Refael Hassin's Articles


  1. R. Hassin
    Generalizations of Hoffman's existence theorem for circulations
    Networks 11(1981), 243-254.


  2. R. Hassin
    Maximum flow in (s,t ) planar networks
    Information Processing Letters 13 (1981), 107.


  3. R. Hassin
    On maximizing functions by Fibonacci search
    The Fibonacci Quarterly 19 (1981), 347-351.


  4. R. Hassin
    Minimum cost flow with set constraints
    Networks 12 (1982), 1-21.


  5. A. Glazer and R. Hassin
    On the economics of subscriptions
    European Economic Review 19 (1982), 343-356.


  6. R. Hassin and Z. Lieber
    Optimal consumption with a stochastic income stream and two interest rates
    Southern Economic Journal 49 (1982), 482-493.


  7. A. Glazer and R. Hassin
    The economics of cheating in the taxi market
    Transportation Research, 17A (1983), 25-31.
    8. ** What fare is fair? **

  8. R. Hassin
    The minimum cost flow problem: A unifying approach to existing algorithms and a new tree search algorithm
    Mathematical Programming, 25 (1983), 228-239.


  9. A. Glazer and R. Hassin
    ?/M/1: On the equilibrium distribution of customer arrivals
    European Journal of Operational Research 13 (1983), 146-150.


  10. R. Hassin
    A dichotomous search for a geometric random variable
    Operations Research 32 (1984), 432-439.


  11. R. Hassin and M. Henig
    A dichotomous search for random objects on an interval
    Mathematics of Operations Research, 9 (1984), 301-308.


  12. R. Hassin
    On multicommodity flows in planar graphs
    Networks 14 (1984), 225-235.


  13. R. Hassin and D.B. Johnson
    An O(nlog2n) algorithm for maximum flow in undirected planar networks
    SIAM Journal on Computing 14 (1985), 612-624.


  14. R. Hassin and N. Megiddo
    An optimal algorithm for finding all the jumps of a monotone step-function
    Journal of Algorithms 6 (1985), 265-274.


  15. R. Hassin and E. Zemel
    On shortest paths in graphs with random weights
    Mathematics of Operations Research 10 (1985), 557-564.
    Also: A computing scheme for network problems with random edge lengths
    Stochastics in Combinatorial Optimization, G. Andreatta, F. Mason, P. Serafini (Eds.), (1987), 228-232.


  16. R. Hassin
    On the optimality of first-come last-served queues
    Econometrica 53 (1985), 201-202.
    17. ** The thirst to go first ** ** The first one now will later be last ** ** Are you being served? **

  17. A. Glazer and R. Hassin
    Stable priority purchasing in queues
    Operations Research Letters 4 (1986), 285-288.


  18. R. Hassin
    Consumer information in markets with random product quality: the case of queues and balking
    Econometrica 54 (1986), 1185-1195.


  19. R. Hassin and A. Tamir
    Efficient algorithms for optimization and selection on series-parallel graphs
    SIAM Journal on Algebraic and Discrete Methods 7 (1986), 379-389.


  20. F. Granot and R. Hassin
    Multi-terminal maximum flows in node-capacitated networks
    Discrete Applied Mathematics 13 (1986), 157-167.


  21. R. Hassin and M. Henig
    Control of arrivals and departures in a state-dependent input-output system
    Operations Research Letters 5 (1986), 33-36.


  22. A. Glazer and R. Hassin
    A deterministic single item inventory model with seller holding cost and buyer holding and shortage costs
    Operations Research 34 (1986), 613-618.


  23. A. Glazer and R. Hassin
    Equilibrium arrivals in queues with bulk service at scheduled times
    Transportation Science 21 (1987), 273-278.


  24. R. Hassin and E. Zemel
    Probabilistic analysis of the capacitated transportation problem
    Mathematics of Operations Research 13 (1988), 80-89.


  25. R. Hassin
    Solution bases of multiterminal cut problems
    Mathematics of Operations Research 13 (1988), 535-542.


  26. A. Glazer and R. Hassin
    Optimal contests
    Economic Inquiry 26 (1988), 133-143.


  27. R. Hassin and A. Tamir
    Maximizing classes of two parameter objectives over matroids
    Mathematics of Operations Research 14 (1989), 362-375.


  28. R. Hassin and N. Megiddo
    On orientations and shortest paths
    Linear Algebra and Its Applications 114/115(1989), 589-602.


  29. S. Anily and R. Hassin
    Ranking the best binary trees
    SIAM Journal on Computing 18 (1989), 882-892.


  30. R. Hassin
    An algorithm for computing maximum solution bases
    Operations Research Letters 9 (1990), 315-318.
    overview: SIGAL 1990, LNCS 450, 288-299.


  31. A. Glazer and R. Hassin
    Optimal sales to consumers who hold inventory
    Economics Letters 34 (1990), 215-220.


  32. R. Hassin and N. Megiddo
    Approximation algorithms for hitting objects by straight lines
    Discrete Applied Mathematics 30 (1991), 29-42.


  33. R. Hassin and N. Megiddo
    Exact computation of optimal inventory policies over an unbounded horizon
    Mathematics of Operations Research 16 (1991), 534-546.


  34. R. Hassin
    Multiterminal xcut problems
    Annals of Operations Research 33 (1991), 215-225.


  35. Refael Hassin and Arie Tamir
    Improved complexity bounds for location problems on the real line
    Operations Research Letters 10 (1991), 395-402.


  36. R. Hassin
    Approximation schemes for the restricted shortest path problem
    Mathematics of Operations Research 17 (1992), 36-42.


  37. R. Hassin and R. Hotovely
    Asymptotic analysis of dichotomous search with search and travel costs
    European Journal of Operational Research 58 (1992), 78-89.


  38. R. Hassin and M. Haviv
    Mean passage times and nearly uncoupled Markov chains
    SIAM Journal on Discrete Mathematics 5 (1992), 386-397.


  39. R. Hassin and A. Tamir
    Minimal length curves that are not embedded in an open planar set: The problem of a lost swimmer with a
    compass
    SIAM Journal on Control and Optimization 30 (1992), 695-703.


  40. S. Anily and R. Hassin
    The swapping problem
    Networks 22 (1992), 419-433.


  41. R. Hassin
    Algorithms for the minimum cost circulation problem based on maximizing the mean improvement
    Operations Research Letters 12 (1992), 227-233.


  42. R. Hassin and M. Henig
    Monotonicity and efficient computation of optimal dichotomous search
    Discrete Applied Mathematics 46 (1993), 221-234.


  43. E. Arkin, M. Halldorsson, and R. Hassin
    Approximating the tree and tour covers of a graph
    Information Processing Letters 47 (1993), 275-282.


  44. E. Arkin and R. Hassin
    Approximation algorithms for the geometric covering salesman problem
    Discrete Applied Mathematics 55 (1994), 197-218.


  45. R. Hassin and M. Haviv
    Equilibrium strategies and the value of information in a two line queueing system with partial jockeying
    Communications in Statistics - Stochastic Models 10 (1994), 415-436.


  46. R. Hassin and S. Rubinstein
    Approximations for the maximum acyclic subgraph problem
    Information Processing Letters 51 (1994), 133-140.


  47. R. Hassin and S. Lahav
    Maximizing the number of unused colors in the vertex coloring problem
    Information Processing Letters 52 (1994), 87-90.


  48. R. Hassin and A. Tamir
    On the minimum diameter spanning tree problem
    Information Processing Letters 53 (1995), 109-111.


  49. R. Hassin
    Decentralized regulation of a queue
    Management Science 41 (1995), 163-173.


  50. R. Hassin and M. Haviv
    Equilibrium strategies for queues with impatient customers
    Operations Research Letters 17 (1995), 41-45.


  51. M. Benelli and R. Hassin
    Optimal separable partitioning in the plane
    Discrete Applied Mathematics 59 (1995), 215-224.


  52. R. Hassin and M. Haviv
    On optimal and equilibrium retrial rates in a queueing system
    Probability in the Engineering and Informational Sciences 10 (1996), 223-227.
    Also: Proceedings of The 13th National Conference of the Australian Society for Operations Research (1995), 197-224.


  53. R. Hassin
    A flow algorithm for network synchronization
    Operations Research 44 (1996), 570-579.


  54. R. Hassin
    On the advantage of being the first server
    Management Science 42 (1996), 618-623.


  55. R. Hassin and M. Haviv
    Equilibrium threshold strategies: The case of queues with priorities
    Operations Research 45 (1997), 966-973.


  56. N. Guttmann-Beck and R. Hassin
    Approximation algorithms for min-max tree partition
    Journal of Algorithms 24 (1997), 266-286.


  57. E. Arkin, R. Hassin, and L. Klein
    Restricted delivery problems on a network
    Networks 29(1997), 205-216.


  58. R. Hassin and S. Rubinstein
    An approximation algorithm for maximum packing of 3-edge paths
    Information Processing Letters 63 (1997), 63-67.


  59. R. Hassin, S. Rubinstein and A. Tamir
    Approximation algorithms for maximum dispersion
    Operations Research Letters 21 (1997), 133-137.


  60. A. Glazer and R. Hassin
    Government failures in evaluating policy
    Public Choice 94 (1998), 105-115.


  61. S. Anily, C.A. Glass and R. Hassin
    The scheduling of maintenance service
    Discrete Applied Mathematics 82 (1998), 27-42.


  62. A. Glazer and R. Hassin
    Optimal allocation of quotas
    Economics Letters 58 (1998), 55-61.


  63. E. Arkin and R. Hassin
    On local search for weighted packing problems
    Mathematics of Operations Research 23 (1998), 640-648.
    Also: Proceedings of ESA 1997, LNCS 1284, 13-22.


  64. E. Arkin and R. Hassin
    Graph partitions with minimum degree constraints
    Discrete Mathematics 190 (1998), 55-65.


  65. N. Guttmann-Beck and R. Hassin
    Approximation algorithms for minimum tree partition
    Discrete Applied Mathematics 87 (1998), 117-137.


  66. N. Guttmann-Beck and R. Hassin
    Approximation algorithms for min-sum p-clustering
    Discrete Applied Mathematics 89 (1998), 125-142.


  67. R. Hassin and S. Rubinstein
    An approximation algorithm for the maximum traveling salesman problem
    Information Processing Letters 67 (1998), 125-130.


  68. S. Anily, C.A. Glass, and R. Hassin
    Scheduling maintenance service to three machines
    Annals of Operations Research 86 (1999), 375-391.


  69. A. Glazer and R. Hassin
    Sequential rent seeking
    Public Choice 102 (2000), 219-228.


  70. R. Hassin and S. Rubinstein
    Better approximations for Max TSP
    Information Processing Letters 75 (2000), 181-186.


  71. N. Guttmann-Beck and R. Hassin
    Approximation algorithms for minimum K-cut
    Algorithmica 27 (2000), 198-207.


  72. N. Guttmann-Beck, R. Hassin, S. Khuller, and B. Raghavachari
    Approximation algorithms with bounded performance guarantees for the clustered traveling salesman problem
    Algorithmica 28 (2000), 422-437.
    Also: Proceedings of FSTTCS 1998, LNCS 1530 6-17.


  73. E. Arkin and R. Hassin
    Minimum diameter covering problems
    Networks 36 (2000), 147-155.


  74. E. Arkin, R. Hassin and M. Sviridenko
    Approximating the maximum quadratic assignment problem
    Information Processing Letters 77 (2001), 13-16.
    Also: Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2000), 889-890.


  75. A. Ageev, R. Hassin and M. Sviridenko
    An 0.5-approximation algorithm for the MAX DICUT with Given Sides of Parts
    SIAM Journal on Discrete Mathematics 14 (2001), 246-255.
    Also : Proceedings of APPROX 2000, LNCS 1463, 34-41.


  76. R. Hassin and S. Rubinstein
    Approximation algorithms for maximum linear arrangement
    Information Processing Letters 80 (2001), 171-177.
    Also: Proceedings of SWAT 2000, LNCS 1851, 231-236.


  77. A. Glazer and R. Hassin
    The calculus of stonewalling
    Journal of Theoretical Politics 13 (2001), 413-424.


  78. E. Arkin and R. Hassin
    Orientations of mixed graphs
    Discrete Applied Mathematics 116 (2001), 313-320.


  79. R. Hassin and S. Khuller
    z-approximations
    Journal of Algorithms 41, (2001) 429-442.


  80. R. Hassin and S. Rubinstein
    A 7/8-approximation algorithm for metric Max TSP
    Information Processing Letters 81 (2002), 247-251.
    Also: Proceedings of WADS 2001, LNCS 2125, 205-209.


  81. B. Chen, R. Hassin, and M. Tzur
    Allocation of bandwidth and storage
    IIE Transactions 34 (2002), 501-507.


  82. Y. Asahiro, R. Hassin, and K. Iwama
    Complexity of finding dense subgraphs
    Discrete Applied Mathematics 121 (2002), 15-26.


  83. E. Arkin, R. Hassin, and S. Shachar
    Increasing digraph connectivity by arc addition, reversal and complement
    Discrete Applied Mathematics 122 (2002), 13-22.


  84. R. Hassin and M. Haviv
    Nash equilibrium and subgame perfection in observable queues
    Annals of Operations Research 113 (2002), 15-26.


  85. R. Hassin and A. Levin
    Approximation algorithms for constructing wavelength routing networks
    Networks 40 (2002), 32-37.


  86. R. Hassin and S. Rubinstein
    Robust matchings
    SIAM Journal on Discrete Mathematics 15 (2002), 530-537.
    Also: Proceedings of SWAT 2000, LNCS 1851 , 251-258.


  87. R. Hassin and A. Levin
    Subgraphs decomposable into two trees and k-edge-connected graphs
    Discrete Applied Mathematics 126 (2003), 181-195.


  88. R. Hassin and S. Rubinstein
    Approximation algorithms for the metric maximum clustering problem with given cluster sizes
    Operations Research Letters 31 (2003), 179-184.


  89. M. Conforti, R. Hassin, and R. Ravi
    Reconstructing flow paths
    Operations Research Letters 31 (2003), 273-276.


  90. R. Hassin and A. Levin
    Minimum spanning tree with hop restrictions
    Journal of Algorithm 48 (2003), 220-238.


  91. S. Guha, R. Hassin, S. Khuller, and E. Or
    Capacitated vertex covering with applications
    Journal of Algorithms 48 (2003), 257-270.
    Also: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002), 858-865.


  92. R. Hassin, R. Ravi, and F. S. Salman
    Approximation algorithms for the network design problem with a single cable type
    Algorithmica 38 (2003), 417-431.
    Also: Proceedings of APPROX 2000, LNCS 1913, 167-176.


  93. R. Hassin, A. Levin and D. Morad
    Lexicographic local search and the p-center problem
    European Journal of Operational Research 151 (2003), 265-279.


  94. R. Hassin and A. Levin
    Minimum restricted diameter spanning trees
    Discrete Applied Mathematics 137 (2004),
    Also: Proceedings of APPROX 2002, LNCS 2462, 175-184.


  95. R. Hassin and A. Levin
    Synthesis of 2-commodity flow networks
    Mathematics of Operations Research 29 (2004), 280-288.


  96. R. Hassin and A. Levin
    An efficient polynomial time approximation scheme for the constrained minimum spanning tree problem using matroid intersection
    SIAM Journal on Computing 33 (2004), 261-268.


  97. E. Arkin, R. Hassin, S. Rubinstein and M. Sviridenko
    Approximations for maximum transportation problem with permutable supply vector and other capacitated star packing problems
    Algorithmica 39 (2004), 175-187.
    Also: Proceedings of SWAT 2002, LNCS 2368, 280-287.


  98. R. Hassin and S. Rubinstein
    On the complexity of the vehicle routing problem
    Operations Research Letters 33 (2004), 71-76.


  99. R. Hassin and M. Shani
    Machine scheduling with earliness, tardiness and non-execution penalties
    Computers and Operations Research 32 (2005), 683-705.


  100. C. Bazgan, R. Hassin, and J. Monnot
    Approximation algorithms for some routing problems
    Discrete Applied Mathematics 146 (2005), 27-42.
    Also: CIAC 2003, LNCS 2653, 277-288.


  101. R. Hassin and J. Monnot
    The maximum saving partition problem
    Operations Research Letters 33 (2005), 242-248.


  102. R. Hassin and A. Levin
    Approximation algorithms for quickest spanning tree problems
    Algorithmica 41 (2005), 43-52.
    Also: Proceedings of ESA 2004, LNCS 3221, 403-413.


  103. A. Veltman and R. Hassin
    Equilibrium in queueing systems with complementary products
    Queueing Systems 50 (2005), 325-342.


  104. O. Einstein and R. Hassin
    The number of solutions sufficient for solving a family of problems
    Mathematics of Operations Research 30 (2005), 880-896.


  105. R. Hassin and A. Levin
    A better-than-greedy approximation algorithm for the minimum set cover problem
    SIAM Journal on Computing 35 (2005), 189-200.


  106. E. Arkin, R. Hassin, and A. Levin
    Approximation algorithms for minimum and min-max vehicle routing
    Journal of Algorithms 59 (2006), 1-18.


  107. R. Hassin and M. Haviv
    Who should be given priority in a queue?
    Operations Research Letters 34 (2006), 191-198.


  108. R. Hassin and D. Segev
    Robust subgraphs for trees and paths
    ACM Transactions on Algorithms 2 (2006), 263-281.
    Also: Proceedings of SWAT 2004, LNCS 3111, 51-63.


  109. R. Hassin and A. Levin
    The minimum generalized vertex cover problem
    ACM Transactions on Algorithms 2 (2006), 66-78.

    110. Also: Proceedings of ESA 2003, LNCS 2832, 289-300.

  110. Z. Agur, R. Hassin and S. Levy
    Optimizing chemotherapy scheduling using local search heuristics
    Operations Research. 54 (2006), 829-846.


  111. R. Hassin and S. Rubinstein
    An improved approximation algorithm for the metric maximum clustering problem with given cluster sizes
    Information Processing Letters 98 (2006), 92-95.


  112. R. Hassin and S. Rubinstein
    An approximation algorithm for maximum triangle packing
    Discrete Applied Mathematics 154 (2006), 971-979.
    Erratum: Discrete Applied Mathematics 154 (2006), 2620.
    Also: Proceedings of ESA 2004, LNCS 3221, 395-402.


  113. R. Hassin
    Information and uncertainty in a queueing system

    114.  Probability in the Engineering and Informational Sciences 21 (2007), 361-380.

  114. R. Hassin and A. Levin
    Flow trees for vertex-capacitated networks

    115. Discrete Applied Mathematics 155 (2007), 572-578.

  115. R. Hassin, J. Monnot and D. Segev
    Approximation algorithms and hardness results for labeled connectivity problems
    Journal of Combinatorial Optimization 14 (2007), 437-453.
    Also: Proceedings of MFCS 2006, LNCS 4162 480-491.


  116. E. Arkin and R. Hassin
    The k-path tree matroid and its application to survivable network design

    117. Journal of Combinatorial Optimization. Special Issue in Memory of George Dantzig. 5 (2008), 314-322.

  117. R. Hassin and A. Keinan,
    Greedy heuristics with regret, with application to the cheapest insertion algorithm for the TSP
    Operations Research Letters 36 (2008), 243-246.


  118. R. Hassin and
    119. D. Segev
    Rounding to an Integral Program
    Operations Research Letters 36 (2008), 321-326.
    Also: WEA 2005, LNCS 3503, 44-54.

  119. R. Hassin and S. Mendel
    Scheduling arrivals to queues: a model with no-shows

    120. Management Science 54 (2008), 565-572.

  120. R. Hassin, J. Puerto and Francisco R. Fernandez
    The use of relative priorities in optimizing the performance of a queueing system

    121.  European Journal of Operational Research 193 (2009), 476-483.

  121. R. Hassin, A. Levin and S. Rubinstein
    Approximation algorithms for maximum latency and partial cycle cover
    Discrete Optimization 6 (2009), 197-205.

    122. Also: ESA 2005, LNCS 3669 726-733.

  122. R. Hassin
    Equilibrium customers choice between FCFS and Random servers
    Queueing Systems 62 (2009), 243-254.

  123. R. Hassin, A. Levin and M. Sviridenko
    Approximating the minimum quadratic assignment problems
    ACM Transactions on Algorithms 6(1) (2009)


  124. N. Guttmann-Beck and R. Hassin
    On two restricted ancestors tree problems
    Information Processing Letters 110 (2010), 570-575.

  125. R. Hassin, J. Monnot, and D. Segev
    The complexity of bottleneck labeled graph problems
    Algorithmica 58 (2010), 245-262.
    Also: WG 2007, LNCS 4769, 328-340.


  126. A. Glazer and R. Hassin
    Inducing search by periodic advertising
    Information Economics and Policy 22 (2010), 276-286.

  127. R. Hassin and E. Or
    Min sum clustering with penalties
    European Journal of Operational Research 206 (2010), 547-554.
    Also: ESA 2005, LNCS 3669 167-178.


  128. R. Hassin and G. Goraly
    Multi-color pebble motion on graphs
    Algorithmica 58 (2010) 610-636.

  129. R. Hassin and Y. Kleiner
    Equilibrium and optimal arrival patterns to a server with opening and closing times
    IIE Transactions 43 (2011) 164-175.

  130. P. Guo and R. Hassin
    Strategic behavior and social optimization in Markovian vacation queues
    Operations Research 59 (2011) 986-997.

  131. N. Guttmann-Beck and R. Hassin
    On coloring the arcs of a tournament, covering shortest paths, and reducing the diameter of a graph
    Discrete Optimization 8 (2011) 302-314.

  132. N. Guttmann-Beck> and R. Hassin
    Series-parallel orientations preserving the cycle-radius
    Information Processing Letters 112 (2012) 153-160.

  133. P. Guo and R. Hassin
    Strategic behavior and social optimization in Markovian vacation queues: the case of heterogeneous customers

    134. European Journal of Operational Research (2012) 278-286.

  134. E. Arkin, N. Guttmann-Beck and R. Hassin,
    The (K,k)-Capacitated Spanning Tree Problem
    Discrete Optimization 9 (2012) 258-266.
    Also: AAIM 2005, LNCS 6124 25-34.


  135. P. Guo and R. Hassin
    On the advantage of leadership in service pricing competition
    Operations Research Letters 41 (2013) 397-402.

  136. S. Anily and R. Hassin
    Optimal pricing, replenishment, and timing of selling in a market with heterogeneous customers
    International Journal of Production Economics

  137. G. Gilboa-Freedman, R. Hassin, and Y. Kerner
    The price of anarchy in the Markovian single server queue
    IEEE Transactions on Automatic Control

  138. R. Hassin and O. Schneider
    A local search algorithm for binary maximum 2-path partitioning
    Discrete Optimization


Submitted for publication

  1. A. Glazer R. Hassin, and I. Milchtaich
    Time-differentiated monopolies


  2. G. Gilboa-Freedman and R. Hassin
    The regulation of a queue with a mixed population of cooperative and non-cooperative customers

  3. J. Erlichman and R. Hassin
    Strategic overtaking in a monopolistic observable M/M/1 queue
    Early version in VALUETOOLS 2009.


  4. R. Hassin, R. Ravi and F. S. Salman
    Facility location on a network with a linear reliability order of links
    Early versions: Proceedings of INOC 2009; Proceedings of ESA 2009, LNCS 5757, 275-276.


  5. R. Roet-Green and R. Hassin
    Equilibrium in a two-dimensional queueing game: when inspecting the queue is costly

  6. R. Hassin Y. Shaki, and U. Yovel
    Optimal service-capacity allocation in a loss system

  7. P. Guo and R. Hassin
    Equilibrium and optimal strategies for placing duplicate orders in queues


  8. I. Elem, , R. Hassin, and J. Monnot
    The minimum Gc cut problem

  9. Jian Chen, Shuo Huang, R. Hassin, and Nan Zhang,
    Compensation mechanisms design in inventory systems with strategic customers


Other reports

  1. E. Altman and R. Hassin
    Non-threshold equilibrium for customers joining an M/G/1 queue
    Proceedings of the 10th International Symposium of Dynamic Games and Applications, 2002.


  2. R. Hassin and D. Segev
    The multi-radius cover problem
    WADS 2005, LNCS 3608 24-35.


  3. R. Hassin and D. Segev
    The set cover with pairs problem
    FSTTCS 2005, LNCS 3821 164-176.


  4. R. Hassin and A. Levin
    An approximation algorithm for the minimum latency set cover problem

    5. ESA 2005.

  5. R. Hassin and E. Or
    A maximum profit coverage algorithm with application to small molecules cluster identification
    WEA 2006, LNCS 4007 265-276.


  6. N. Guttmann-Beck and R. Hassin
    Minimum diameter and cycle-diameter orientations in planar graphs

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