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Orthogonal drawings of graphs for the automation of VLSI circuit design

  • Liu Yanpei 
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Abstract

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.

Keywords

graph surface embedding algorithm orthogonal drawing VLSI 

References

  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

Copyright information

© 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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