, Volume 78, Issue 3, pp 914–944 | Cite as

Identification, Location-Domination and Metric Dimension on Interval and Permutation Graphs. II. Algorithms and Complexity

  • Florent Foucaud
  • George B. Mertzios
  • Reza Naserasr
  • Aline ParreauEmail author
  • Petru Valicov


We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted Identifying Code, (Open) Open Locating-Dominating Set and Metric Dimension) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diameter 2 and permutation graphs of diameter 2. While Identifying Code and (Open) Locating-Dominating Set are trivially fixed-parameter-tractable when parameterized by solution size, it is known that in the same setting Metric Dimension is W[2]-hard. We show that for interval graphs, this parameterization of Metric Dimension is fixed-parameter-tractable.


Metric dimension Resolving set Identifying code Locating-dominating set Interval graph Permutation graph Complexity 



We thank Adrian Kosowski for helpful preliminary discussions on the topic of this paper. We are also grateful to the reviewers for their useful comments which subsequently made the paper clearer.


  1. 1.
    Agnarsson, G., Damaschke, P., Halldórsson, M.M.: Powers of geometric intersection graphs and dispersion algorithms. Discrete Appl. Math. 132(1–3), 3–16 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Auger, D.: Minimal identifying codes in trees and planar graphs with large girth. Eur. J. Comb. 31(5), 1372–1384 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Babai, L.: On the complexity of canonical labeling of strongly regular graphs. SIAM J. Comput. 9(1), 212–216 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bampas, E., Bilò, D., Drovandi, G., Gualà, L., Klasing, R., Proietti, G.: Network verification via routing table queries. In: Proceedings of the 18th International Colloquium on Structural Information and Communication Complexity, SIROCCO 2011, LNCS 6796: 270–281 (2011)Google Scholar
  5. 5.
    Beerliova, Z., Eberhard, F., Erlebach, T., Hall, A., Hoffmann, M., Mihalák, M., Ram, L.S.: Network discovery and verification. IEEE J. Sel. Areas Commun. 24(12), 2168–2181 (2006)CrossRefzbMATHGoogle Scholar
  6. 6.
    Belmonte, R., Fomin, F.V., Golovach, P.A., Ramanujan, M.S.: Metric dimension of bounded width graphs. In: Proceedings of 40th International Symposium of Mathematical Foundations of Computer Science, MFCS 2015, LNCS 9235: 115–126, (2015)Google Scholar
  7. 7.
    Berger-Wolf, T.Y., Laifenfeld, M., Trachtenberg, A.: Identifying codes and the set cover problem. In: Proceedings of the 44th Annual Allerton Conference on Communication, Control and Computing, Monticello, USA (2006)Google Scholar
  8. 8.
    Bertrand, N., Charon, I., Hudry, O., Lobstein, A.: 1-identifying codes on trees. Australas. J. Comb. 31, 21–35 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bollobás, B., Scott, A.D.: On separating systems. Eur. J. Comb. 28, 1068–1071 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bondy, J.A.: Induced subsets. J. Comb. Theory Ser. B 12(2), 201–202 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bousquet, N., Lagoutte, A., Li, Z., Parreau, A., Thomassé, S.: Identifying codes in hereditary classes of graphs and VC-dimension. SIAM J. Discrete Math. 29(4), 2047–2064 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Brandstädt, A., Le, V.B., Spinrad, J.: Graph classes: a survey. SIAM Monogr. Discrete Math. Appl. (1999)Google Scholar
  14. 14.
    Charon, I., Hudry, O., Lobstein, A.: Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard. Theoret. Comput. Sci. 290(3), 2109–2120 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Charbit, E., Charon, I., Cohen, G., Hudry, O., Lobstein, A.: Discriminating codes in bipartite graphs: bounds, extremal cardinalities, complexity. Adv. Math. Commun. 2(4), 403–420 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chartrand, G., Eroh, L., Johnson, M., Oellermann, O.: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105(1–3), 99–113 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chellali, M.: On locating and differetiating-total domination in trees. Discuss. Math. Graph Theory 28(3), 383–392 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cohen, G., Honkala, I., Lobstein, A., Zémor, G.: On identifyingcodes. In: Proceedings of the DIMACS Workshop on Codes andAssociation Schemes, Series in Discrete Mathematics and Theoretical Computer Science 5697–109, (2001)Google Scholar
  19. 19.
    Colbourn, C., Slater, P.J., Stewart, L.K.: Locating-dominating sets in series-parallel networks. Congr. Numer. 56, 135–162 (1987)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Diaz, J., Pottonen, O., Serna, M., van Leeuwen, E.J.: On the complexity of metric dimension. In: Proceedings of the 20th European Symposium on Algorithms, ESA 2012, LNCS 7501: 419–430 (2012)Google Scholar
  22. 22.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  23. 23.
    Eppstein, D.: Metric dimension parameterized by max leaf number. J. Graph Algorithms Appl. 19(1), 313–323 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Epstein, L., Levin, A., Woeginger, G.J.: The (weighted) metric dimension of graphs: hard and easy cases. Algorithmica 72(4), 1130–1171 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Fernau, H., Heggernes, P., van’t Hof, P., Meister, D., Saei, R.: Computing the metric dimension for chain graphs. Inf. Process. Lett. 115, 671–676 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Foucaud, F.: Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes. J. Discrete Algorithms 31, 48–68 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Foucaud, F., Gravier, S., Naserasr, R., Parreau, A., Valicov, P.: Identifying codes in line graphs. J. Graph Theory 73(4), 425–448 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Foucaud, F., Mertzios, G., Naserasr, R., Parreau, A., Valicov, P.: Algorithms and complexity for metric dimension and location-domination on interval and permutation graphs. In: Proceedings of the 41st International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2015, LNCS, to appearGoogle Scholar
  29. 29.
    Foucaud, F., Mertzios, G., Naserasr, R., Parreau, A., Valicov, P.: Identification, location-domination and metric dimension on interval and permutation graphs. I. Bounds (2015).
  30. 30.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman, New York (1979)zbMATHGoogle Scholar
  31. 31.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Elsevier, New York (2004)zbMATHGoogle Scholar
  32. 32.
    Gravier, S., Klasing, R., Moncel, J.: Hardness results and approximation algorithms for identifying codes and locating-dominating codes in graphs. Algorithm. Oper. Res. 3(1), 43–50 (2008)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Habib, M., Paul, C.: A simple linear time algorithm for cograph recognition. Discrete Appl. Math. 145(2), 183–197 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Harary, F., Melter, R.A.: On the metric dimension of a graph. Ars Comb. 2, 191–195 (1976)zbMATHGoogle Scholar
  35. 35.
    Hartung, S.: Exploring parameter spaces in coping with computational intractability. PhD Thesis, TU Berlin, Germany (2014)Google Scholar
  36. 36.
    Hartung, S., Nichterlein, A.: On the parameterized and approximation hardness of metric dimension. Proc. IEEE Conf. Comput. Complex. CCC 2013, 266–276 (2013)MathSciNetGoogle Scholar
  37. 37.
    Henning, M.A., Rad, N.J.: Locating-total domination in graphs. Discrete Appl. Math. 160, 1986–1993 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Henning, M.A.H., Yeo, A.: Distinguishing-transversal in hypergraphs and identifying open codes in cubic graphs. Graphs Comb. 30(4), 909–932 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Hoffmann, S., Wanke, E.: Metric dimension for Gabriel unit diskgraphs is NP-Complete. In: Proceedings of ALGOSENSORS 2012: 90–92 (2012)Google Scholar
  40. 40.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)CrossRefGoogle Scholar
  41. 41.
    Karpovsky, M.G., Chakrabarty, K., Levitin, L.B.: On a new class of codes for identifying vertices in graphs. IEEE Trans. Inf. Theory 44, 599–611 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discrete Appl. Math. 70(3), 217–229 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Kim, J.H., Pikhurko, O., Spencer, J., Verbitsky, O.: How complex are random graphs in First Order logic? Random Struct. Algorithms 26(1–2), 119–145 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Kloks, T.: Treewidth, Computations and Approximations. Springer, New York (1994)zbMATHGoogle Scholar
  45. 45.
    Manuel, P., Rajan, B., Rajasingh, I., Chris Monica, M.: On minimum metric dimension of honeycomb networks. J. Discrete Algorithms 6(1), 20–27 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Moret, B.M.E., Shapiro, H.D.: On minimizing a set of tests. SIAM J. Sci. Stat. Comput. 6(4), 983–1003 (1985)CrossRefGoogle Scholar
  47. 47.
    Müller, T., Sereni, J.-S.: Identifying and locating-dominating codes in (random) geometric networks. Comb. Probab. Comput. 18(6), 925–952 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  49. 49.
    Rényi, A.: On random generating elements of a finite Boolean algebra. Acta Sci. Math. Szeged 22, 75–81 (1961)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Seo, S.J., Slater, P.J.: Open neighborhood locating-dominating sets. Australas. J. Comb. 46, 109–120 (2010)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Slater, P.J.: Leaves of trees. Congr. Numer. 14, 549–559 (1975)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Slater, P.J.: Domination and location in acyclic graphs. Networks 17(1), 55–64 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Slater, P.J.: Dominating and reference sets in a graph. J. Math. Phys. Sci. 22(4), 445–455 (1988)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Spinrad, J.: Bipartite permutation graphs. Discrete Appl. Math. 18(3), 279–292 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Suomela, J.: Approximability of identifying codes and locating-dominating codes. Inf. Process. Lett. 103(1), 28–33 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Ungrangsi, R., Trachtenberg, A., Starobinski, D.: An implementation of indoor location detection systems based on identifying codes. In: Proceedings of Intelligence in Communication Systems, INTELLCOMM 2004, LNCS 3283:175–189 (2004)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Florent Foucaud
    • 1
  • George B. Mertzios
    • 2
  • Reza Naserasr
    • 3
  • Aline Parreau
    • 4
    Email author
  • Petru Valicov
    • 5
  1. 1.LIMOS - CNRS UMR 6158Université Blaise PascalClermont-FerrandFrance
  2. 2.School of Engineering and Computing SciencesDurham UniversityDurhamUK
  3. 3.CNRS - IRIF, Université Paris DiderotParisFrance
  4. 4.University Lyon, Université Claude Bernard Lyon 1, CNRS, LIRIS, UMR 5205VilleurbanneFrance
  5. 5.Aix-Marseille Université, CNRS, LIF, UMR 7279MarseilleFrance

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