Orthogonal drawings of graphs for the automation of VLSI circuit design



This article shows the recent developments on orthogonal drawings of graphs which have applications for the automation of VLSI circuit design. Meanwhile, a number of problem are posed for further research.


graph surface embedding algorithm orthogonal drawing VLSI 


  1. [1]
    Kuratowski K. Sur le Problem des Coubes Gauches en Topologie.Fund. Math., 1930, 15: 271–283.MATHGoogle Scholar
  2. [2]
    MacLane S. A structural characterization of planar combinatorial graphs.Duke Math. J., 1937, 3: 460–472.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    MacLane S. A combinatorial condition for planar graphs.Fund. Math., 1937, 28: 22–32.MATHGoogle Scholar
  4. [4]
    Whitney H. Non-separable and planar graph.Trans. AMS, 1932, 34: 339–162.CrossRefMathSciNetGoogle Scholar
  5. [5]
    Whitney H. Planar graphs.Fund. Math., 1933, 21: 73–84.Google Scholar
  6. [6]
    Whitney H. On regular closed curves in the plane.Compositio Math., 1937, 4: 276–284.MATHMathSciNetGoogle Scholar
  7. [7]
    Wu W T. The realization of complexies in the Euclidean space.Acta Math. Sinica, 1955, 5: 505–452 (in Chinese),MATHMathSciNetGoogle Scholar
  8. [8]
    Wu W T. A theory of Imbedding, Immersion, and Isotopy of Polytopes in an Euclidean Space. Science Press, Beijing, 1965.Google Scholar
  9. [9]
    Wu W T. Planar embedding of linear graphs.Sci. Bull. (KEXUETONGBAO), 1974, 19(2): 226–228 (in Chinese).Google Scholar
  10. [10]
    Wu W T. Rational Homotopy Type. Lect. Notes in Math. 1246, Springer, New York/Heidelberg/Berlin, 1987.MATHGoogle Scholar
  11. [11]
    Wu W T. Mathematical problems in the design of integrated circuits.Math. Theory Practice, 1973, 1: 20–40 (in Chinese).Google Scholar
  12. [12]
    Wu W T. On the planar embedding of linear graphs I.J. Syst. Sci. Math, 1985, 5: 290–320.MATHGoogle Scholar
  13. [13]
    Wu W T. On the planar embedding of linear graphs II.J. Syst. Sci. Math., 1986, 6: 23–35.MATHGoogle Scholar
  14. [14]
    Wu W T. The Realization of Polytopes in the Euclidean Space. Science Press, Beijing, 1978 (in Chinese).Google Scholar
  15. [15]
    Wu W T. Selected Papers of Wu Wenjun. Shandong Education Press, Jinan, 1986 (in Chinese).Google Scholar
  16. [16]
    Wu W T. On Mathematical Mechanization on by Wu Wenjun. Shandong Education Press, Jinan, 1995 (in Chinese).Google Scholar
  17. [17]
    Liu Y P. Embeddability Theory of Graphs. Science Press, Beijing, 1994, (in Chinese).Google Scholar
  18. [18]
    Liu Y P. Embeddability in Graphs. Kluwer, Dordrecht/Boston/London, 1995.MATHGoogle Scholar
  19. [19]
    Tutte W T. A class of Abelian groups.Canad. J. Math., 1956, 8: 13–28.MATHMathSciNetGoogle Scholar
  20. [20]
    Tutte W T. Toward a theory of crossing numbers.J. Comb. Theory, 1970, 8: 45–53.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Liu Y P. Module 2 programming and planar embedding.Acta Math. Appl Sinica, 1978, 1: 395–406 (in Chinese).Google Scholar
  22. [22]
    Liu Y P. On the linearity of testing planarity of a graph. Comb. Optim. CORR84-4, University of Waterloo, 1984; Also inChinese Ann. Math., 1986, 7B: 425–434.Google Scholar
  23. [23]
    Liu Y P. A new approach to the linearity of testing planarity of graphs. Report, Rutgers University, 1984; Also inActa Math. Appl. Sinica, Eng. Series, 1988, 4: 257–265.Google Scholar
  24. [24]
    Rosenstiehl P. Preuve algebrique du critere de planarite de Wu(Wenjun)-Liu(Yanpei).Ann. Discrete Math., 1980, 9: 67–78.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    Cook S A. The complexity of theorem proving procedures. InProc. 3rd ACM Symp. Comput., 1971, pp.151–158.Google Scholar
  26. [26]
    Garey M R, Johnson D S. Computer and Intractability-A Guide to the Theory of NP-Completeness, Freeman W H (eds.), San Francisco, 1979.Google Scholar
  27. [27]
    Lipton R J, Tarjan R E. A separator theorem for planar graphs.SIAM J. Appl. Math., 1979, 36: 177–189.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    Lipton R J, Tarjan R E. Applications of a planar separator theorem.SIAM J. Comput., 1980, 9: 615–627.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    Liu Y P. Theory of Rectilinear Layouts. China Railway Publishing House, Beijing, 1997 (in Chinese).Google Scholar
  30. [30]
    Hopcroft J, Tarjan R. Isomorphism of planar graphs. InComplexity of Computer Computations, Miller Ret al. (eds.), Plenum, 1972, pp.131–152.Google Scholar
  31. [31]
    Hopcroft J, Tarjan R. Dividing a graph into triconnected components.SIAM J. Comput., 1973, 2: 135–158.CrossRefMathSciNetGoogle Scholar
  32. [32]
    Hopcroft J, Tarjan R. Efficient planarity testing.J. ACM., 1974, 21: 549–568.MATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    Auslander L, Parter S V. On imbedding graphs in sphere.J. Math. Mech., 1961, 10: 517–523.MATHMathSciNetGoogle Scholar
  34. [34]
    Becker B, Hotz G. On the optimal layout of planar graphs with fixed boundary.SIAM J. Comput., 1987, 16: 946–972.MATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    Dambit Ja. Embedding of a graph into the plane.Latvian Math., 1966, Yearbook 2: 79–93.MATHMathSciNetGoogle Scholar
  36. [36]
    Demoucron G, Malgrange Y, Pertuiset R. Graphe planaires, reconnaissance et construction de representations planaires topologiques.Rev. Francaise Recherche Operationnelle, 1964, 8: 33–47.Google Scholar
  37. [37]
    Deo N. Note on Hopcroft and Tarjan’s planarity algorithm.J. ACM, 1976, 33: 74–75.CrossRefMathSciNetGoogle Scholar
  38. [38]
    Engle W L. An algorithm for embedding graphs in the plane with certain constraints.IEEE Trans. Cir. Theory, 1970, CT-17: 250–252.CrossRefGoogle Scholar
  39. [39]
    Fisher G J, Wing O. Computer recognition and extraction of planar graphs from the incidence matrix.IEEE Trans. Cir. Theory, 1996 CT-23(2): 254–263.Google Scholar
  40. [40]
    Goldstein A J, Schweikert D G. A proper model for testing the planarity of electrical circuits.Bell Syst. Tech. J., 1973, 52: 135–142.MathSciNetGoogle Scholar
  41. [41]
    Hopcroft J. Ann logn algorithm for isomorphism of planar triply connected graphs. InTheory of Machines and Computation, Kohavi Zet al., (eds.), Acad. Press, 1971, pp.189–196.Google Scholar
  42. [42]
    Hopcroft J, Tarjan R. Planarity testing inV logV steps: Extended abstract. InProc. IFIP Cong., 1971, pp.85–90.Google Scholar
  43. [43]
    Hope A K. A planar graph drawing program.Solfware-Practice and Experience, 1971, 1: 82–91.Google Scholar
  44. [44]
    Hotz G. The embedding of graphs in the 2-sphere.Z. Angew. Math. Mech., 1965, 45.Google Scholar
  45. [45]
    Hotz G. Embedding of graphs in the plane.Math. Ann., 1966 167: 214–223.MATHCrossRefMathSciNetGoogle Scholar
  46. [46]
    Inukai T, Weinberg L. Planar, coplanar, and totally planarn-port networks.IEEE Trans. Cir. Syst., 1976, Case-23.Google Scholar
  47. [47]
    Jayakumar R, Thulasiraman K, Swamy M N S. Planar embeddings: Linear time algorithms for vertex placement and edge ordering.IEEE Trans. Cir. Syst., 1988 35 (3): 334–344.MATHCrossRefMathSciNetGoogle Scholar
  48. [48]
    Kirkpatrick D G. Optimal search in planar subdivision.SIAM J. Comput., 1983, 12: 28–35.MATHCrossRefMathSciNetGoogle Scholar
  49. [49]
    Knauer B. Normalformen planar graphen I.Computing, 1972, 10: 121–136.MATHCrossRefMathSciNetGoogle Scholar
  50. [50]
    Knauer B. Normalformen planar graphen II.Computing, 1972, 10: 137–152.MATHCrossRefMathSciNetGoogle Scholar
  51. [51]
    Knauer B. A simple planarity criterion.J. ACM, 1975, 22: 226–230.MATHCrossRefMathSciNetGoogle Scholar
  52. [52]
    Lefschetz S. Planar graphs and related topics. InProc. Nat. Acad. Sci. 1965, 54: 1763–1765.Google Scholar
  53. [53]
    Lempel A, Even S, Cederbaum I. An algorithm for planarity testing of graphs. In Graph Theory Rosenstiehl P (ed.), InProc. Int. Symp., Rome, 1967, p.215.Google Scholar
  54. [54]
    Lenganer T. Hierarchical planarity testing algorithms.J. ACM., 1989, 36: 474–509.CrossRefGoogle Scholar
  55. [55]
    Plesnevic G S. Embedding a graph in the plane.Vycislitellnyes Sistemy, 1963, 6: 45–53.MathSciNetGoogle Scholar
  56. [56]
    Rosenstiehl P. Caracterisation des graphes planaires par une diagonale absreacte.Cong. Numer., 1976, 15: 521–527.MathSciNetGoogle Scholar
  57. [57]
    Rubin F. An algorithm for testing the planarity of a graph.IEEE Computer Group Respositery, R74-73, 1974.Google Scholar
  58. [58]
    Rubin F. An improved algorithm for testing the planarity of a graph.IEEE Trans. on Computers, 1975, C-24: 113–121.MATHCrossRefGoogle Scholar
  59. [59]
    Tutte W T. How to draw a graph. InProc. London Math. Soc., 1963, 13(Ser.3): 743–768.Google Scholar
  60. [60]
    Ulrich J W. A computational theory of planar embedding. InPh. D. Thesis, Univ. Texas, Austin, 1968.Google Scholar
  61. [61]
    Ulrich J W. A characterization of planar oriented graphs.SIAM J. Appl. Math., 1970, 18: 364–371.MATHCrossRefMathSciNetGoogle Scholar
  62. [62]
    Unger P. A theorem on planar graph, components, and subgraphs.J. London Math. Soc., 1951, 26: 256–262.CrossRefMathSciNetGoogle Scholar
  63. [63]
    Weinberg L. Two new characterization of planar graphs. InProc. 5-th Allerton Conf. Cir. Syst., Uni. Ill., 1967.Google Scholar
  64. [64]
    Williamson S G. Embedding graphs in the plane-Algorithm aspects.Am. Discrete Math., 1980, 6: 349–384.MATHCrossRefMathSciNetGoogle Scholar
  65. [65]
    Wing O. On drawing a planar graph.IEEE Trans. Cir. Theory 1966, 13: 112–114.MathSciNetGoogle Scholar
  66. [66]
    Liu Y P, Marchioro P, Petreschi R. At most single-bend embeddings of cubic graphs. Research Report SI-92/01. Dept. Computer Science, Uni. «La Sapienza” of Rome, 1992. Also inApplied Math. (A J. Chinese Unis.), 1994, B9: 127–142.Google Scholar
  67. [67]
    Liu Y P, Marchioro P, Petreschi R, Simeone B. Theoretical results on at most 1-bend embeddability of graphs. Research Report Series A No.3, Department of Statistics, University of Rome “La Sapienza”, 1990; Also inActa Math. Appl. Sinica, Eng., 1992, Series 8: 188–192.Google Scholar
  68. [68]
    Liu Y P, Marchioro P, Petreschi R, Simeone B. On theoretical results of at most 1-embeddability of graphs.Chinese Science Bulletin, 1991, 36: 1054–1055.Google Scholar
  69. [69]
    Liu Y P, Morgana A. Simeone B. On the general theoretical results for rectilinear embeddability of graphs.KEXUE TONGBAO, (Chinese Ed.) 1990, 35: 1513–1514. Or seeChinese Science Bulletin (English Ed.), 1991, 36: 1490.Google Scholar
  70. [70]
    Liu Y P, Morgana A, Simeone B. General theoretical results on rectilinear embeddability of graphs, Research Report Series A, No. 2, Department of Statistics, University of Rome «La Sapienza», 1990; AlsoinActa Math. Appl. Sinica, Eng., 1991, Series 7: 187–192.Google Scholar
  71. [71]
    Liu Y P, Morgana A, Simeone B. A linear time algorithm for 3-bend embeddings of planar graphs in the grid. Research Report, Ser.A, No.1, Dept. Statistics, Uni. «La Sapienza», Rome, 1993. Also inDiscrete Appl. Math., 1998, 81: 69–92.Google Scholar
  72. [72]
    Liu Y P, Morgana A, Simeone B. A graph partition problem. Research Report, No.27, Inst. Appl. Math., Acad. Sinica, 1992. Also inActa Math. Appl. Sinica, Eng., 1996, Series 12: 393–400.Google Scholar
  73. [73]
    Liu Y P, Morgana A, Simeone B. Another linear time algorithm for finding 3-embeddings of a graph. Research Report, No.1, Inst. Appl. Math., Acad. Sinica, 1994.Google Scholar
  74. [74]
    Liu Y P, Morgana A, Simeone B. Characterizations of a kind of orientations of a graph. Research Report, No.2, Inst. Appl. Math., Acad. Sinica, 1994.Google Scholar
  75. [75]
    Liu Y P. On the net-embeddability of graphs.Acta Math. Sinica, 1994, New Series, 8: 413–423.Google Scholar
  76. [76]
    Liu Y P. On the efficient recognition on the net-extensibility of graphs.Chinese Science Bulletin, 1993, 38: 533–536.MATHGoogle Scholar
  77. [77]
    Liu Y P. Combinatorial optimization arising from VLSI circuit design.Applied Math., (JCU), 1993, B8: 218–235.MATHCrossRefGoogle Scholar
  78. [78]
    Fraysseix H, Rosentiehl P. A depth — first search characterization of planarity.Ann. Discrete Math., 1982, 13: 75–80.MATHGoogle Scholar
  79. [79]
    Fraysseix H, Rosenstiehl P. A characterization of planar graphs by Tremaux order.Combinatorica, 1985, 5: 127–155.MATHCrossRefMathSciNetGoogle Scholar
  80. [80]
    Liu Y P. Planarity testing nd planar embeddings of graphs.Acta Math. Appl. Sinica, 1979, 2: 350–365 (in Chinese).MathSciNetGoogle Scholar
  81. [81]
    Liu Y P. Boolean planarity characterization of graphs. RUTCOR Research Report RRR38-87, Rutgers University, 1987; Also inActa Math Sinica, 1988, New Series, 4: 316–329.Google Scholar
  82. [82]
    Liu Y P. Boolean approach to planar embeddings of a graph. RUTCOR Research Report RRR39-87, Rutgers University, 1987; Also inActa Math. Sinica, 1989, New Series, 5: 64–79.Google Scholar
  83. [83]
    Liu Y P. Boolean characterizations of planarity and planar embeddings of graphs.Ann. Operations Research, 1990, 24: 165–174.MATHCrossRefGoogle Scholar
  84. [84]
    Sun X R. On the complexity of testing the planarity by Wu(Wenjun)-Liu (Yanpei) Theorem, (in Chinese with English abstract). Chinese J. Comput., 1989, 12(1): 33–37.Google Scholar
  85. [85]
    Sun X R. Wu (Wenjun)-Liu (Yanpei) Theorem and planarity testing of graphs.Thesis, Inst. Applied Math., Acad. Sinica, 1987 (in Chinese).Google Scholar
  86. [86]
    Xu W X. An efficient algorithm for planarity testing based on Wu (Wenjun)-Liu (Yanpei)’s criterion. InProc. 1-st China-USA Conf. Graph Theory and its Applications, Ann. N. Y. Acad. Sci., 1989, 576: 641–652.Google Scholar
  87. [87]
    Hopcroft J E, Wong J K. Linear time algorithm for isomorphism of planar graphs (extended abstract). In6-th Ann. ACM Symp. Comput., Seattle, 1974.Google Scholar
  88. [88]
    Hopcroft J, Tarjan R. A V2 algorithm for determining isomorphism of planar graphs.Inform. Process. Lett. 1971, 1: 32–34.MATHCrossRefGoogle Scholar
  89. [89]
    Weinberg L. A simple and efficient algorithm for determining isomorphism of planar triply connected graphs.IEEE Trans. Circuit Theory, 1966, CT-13: 142–148.MathSciNetGoogle Scholar
  90. [90]
    Weinberg L. Plane representations and codes for planar graphs. InProc. 3-rd Ann. Allerton Conf. Cir. Syst., 1965, pp.733–744.Google Scholar
  91. [91]
    Weinberg L. Additional simple codes for planar graphs. InProc. 4-th Allerton Conf. Cir. Syst., 1966.Google Scholar
  92. [92]
    Basden A, Nichols K G. New topological method for layout printed circuits. InProc. IEEE, 1973, 120(3): 325–328.Google Scholar
  93. [93]
    Batini C, Talamo M, Tamassia R. Computer aided layout of entity-relationship diagrams.IEEE J. Syst. Software, 1994, 4: 163–173.CrossRefGoogle Scholar
  94. [94]
    Behzao M. A criterion for the planarity of the total graph of a graph. InProc. Cambridge Phil. Soc., 1967, 63: 679–681.Google Scholar
  95. [95]
    Bhatt S N, Leighton F T. A framework for solving VLSI graph layout problems.J. Comput Syst. Scien., 1984, 28: 300–343.MATHCrossRefMathSciNetGoogle Scholar
  96. [96]
    Chen R W, Kajitani Y, Chan S P. A graph- theoretic via minimization algorithm for two — layer printed circuit boards.IEEE Trans. Cir. Syst., 1983, Case-30: 284–299.MATHCrossRefGoogle Scholar
  97. [97]
    Chua L O, Chen L K. On optimally sparse cycle and coboundary basis for a linear graph.IEEE Trans. Cir. Theory, 1973, CT-20: 495–503.Google Scholar
  98. [98]
    Charke E M, Feng Y. Escher-a geometrical layout system for recursively defined circuits.IEEE Trans. CAD, 1988, 7: 908–918.Google Scholar
  99. [99]
    van Cleemput W M. Mathematical models for the circuit layout problem.IEEE Trans. Cir. Syst., 1976, Case-23(12): 759–767.CrossRefGoogle Scholar
  100. [100]
    van Cleemput W M, Linders J G. An improved graph theoretical model for the circuit layout problem. InProc. 11-th Design Automation Workshop, Denver, 1974.Google Scholar
  101. [101]
    Cohoon J P, Heck P L. BEAVER: a computational geometry based tool for switchbox routing.IEEE Trans. CAD, 1988, 7: 684–697.Google Scholar
  102. [102]
    Cong J, Wong D F, Liu C L. A new approach to three or four layer channel routing.IEEE Trans. CAD, 1988, 7: 1094–1104.Google Scholar
  103. [103]
    Du D Z, Liu Y P. Combinatorial Optimization. Handbook of OR Fundamentals. Hui Get al. (eds.), Science Press, Beijing, 1995, pp.125–167.Google Scholar
  104. [104]
    Hu T C, Kuh S E. Theory and concepts of circuit layout. InVLSI Circuit Layout: Theory and Design. IEEE Press, 1985, pp.3–18.Google Scholar
  105. [105]
    Lefschetz S. Applications of Algebraic Topology: Graphs and Networks, Springer, New York/Heidelberg/ Berlin, 1975.MATHGoogle Scholar
  106. [106]
    Liu Y P. Rectilinear Embeddings: Theory and Methods. Beijing: Science Press, 1994 (in Chinese).Google Scholar
  107. [107]
    Rim C Set al. Exact algorithms for multilayer topological via minimization.IEEE Trans. CAD, 1989, 8: 1165–1173.Google Scholar
  108. [108]
    Rose N A, Oldfield J V. Printed — wiring — board layout by computer.Electronics and Power, Oct. 1971, pp.376–379.Google Scholar
  109. [109]
    Alekseev V B, Gonchakov V S. Thickness of arbitrary complete graphs.Math. Sbornik, 1976, 101: 212–230.Google Scholar
  110. [110]
    Beineke L, Harary F. On the thickness of the complete graph.Bull. Amer. Math. Soc., 1964, 70: 618–620.MATHCrossRefMathSciNetGoogle Scholar
  111. [111]
    Beineke L, Harary F. Inequalities involving the genus of a graph and its thickness. InProc. Glasgow Math. Assoc., 1965, 7: 19–21.Google Scholar
  112. [112]
    Beineke L, Harary F. The thickness of the complete graph.Canad. J. Math., 1965, 17: 850–859.MATHMathSciNetGoogle Scholar
  113. [113]
    Beineke L, Harary F, Moon J W. On the thickness of the complete bipartite graph. InProc. Camb. Phil. Soc., 1964, 60: 1–4.Google Scholar
  114. [114]
    Bose N K, Prabhu K A. Thickness of graphs with degree constrained vertices.IEEE Trans. Cir. Syst., 1977, Case-24(4): 184–190.MATHCrossRefMathSciNetGoogle Scholar
  115. [115]
    Chung F R K, Leighton F T, Rosenberg A L. Embedding graphs in books: A graph layout problem with applications to VLSI design.SIAM J. Algeb Discrete Methods, 1987, 8: 33–48.MATHCrossRefMathSciNetGoogle Scholar
  116. [116]
    Hobbs A M. A survey of thickness. InRecent progress in Combinatorics, Tutte W T (ed.), 1969, pp.255–264.Google Scholar
  117. [117]
    Hobbs A M, Grossman G W. A class of thickness-minimal graphs.J. Res. Nat. Bur. Standards, 1968, 72B: 145–153.MathSciNetGoogle Scholar
  118. [118]
    Hobbs A M, Grossman G W. Thickness and connectivity in graphs.J. Res. Nat. Bur. Standards, 1968, 72B: 239–244.MathSciNetGoogle Scholar
  119. [119]
    Jayakumar Ret al O(n 2) algorithms for graph planarization.IEEE Trans. CAD, 1989, 8: 257–267.Google Scholar
  120. [120]
    Kleinert M. The thickness of then-dimensional cube.J. Comb, Theory, 1967, 3: 10–15.MATHCrossRefMathSciNetGoogle Scholar
  121. [121]
    Lerda F, Majoranic E. An algorithm for connectingn points with a minimum number of crossings.Calcolo, 1964, 1: 257–365.MATHCrossRefMathSciNetGoogle Scholar
  122. [122]
    Levow R B. On Tutte’s algebraic approach to the theory of crossing numbers. InProc. 3-rd S-E Conf. Comb. Graph Theory Comput., 1972, pp.315–324.Google Scholar
  123. [123]
    Lin P M. On Methods of deleting planar graphs. InProc. 8-th Midwest Symp. Cir. Theory, Colorado State Univ., Boulder, 1965, pp.11–19Google Scholar
  124. [124]
    Liu T Y, Liu Y P. On the crossing number of circular graphs.OR Transactions, 1998, 2(4): 32–38.Google Scholar
  125. [125]
    Liu Y P. Boolean approaches to graph embeddings related to VLSI.Discrete Applied Math., accepted and to appear.Google Scholar
  126. [126]
    Liu Y P. Planarity theory and routing automation. Plenary Report, Symposium on Mathematical Mechanization, Beijing, 1999 (in Chinese).Google Scholar
  127. [127]
    Melikhov A N, Kuleychik V M, Lisyak V V. Partition of a graph into plane subgraphs (in Rassian).Cybernetics, 1972, 10: 1087–1090.Google Scholar
  128. [128]
    Richter B. Cubic graphs with crossing number two.J. Graph Theory, 1988, 12(3): 363–374.MATHCrossRefMathSciNetGoogle Scholar
  129. [129]
    Rubin F. A note on Lerda and Majoronic’s minimum corssing algorithm.Calcolo, 1974, 11: 201–203.MATHCrossRefMathSciNetGoogle Scholar
  130. [130]
    Yannakakis M. Embedding planar graphs in four pages.J. Comput. Syst. Sci., 1989, 38: 36–67.MATHCrossRefMathSciNetGoogle Scholar
  131. [131]
    Auslander L, Brown T, Youngs J W T. The embedding of graphs in manifolds.J. Math. Mech., 1963, 12(4): 229–234.MathSciNetGoogle Scholar
  132. [132]
    Liu Y P. Transportation networks: Old and new.Applied Math., (JCU), 1996, B11: 251–272.MATHCrossRefGoogle Scholar
  133. [133]
    Liu Y P. Transportation Networks: Theory and Methods. Shanghai Jiaotong University Press, Shanghai, 1998 (in Chinese).Google Scholar
  134. [134]
    Kaufmann M. A linear time algorithm for routing in a convex grid.IEEE Trans. CAD, 1990, 9: 180–184.MathSciNetGoogle Scholar
  135. [135]
    Lawrencenko S, Liu X, Liu Y P. An algorithm for constructing a rectilinear embedding of a given graph in the plane (with Lawrencenko S, Liu X).Comb. Graph Theory’95, World Scien. Pub., 1995, pp.205–217.Google Scholar
  136. [136]
    Storer J A. The node cost measure for embedding graphs in the planar grid. InProc. 12-th ACM Symp. Comput., 1980, pp.201–210.Google Scholar
  137. [137]
    Liu Y P Enumerative Theory of Maps. Kluwer, Dordrecht/Boston/London, 1999.MATHGoogle Scholar

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© Science Press, Beijing China and Allerton Press Inc. 1999

Authors and Affiliations

  1. 1.Institute of MathematicsNorthern Jiaotong UniversityBeijingP.R. China

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