Math. (in press)Google Scholar
  2. 2.
    Berger-Wolf, T.Y., Laifenfeld, M., Trachtenberg, A.: Identifying codes and the set cover problem. In: Proc. 44th Annual Allerton Conference on Communication, Control and Computing (September 2006)Google Scholar
  3. 3.
    Bondy, J.A.: Induced subsets. J. Comb. Theory B 12(2), 201–202 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Charon, I., Cohen, G., Hudry, O., Lobstein, A.: Discriminating codes in bipartite graphs. Adv. Math. Commun. 4(2), 403–420 (2008)MathSciNetGoogle Scholar
  5. 5.
    Charon, I., Honkala, I., Hudry, O., Lobstein, A.: Structural properties of twin-free graphs. Electron. J. Comb. 14, R16 (2007)MathSciNetGoogle Scholar
  6. 6.
    Charon, I., Hudry, O., Lobstein, A.: Identifying and locating-dominating codes: NP-completeness results for directed graphs. IEEE T. Inform. Theory 48(8), 2192–2200 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chlebík, M., Chlebíková, J.: Complexity of approximating bounded variants of optimization problems. Theor. Comput. Sci. 354, 320–338 (2006)zbMATHCrossRefGoogle Scholar
  8. 8.
    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. B 98, 477–491 (2003)zbMATHCrossRefGoogle Scholar
  9. 9.
    Foucaud, F., Guerrini, E., Kovše, M., Naserasr, R., Parreau, A., Valicov, P.: Extremal graphs for the identifying code problem. Eur. J. Combin. 32(4), 628–638 (2011)zbMATHCrossRefGoogle Scholar
  10. 10.
    Foucaud, F., Gravier, S., Naserasr, R., Parreau, A., Valicov, P.: Identifying codes in line graphs. To appear in J. Graph Theor.Google Scholar
  11. 11.
    Foucaud, F., Naserasr, R., Parreau, A.: Characterizing extremal digraphs for identifying codes and extremal cases of Bondy’s theorem on induced subsets. Graphs Comb. (in press)Google Scholar
  12. 12.
    Frieze, A., Martin, R., Moncel, J., Ruszinkó, M., Smyth, C.: Codes identifying sets of vertices in random networks. Discrete Math. 307(9-10), 1094–1107 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)Google Scholar
  14. 14.
    Honkala, I., Karpovsky, M., Litsyn, S.: Cycles identifying vertices and edges in binary hypercubes and 2-dimensional tori. Discrete Appl. Math. 129(2-3), 409–419 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)zbMATHCrossRefGoogle Scholar
  16. 16.
    Karpovsky, M., Chakrabarty, K., Levitin, L.B.: On a new class of codes for identifying vertices in graphs. IEEE T. Inform. Theory 44, 599–611 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Laifenfeld, M., Trachtenberg, A., Cohen, R., Starobinski, D.: Joint monitoring and routing in wireless sensor networks using robust identifying codes. In: Proc. IEEE Broadnets 2007, pp. 197–206 (September 2007)Google Scholar
  18. 18.
    Moret, B.M.E., Shapiro, H.D.: On minimizing a set of tests. SIAM J. Sci. Stat. Comp. 6(4), 983–1003 (1985)CrossRefGoogle Scholar
  19. 19.
    Papadimitriou, C.: Computational Complexity. Addison-Wesley (1994)Google Scholar
  20. 20.
    Rosendahl, P.: On the identification of vertices using cycles. Electron. J. Comb. 10, P1 (2003)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Florent Foucaud
    • 1
    • 2
  • Matjaž Kovše
    • 3
    • 4
  1. 1.Univ. Bordeaux, LaBRI, UMR5800TalenceFrance
  2. 2.CNRS, LaBRI, UMR5800TalenceFrance
  3. 3.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
  4. 4.Bioinformatics Group, Department of Computer Science and Interdisciplinary Center for BioinformaticsUniv. of LeipzigLeipzigGermany

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