Network Decontamination

  • Nicolas Nisse
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11340)


The Network Decontamination problem consists of coordinating a team of mobile agents in order to clean a contaminated network. The problem is actually equivalent to tracking and capturing an invisible and arbitrarily fast fugitive. This problem has natural applications in network security in computer science or in robotics for search or pursuit-evasion missions. Many different objectives have been studied: the main one being the minimization of the number of mobile agents necessary to clean a contaminated network.

Many environments (continuous or discrete) have also been considered. In this Chapter, we focus on networks modeled by graphs. In this context, the optimization problem that consists of minimizing the number of agents has a deep graph-theoretical interpretation. Network decontamination and, more precisely, graph searching models, provide nice algorithmic interpretations of fundamental concepts in the Graph Minors theory by Robertson and Seymour.

For all these reasons, graph searching variants have been widely studied since their introduction by Breish (1967) and mathematical formalizations by Parsons (1978) and Petrov (1982). This chapter consists of an overview of the algorithmic results on graph decontamination and graph searching.


Graph searching Path- and tree-decompositions (Distributed) graph algorithms Computational complexity 


  1. [ABC+15]
    Ames, B.P.W., et al.: A leapfrog strategy for pursuit-evasion in a polygonal environment. Int. J. Comput. Geom. Appl. 25(2), 77–100 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  2. [ACN15]
    Amini, O., Coudert, D., Nisse, N.: Non-deterministic graph searching in trees. Theor. Comput. Sci. 580, 101–121 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  3. [ADHY07]
    Alspach, B., Dyer, D., Hanson, D., Yang, B.: Arc searching digraphs without jumping. In: Dress, A., Xu, Y., Zhu, B. (eds.) COCOA 2007. LNCS, vol. 4616, pp. 354–365. Springer, Heidelberg (2007). Scholar
  4. [Adl07]
    Adler, I.: Directed tree-width examples. J. Comb. Theory Ser. B 97(5), 718–725 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  5. [AKR16]
    Amiri, S.A., Kreutzer, S., Rabinovich, R.: Dag-width is PSPACE-complete. Theor. Comput. Sci. 655, 78–89 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  6. [Als04]
    Alspach, B.: Searching and sweeping graphs: a brief survey. Mathematiche 59, 5–37 (2004)MathSciNetzbMATHGoogle Scholar
  7. [Bar06]
    Barát, J.: Directed path-width and monotonicity in digraph searching. Graphs Comb. 22(2), 161–172 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [BBM+13]
    Berthomé, P., Bouvier, T., Mazoit, F., Nisse, N., Soares, R.P.: An unified FPT algorithm for width of partition functions. Research Report RR-8372, INRIA, September 2013Google Scholar
  9. [BBN12]
    Blin, L., Burman, J., Nisse, N.: Brief announcement: distributed exclusive and perpetual tree searching. In: Aguilera, M.K. (ed.) DISC 2012. LNCS, vol. 7611, pp. 403–404. Springer, Heidelberg (2012). Scholar
  10. [BBN17]
    Blin, L., Burman, J., Nisse, N.: Exclusive graph searching. Algorithmica 77(3), 942–969 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  11. [BDH+12]
    Berwanger, D., Dawar, A., Hunter, P., Kreutzer, S., Obdrzálek, J.: The DAG-width of directed graphs. J. Comb. Theory Ser. B 102(4), 900–923 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  12. [BDK15]
    Borowiecki, P., Dereniowski, D., Kuszner, L.: Distributed graph searching with a sense of direction. Distrib. Comput. 28(3), 155–170 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  13. [BF02]
    Bodlaender, H.L., Fomin, F.V.: Approximation of pathwidth of outerplanar graphs. J. Algorithms 43(2), 190–200 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  14. [BFF+12]
    Barrière, L., et al.: Connected graph searching. Inf. Comput. 219, 1–16 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  15. [BFFS02]
    Barrière, L., Flocchini, P., Fraigniaud, P., Santoro, N.: Capture of an intruder by mobile agents. In: Proceedings of the 14th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pp. 200–209 (2002)Google Scholar
  16. [BFL+09]
    Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) kernelization. In: 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 629–638. IEEE Computer Society (2009)Google Scholar
  17. [BFNV08]
    Blin, L., Fraigniaud, P., Nisse, N., Vial, S.: Distributed chasing of network intruders. Theor. Comput. Sci. 399(1–2), 12–37 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. [BFST03]
    Barrière, L., Fraigniaud, P., Santoro, N., Thilikos, D.M.: Searching is not jumping. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 34–45. Springer, Heidelberg (2003). Scholar
  19. [BGTZ16]
    Best, M.J., Gupta, A., Thilikos, D.M., Zoros, D.: Contraction obstructions for connected graph searching. Discrete Appl. Math. 209, 27–47 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  20. [BH06]
    Brandenburg, F.J., Herrmann, S.: Graph searching and search time. In: Wiedermann, J., Tel, G., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2006. LNCS, vol. 3831, pp. 197–206. Springer, Heidelberg (2006). Scholar
  21. [Bie91]
    Bienstock, D.: Graph searching, path-width, tree-width and related problems (a survey). In: Proceedings of Reliability Of Computer And Communication Networks, a DIMACS Workshop. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 5, pp. 33–50. DIMACS/AMS (1991)Google Scholar
  22. [BK96]
    Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21(2), 358–402 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  23. [BKIS12]
    Bhadauria, D., Klein, K., Isler, V., Suri, S.: Capturing an evader in polygonal environments with obstacles: the full visibility case. Int. J. Robot. Res. 31(10), 1176–1189 (2012)CrossRefGoogle Scholar
  24. [BLS99]
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. Society for Industrial and Applied Mathematics, Philadelphia (1999)zbMATHCrossRefGoogle Scholar
  25. [BM93]
    Bodlaender, H.L., Möhring, R.H.: The pathwidth and treewidth of cographs. SIAM J. Discrete Math. 6(2), 181–188 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  26. [BMM10]
    Blair, J., Manne, F., Mihai, R.: Efficient self-stabilizing graph searching in tree networks. In: Dolev, S., Cobb, J., Fischer, M., Yung, M. (eds.) SSS 2010. LNCS, vol. 6366, pp. 111–125. Springer, Heidelberg (2010). Scholar
  27. [BN11]
    Bonato, A., Nowakovski, R.J.: The Game of Cops and Robber on Graphs. American Mathematical Society, Providence (2011)CrossRefGoogle Scholar
  28. [Bod98]
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209(1–2), 1–45 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  29. [Bre67]
    Breisch, R.L.: An intuitive approach to speleotopology. Southwest. Cavers 6, 72–78 (1967)Google Scholar
  30. [Bre12]
    Breish, R.L.: Lost in a Cave: Applying Graph Theory to Cave Exploration. Greyhound Press, Dallas (2012)Google Scholar
  31. [BRST91]
    Bienstock, D., Robertson, N., Seymour, P.D., Thomas, R.: Quickly excluding a forest. J. Comb. Theory Ser. B 52(2), 274–283 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  32. [BS91]
    Bienstock, D., Seymour, P.D.: Monotonicity in graph searching. J. Algorithms 12(2), 239–245 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  33. [BT04]
    Bodlaender, H.L., Thilikos, D.M.: Computing small search numbers in linear time. In: Downey, R., Fellows, M., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 37–48. Springer, Heidelberg (2004). Scholar
  34. [BTK11]
    Borie, R.B., Tovey, C.A., Koenig, S.: Algorithms and complexity results for graph-based pursuit evasion. Auton. Robots 31(4), 317–332 (2011)CrossRefGoogle Scholar
  35. [CCM+11]
    Cohen, N., Coudert, D., Mazauric, D., Nepomuceno, N., Nisse, N.: Tradeoffs in process strategy games with application in the WDM reconfiguration problem. Theor. Comput. Sci. 412(35), 4675–4687 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  36. [CDH+16]
    Corneil, D.G., Dusart, J., Habib, M., Mamcarz, A., de Montgolfier, F.: A tie-break model for graph search. Discrete Appl. Math. 199, 89–100 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  37. [CFK+15]
    Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015)zbMATHCrossRefGoogle Scholar
  38. [CHI11]
    Chung, T.H., Hollinger, G.A., Isler, V.: Search and pursuit-evasion in mobile robotics - a survey. Auton. Robots 31(4), 299–316 (2011)CrossRefGoogle Scholar
  39. [CHM+09]
    Coudert, D., Huc, F., Mazauric, D., Nisse, N., Sereni, J.-S.: Reconfiguration of the routing in WDM networks with two classes of services. In: Conference on Optical Network Design and Modeling (ONDM), Braunschweig, Germany (2009)Google Scholar
  40. [CHM12]
    Coudert, D., Huc, F., Mazauric, D.: A distributed algorithm for computing the node search number in trees. Algorithmica 63(1–2), 158–190 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  41. [CHS07]
    Coudert, D., Huc, F., Sereni, J.-S.: Pathwidth of outerplanar graphs. J. Graph Theory 55(1), 27–41 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  42. [CK06]
    Chandran, L.S., Kavitha, T.: The treewidth and pathwidth of hypercubes. Discrete Math. 306(3), 359–365 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  43. [CM93]
    Courcelle, B., Mosbah, M.: Monadic second-order evaluations on tree-decomposable graphs. Theor. Comput. Sci. 109(1&2), 49–82 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  44. [CMN16]
    Coudert, D., Mazauric, D., Nisse, N.: Experimental evaluation of a branch-and-bound algorithm for computing pathwidth and directed pathwidth. ACM J. Exp. Algorithmics 21(1), 1.3:1–1.3:23 (2016)MathSciNetzbMATHGoogle Scholar
  45. [Cou16]
    Coudert, D.: A note on integer linear programming formulations for linear ordering problems on graphs. Research report, Inria, I3S, Universite Nice Sophia Antipolis, CNRS, February 2016Google Scholar
  46. [CS11]
    Coudert, D., Sereni, J.-S.: Characterization of graphs and digraphs with small process numbers. Discrete Appl. Math. 159(11), 1094–1109 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  47. [DD13]
    Dereniowski, D., Dyer, D.: On minimum cost edge searching. Theor. Comput. Sci. 495, 37–49 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  48. [DDD13]
    Dereniowski, D., Diner, Ö.Y., Dyer, D.: Three-fast-searchable graphs. Discrete Appl. Math. 161(13–14), 1950–1958 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  49. [Der09]
    Dereniowski, D.: Maximum vertex occupation time and inert fugitive: recontamination does help. Inf. Process. Lett. 109(9), 422–426 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  50. [Der11]
    Dereniowski, D.: Connected searching of weighted trees. Theor. Comput. Sci. 412(41), 5700–5713 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  51. [Der12a]
    Dereniowski, D.: Approximate search strategies for weighted trees. Theor. Comput. Sci. 463, 96–113 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  52. [Der12b]
    Dereniowski, D.: From pathwidth to connected pathwidth. SIAM J. Discrete Math. 26(4), 1709–1732 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  53. [DFZ10]
    Daadaa, Y., Flocchini, P., Zaguia, N.: Network decontamination with temporal immunity by cellular automata. In: Bandini, S., Manzoni, S., Umeo, H., Vizzari, G. (eds.) ACRI 2010. LNCS, vol. 6350, pp. 287–299. Springer, Heidelberg (2010). Scholar
  54. [DH08]
    Demaine, E.D., Hajiaghayi, M.T.: The bidimensionality theory and its algorithmic applications. Comput. J. 51(3), 292–302 (2008)CrossRefGoogle Scholar
  55. [Die12]
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2012)zbMATHGoogle Scholar
  56. [DJS16]
    Daadaa, Y., Jamshed, A., Shabbir, M.: Network decontamination with a single agent. Graphs Comb. 32(2), 559–581 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  57. [DKL87]
    Deo, N., Krishnamoorthy, M.S., Langston, M.A.: Exact and approximate solutions for the gate matrix layout problem. IEEE Trans. CAD Integr. Circuits Syst. 6(1), 79–84 (1987)CrossRefGoogle Scholar
  58. [DKT97]
    Dendris, N.D., Kirousis, L.M., Thilikos, D.M.: Fugitive-search games on graphs and related parameters. Theor. Comput. Sci. 172(1–2), 233–254 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  59. [DKZ15]
    Dereniowski, D., Kubiak, W., Zwols, Y.: The complexity of minimum-length path decompositions. J. Comput. Syst. Sci. 81(8), 1715–1747 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  60. [DNN17]
    D’Angelo, G., Navarra, A., Nisse, N.: A unified approach for gathering and exclusive searching on rings under weak assumptions. Distrib. Comput. 30(1), 17–48 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  61. [DOR18]
    Dereniowski, D., Osula, D., Rzazewski, P.: Finding small-width connected path decompositions in polynomial time. CoRR, abs/1802.05501 (2018)Google Scholar
  62. [DSN+15]
    D’Angelo, G., Di Stefano, G., Navarra, A., Nisse, N., Suchan, K.: Computing on rings by oblivious robots: a unified approach for different tasks. Algorithmica 72(4), 1055–1096 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  63. [DU16]
    Dereniowski, D., Urbanska, D.: Distributed searching of partial grids. CoRR, abs/1610.01458 (2016)Google Scholar
  64. [DYY08]
    Dyer, D., Yang, B., Yaşar, Ö.: On the fast searching problem. In: Fleischer, R., Xu, J. (eds.) AAIM 2008. LNCS, vol. 5034, pp. 143–154. Springer, Heidelberg (2008). Scholar
  65. [EHS13]
    Evans, W., Hunter, P., Safari, M.A.: D-width and cops and robbers. Research report (2013, unpublished)Google Scholar
  66. [EM04]
    Ellis, J.A., Markov, M.: Computing the vertex separation of unicyclic graphs. Inf. Comput. 192(2), 123–161 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  67. [EST87]
    Ellis, J.A., Sudborough, I.H., Turner, J.S.: Graph separation and search number. Technical report, Report Number: WUCS-87-11 (1987)Google Scholar
  68. [EST94]
    Ellis, J.A., Sudborough, I.H., Turner, J.S.: The vertex separation and search number of a graph. Inf. Comput. 113(1), 50–79 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  69. [FFN09]
    Fomin, F.V., Fraigniaud, P., Nisse, N.: Nondeterministic graph searching: from pathwidth to treewidth. Algorithmica 53(3), 358–373 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  70. [FG00]
    Fomin, F.V., Golovach, P.A.: Graph searching and interval completion. SIAM J. Discrete Math. 13(4), 454–464 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  71. [FHL07]
    Flocchini, P., Huang, M.J., Luccio, F.L.: Decontaminating chordal rings and tori using mobile agents. Int. J. Found. Comput. Sci. 18(3), 547–563 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  72. [FHL08]
    Flocchini, P., Huang, M.J., Luccio, F.L.: Decontamination of hypercubes by mobile agents. Networks 52(3), 167–178 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  73. [FHM10]
    Fomin, F.V., Heggernes, P., Mihai, R.: Mixed search number and linear-width of interval and split graphs. Networks 56(3), 207–214 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  74. [FHT05]
    Fomin, F.V., Heggernes, P., Telle, J.A.: Graph searching, elimination trees, and a generalization of bandwidth. Algorithmica 41(2), 73–87 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  75. [FIP06]
    Fraigniaud, P., Ilcinkas, D., Pelc, A.: Oracle size: a new measure of difficulty for communication tasks. In: Proceedings of the Twenty-Fifth Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 179–187. ACM (2006)Google Scholar
  76. [FL94]
    Fellows, M.R., Langston, M.A.: On search, decision, and the efficiency of polynomial-time algorithms. J. Comput. Syst. Sci. 49(3), 769–779 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  77. [FLPS16]
    Flocchini, P., Luccio, F., Pagli, L., Santoro, N.: Network decontamination under m-immunity. Discrete Appl. Math. 201, 114–129 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  78. [FLS05]
    Flocchini, P., Luccio, F.L., Song, L.X.: Size optimal strategies for capturing an intruder in mesh networks. In: Proceedings of the International Conference on Communications in Computing (CIC), pp. 200–206. CSREA Press (2005)Google Scholar
  79. [FLS18]
    Fomin, F.V., Lokshtanov, D., Saurabh, S.: Excluded grid minors and efficient polynomial-time approximation schemes. J. ACM 65(2), 10:1–10:44 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  80. [FMS08]
    Flocchini, P., Mans, B., Santoro, N.: Tree decontamination with temporary immunity. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 330–341. Springer, Heidelberg (2008). Scholar
  81. [FN08]
    Fraigniaud, P., Nisse, N.: Monotony properties of connected visible graph searching. Inf. Comput. 206(12), 1383–1393 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  82. [FNS07]
    Flocchini, P., Nayak, A., Schulz, A.: Decontamination of arbitrary networks using a team of mobile agents with limited visibility. In: 6th Annual IEEE/ACIS International Conference on Computer and Information Science (ICIS), pp. 469–474. IEEE Computer Society (2007)Google Scholar
  83. [Fom98]
    Fomin, F.V.: Helicopter search problems, bandwidth and pathwidth. Discrete Appl. Math. 85(1), 59–70 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  84. [Fom99]
    Fomin, F.V.: Note on a helicopter search problem on graphs. Discrete Appl. Math. 95(1–3), 241–249 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  85. [Fom04]
    Fomin, F.V.: Searching expenditure and interval graphs. Discrete Appl. Math. 135(1–3), 97–104 (2004)MathSciNetCrossRefGoogle Scholar
  86. [FS06]
    Flocchini, P., Santoro, N.: Distributed security algorithms by mobile agents. In: Chaudhuri, S., Das, S.R., Paul, H.S., Tirthapura, S. (eds.) ICDCN 2006. LNCS, vol. 4308, pp. 1–14. Springer, Heidelberg (2006). Scholar
  87. [FT08]
    Fomin, F.V., Thilikos, D.M.: An annotated bibliography on guaranteed graph searching. Theor. Comput. Sci. 399(3), 236–245 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  88. [FTT05]
    Fomin, F.V., Thilikos, D.M., Todinca, I.: Connected graph searching in outerplanar graphs. Electron. Notes Discrete Math. 22, 213–216 (2005)zbMATHCrossRefGoogle Scholar
  89. [GHK+16]
    Ganian, R., et al.: Are there any good digraph width measures? J. Comb. Theory Ser. B 116, 250–286 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  90. [GHM12]
    Golovach, P.A., Heggernes, P., Mihai, R.: Edge search number of cographs. Discrete Appl. Math. 160(6), 734–743 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  91. [GHT12]
    Giannopoulou, A.C., Hunter, P., Thilikos, D.M.: LIFO-search: a min-max theorem and a searching game for cycle-rank and tree-depth. Discrete Appl. Math. 160(15), 2089–2097 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  92. [GLL+99]
    Guibas, L.J., Latombe, J.-C., LaValle, S.M., Lin, D., Motwani, R.: A visibility-based pursuit-evasion problem. Int. J. Comput. Geometry Appl. 9(4/5), 471–494 (1999)MathSciNetCrossRefGoogle Scholar
  93. [Gol89a]
    Golovach, P.A.: Equivalence of two formalizations of a search problem on a graph. Vestnik Leningrad Univ. Math 22, 13–19 (1989)MathSciNetzbMATHGoogle Scholar
  94. [Gol89b]
    Golovach, P.A.: A topological invariant in pursuit problems. Differ. Equ. 25, 657–661 (1989)MathSciNetzbMATHGoogle Scholar
  95. [Gus93]
    Gustedt, J.: On the pathwidth of chordal graphs. Discrete Appl. Math. 45(3), 233–248 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  96. [HK08]
    Hunter, P., Kreutzer, S.: Digraph measures: kelly decompositions, games, and orderings. Theor. Comput. Sci. 399(3), 206–219 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  97. [HM08]
    Heggernes, P., Mihai, R.: Mixed search number of permutation graphs. In: Preparata, F.P., Wu, X., Yin, J. (eds.) FAW 2008. LNCS, vol. 5059, pp. 196–207. Springer, Heidelberg (2008). Scholar
  98. [INS09]
    Ilcinkas, D., Nisse, N., Soguet, D.: The cost of monotonicity in distributed graph searching. Distrib. Comput. 22(2), 117–127 (2009)zbMATHCrossRefGoogle Scholar
  99. [ISZ07]
    Imani, N., Sarbazi-Azad, H., Zomaya, A.Y.: Capturing an intruder in product networks. J. Parallel Distrib. Comput. 67(9), 1018–1028 (2007)zbMATHCrossRefGoogle Scholar
  100. [ISZ08]
    Imani, N., Sarbazi-Azad, H., Zomaya, A.Y.: Intruder capturing in mesh and torus networks. Int. J. Found. Comput. Sci. 19(4), 1049–1071 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  101. [JRST01]
    Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. J. Comb. Theory Ser. B 82(1), 138–154 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  102. [Kin92]
    Kinnersley, N.G.: The vertex separation number of a graph equals its pathwidth. Inf. Process. Lett. 42, 345–350 (1992)zbMATHCrossRefGoogle Scholar
  103. [KKK15]
    Kintali, S., Kothari, N., Kumar, A.: Approximation algorithms for digraph width parameters. Theor. Comput. Sci. 562, 365–376 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  104. [KKK+16]
    Kitsunai, K., Kobayashi, Y., Komuro, K., Tamaki, H., Tano, T.: Computing directed pathwidth in o(1.89\({}^{\text{ n }}\)) time. Algorithmica 75(1), 138–157 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  105. [KO11]
    Kreutzer, S., Ordyniak, S.: Digraph decompositions and monotonicity in digraph searching. Theor. Comput. Sci. 412(35), 4688–4703 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  106. [KP85]
    Kirousis, L.M., Papadimitriou, C.H.: Interval graphs and searching. Discrete Math. 55(2), 181–184 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  107. [KP86]
    Kirousis, L.M., Papadimitriou, C.H.: Searching and pebbling. Theor. Comput. Sci. 47(3), 205–218 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  108. [KP16]
    Kinnersley, W.B., Pralat, P.: Game brush number. Discrete Appl. Math. 207, 1–14 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  109. [KS15]
    Klein, K., Suri, S.: Pursuit evasion on polyhedral surfaces. Algorithmica 73(4), 730–747 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  110. [LaP93]
    LaPaugh, A.S.: Recontamination does not help to search a graph. J. ACM 40(2), 224–245 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  111. [LPS06]
    Fabrizio, L., Pagli, L., Santoro, N.: Network decontamination with local immunization. In: Proceedings of 20th International Parallel and Distributed Processing Symposium (IPDPS). IEEE (2006)Google Scholar
  112. [Luc09]
    Luccio, F.L.: Contiguous search problem in Sierpinski graphs. Theory Comput. Syst. 44(2), 186–204 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  113. [Mal18]
    Mallach, S.: Linear ordering based MIP formulations for the vertex separation or pathwidth problem. In: Brankovic, L., Ryan, J., Smyth, W.F. (eds.) IWOCA 2017. LNCS, vol. 10765, pp. 327–340. Springer, Cham (2018). Scholar
  114. [MHG+88]
    Megiddo, N., Hakimi, S.L., Garey, M.R., Johnson, D.S., Papadimitriou, C.H.: The complexity of searching a graph. J. ACM 35(1), 18–44 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  115. [MM09]
    Mihai, R., Mjelde, M.: A self-stabilizing algorithm for graph searching in trees. In: Guerraoui, R., Petit, F. (eds.) SSS 2009. LNCS, vol. 5873, pp. 563–577. Springer, Heidelberg (2009). Scholar
  116. [MN08]
    Mazoit, F., Nisse, N.: Monotonicity of non-deterministic graph searching. Theor. Comput. Sci. 399(3), 169–178 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  117. [MNP08]
    Messinger, M.-E., Nowakowski, R.J., Pralat, P.: Cleaning a network with brushes. Theor. Comput. Sci. 399(3), 191–205 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  118. [MNP17]
    Markou, E., Nisse, N., Pérennes, S.: Exclusive graph searching vs. pathwidth. Inf. Comput. 252, 243–260 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  119. [MS88]
    Monien, B., Sudborough, I.H.: Min cut is NP-complete for edge weighted trees. Theor. Comput. Sci. 58(1), 209–229 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  120. [MT09]
    Mihai, R., Todinca, I.: Pathwidth is NP-hard for weighted trees. In: Deng, X., Hopcroft, J.E., Xue, J. (eds.) FAW 2009. LNCS, vol. 5598, pp. 181–195. Springer, Heidelberg (2009). Scholar
  121. [MTV10]
    Meister, D., Telle, J.A., Vatshelle, M.: Recognizing digraphs of Kelly-width 2. Discrete Appl. Math. 158(7), 741–746 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  122. [NdM08]
    Nesetril, J., de Mendez, P.O.: Grad and classes with bounded expansion i. Decompositions. Eur. J. Comb. 29(3), 760–776 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  123. [Nis09]
    Nisse, N.: Connected graph searching in chordal graphs. Discrete Appl. Math. 157(12), 2603–2610 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  124. [Nis14]
    Nisse, N.: Algorithmic complexity: between structure and knowledge how pursuit-evasion games help. Habilitation à Diriger des Recherches, Université Nice Sophia-Antipolis (2014).
  125. [NS09]
    Nisse, N., Soguet, D.: Graph searching with advice. Theor. Comput. Sci. 410(14), 1307–1318 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  126. [NS16]
    Nisse, N., Soares, R.P.: On the monotonicity of process number. Discrete Appl. Math. 210, 103–111 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  127. [Par78a]
    Parsons, T.D.: Pursuit-evasion in a graph. In: Alavi, Y., Lick, D.R. (eds.) Theory and Applications of Graphs. LNM, vol. 642, pp. 426–441. Springer, Berlin (1978). Scholar
  128. [Par78b]
    Parsons, T.D.: The search number of a connected graph. In: 9th Southeastern Conference on Combinatorics, Graph Theory and Computing, Congress. Numer., vol. XXI, pp. 549–554. Utilitas Mathematica (1978)Google Scholar
  129. [Pet82]
    Petrov, N.N.: A problem of pursuit in the absence of information on the pursued. Differ. Uravn. 18, 1345–1352 (1982)MathSciNetGoogle Scholar
  130. [PHH+00]
    Peng, S.-L., Ho, C.-W., Hsu, T., Ko, M.-T., Tang, C.Y.: Edge and node searching problems on trees. Theor. Comput. Sci. 240(2), 429–446 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  131. [PSS13]
    Penuel, J., Cole Smith, J., Shen, S.: Integer programming models and algorithms for the graph decontamination problem with mobile agents. Networks 61(1), 1–19 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  132. [PTK+00]
    Peng, S.-L., Tang, C.Y., Ko, M.-T., Ho, C.-W., Hsu, T.: Graph searching on some subclasses of chordal graphs. Algorithmica 27(3), 395–426 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  133. [PY07]
    Peng, S.-L., Yang, Y.-C.: On the treewidth and pathwidth of biconvex bipartite graphs. In: Cai, J.-Y., Cooper, S.B., Zhu, H. (eds.) TAMC 2007. LNCS, vol. 4484, pp. 244–255. Springer, Heidelberg (2007). Scholar
  134. [RS83]
    Robertson, N., Seymour, P.D.: Graph minors. I. Excluding a forest. J. Comb. Theory Ser. B 35(1), 39–61 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  135. [RS90]
    Robertson, N., Seymour, P.D.: Graph minors. IV. Tree-width and well-quasi-ordering. J. Comb. Theory Ser. B 48(2), 227–254 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  136. [RS95]
    Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory Ser. B 63(1), 65–110 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  137. [RS04]
    Robertson, N., Seymour, P.D.: Graph minors. XX. Wagner’s conjecture. J. Comb. Theory Ser. B 92(2), 325–357 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  138. [RT11]
    Richerby, D., Thilikos, D.M.: Searching for a visible, lazy fugitive. SIAM J. Discrete Math. 25(2), 497–513 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  139. [Saf05]
    Safari, M.A.: D-width: a more natural measure for directed tree width. In: Jędrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 745–756. Springer, Heidelberg (2005). Scholar
  140. [SIS06]
    Shareghi, P., Imani, N., Sarbazi-Azad, H.: Capturing an intruder in the pyramid. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 580–590. Springer, Heidelberg (2006). Scholar
  141. [Sko03]
    Skodinis, K.: Construction of linear tree-layouts which are optimal with respect to vertex separation in linear time. J. Algorithms 47(1), 40–59 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  142. [ST93]
    Seymour, P.D., Thomas, R.: Graph searching and a min-max theorem for tree-width. J. Comb. Theory Ser. B 58(1), 22–33 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  143. [ST07]
    Suchan, K., Todinca, I.: Pathwidth of circular-arc graphs. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 258–269. Springer, Heidelberg (2007). Scholar
  144. [SV09]
    Suchan, K., Villanger, Y.: Computing pathwidth faster than 2n. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 324–335. Springer, Heidelberg (2009). Scholar
  145. [SY09]
    Stanley, D., Yang, B.: Lower bounds on fast searching. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 964–973. Springer, Heidelberg (2009). Scholar
  146. [SY11]
    Stanley, D., Yang, B.: Fast searching games on graphs. J. Comb. Optim. 22(4), 763–777 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  147. [Thi00]
    Thilikos, D.M.: Algorithms and obstructions for linear-width and related search parameters. Discrete Appl. Math. 105(1–3), 239–271 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  148. [TUK95]
    Takahashi, A., Ueno, S., Kajitani, Y.: Mixed searching and proper-path-width. Theor. Comput. Sci. 137(2), 253–268 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  149. [WAPL14]
    Yu, W., Austrin, P., Pitassi, T., Liu, D.: Inapproximability of treewidth and related problems. J. Artif. Intell. Res. 49, 569–600 (2014)zbMATHCrossRefGoogle Scholar
  150. [XY17]
    Xue, Y., Yang, B.: The fast search number of a Cartesian product of graphs. Discrete Appl. Math. 224, 106–119 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  151. [XYZZ16]
    Xue, Y., Yang, B., Zhong, F., Zilles, S.: Fast searching on complete k-partite graphs. In: Chan, T.-H.H., Li, M., Wang, L. (eds.) COCOA 2016. LNCS, vol. 10043, pp. 159–174. Springer, Cham (2016). Scholar
  152. [Yan07]
    Yang, B.: Strong-mixed searching and pathwidth. J. Comb. Optim. 13(1), 47–59 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  153. [Yan11]
    Yang, B.: Fast edge searching and fast searching on graphs. Theor. Comput. Sci. 412(12–14), 1208–1219 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  154. [Yan13]
    Yang, B.: Fast-mixed searching and related problems on graphs. Theor. Comput. Sci. 507, 100–113 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  155. [YC07a]
    Yang, B., Cao, Y.: Directed searching digraphs: monotonicity and complexity. In: Cai, J.-Y., Cooper, S.B., Zhu, H. (eds.) TAMC 2007. LNCS, vol. 4484, pp. 136–147. Springer, Heidelberg (2007). Scholar
  156. [YC07b]
    Yang, B., Cao, Y.: Monotonicity of strong searching on digraphs. J. Comb. Optim. 14(4), 411–425 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  157. [YC08a]
    Yang, B., Cao, Y.: Digraph searching, directed vertex separation and directed pathwidth. Discrete Appl. Math. 156(10), 1822–1837 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  158. [YC08b]
    Yang, B., Cao, Y.: Monotonicity in digraph search problems. Theor. Comput. Sci. 407(1–3), 532–544 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  159. [YC08c]
    Yang, B., Cao, Y.: On the monotonicity of weak searching. In: Hu, X., Wang, J. (eds.) COCOON 2008. LNCS, vol. 5092, pp. 52–61. Springer, Heidelberg (2008). Scholar
  160. [YC09]
    Yang, B., Cao, Y.: Standard directed search strategies and their applications. J. Comb. Optim. 17(4), 378–399 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  161. [YDA09]
    Yang, B., Dyer, D., Alspach, B.: Sweeping graphs with large clique number. Discrete Math. 309(18), 5770–5780 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  162. [YZC07]
    Yang, B., Zhang, R., Cao, Y.: Searching cycle-disjoint graphs. In: Dress, A., Xu, Y., Zhu, B. (eds.) COCOA 2007. LNCS, vol. 4616, pp. 32–43. Springer, Heidelberg (2007). Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université Côte d’Azur, Inria, CNRS, I3SSophia AntipolisFrance

Personalised recommendations