A Linear-Time Algorithm for Bend-Optimal Orthogonal Drawings of Biconnected Cubic Plane Graphs

Extended Abstract
  • Shin-ichi Nakano
  • Makiko Yoshikawa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1984)


An orthogonal drawing of a plane graph G is a drawing of G with the given planar embedding in which each vertex is mapped to a point, each edge is drawn as a sequence of alternate horizontal and vertical line segments, and any two edges do not cross except at their common end. Observe that only a planar graph with the maximum degree four or less has an orthogonal drawing. The best known algorithm to find an orthogonal drawing runs in time O(n7/4√log n) for any plane graph with n vertices. In this paper we give a linear-time algorithm to find an orthogonal drawing of a given biconnected cubic plane graph with the minimum number of bends.


  1. [BLV93]
    G. Di Battista, G. Liotta and F. Vargiu, Spirality of orthogonal representations and optimal drawings of series-parallel graphs and 3-planar graphs, Proc. of Workshop on Algorithms and Data structures, LNCS 709, Springer (1993) 151–162.Google Scholar
  2. [GT94]
    A. Garg and R. Tamassia, On the computational complexity of upward and rectilinear planarity testing, Proc. of Graph Drawing’94, LNCS 894, Springer (1995) 286–297.Google Scholar
  3. [GT96]
    A. Garg and R. Tamassia, A new minimum cost flow algorithm with applications to graph drawing, Proc. of Graph Drawing’96, LNCS 1190, Springer (1997) 201–226.Google Scholar
  4. [K96]
    G. Kant, Drawing planar graphs using the canonical ordering, Algorithmica, 16 (1996) 4–32.MATHMathSciNetCrossRefGoogle Scholar
  5. [KH94]
    G. Kant and X. He, Two algorithms for finding rectangular duals of planargraphs, Proc. of WG’93, LNCS 790, Springer (1994) 396–410.Google Scholar
  6. [RNN96]
    M. S. Rahman, S. Nakano and T. Nishizeki, Rectangular grid drawings of plane graphs, Proc. of COCOON’96, LNCS 1090, Springer (1996) 92–105. Also, Computational Geometry: Theory and Applications, 10 (1998) 203-220.Google Scholar
  7. [RNN99]
    M. S. Rahman, S. Nakano and T. Nishizeki, A linear algorithm for bend-optimal orthogonal drawings of triconnected cubic plane graphs, Journal of Graph Algorithms and Applications, 3 (1999) 31–62.MATHMathSciNetGoogle Scholar
  8. [RNN00]
    M. S. Rahman, S. Nakano and T. Nishizeki, Rectangular Drawings of Plane Graphs without Designated Corners, Proc. of COCOON’00, LNCS 1858, Springer (2000) 85–94.Google Scholar
  9. [T87]
    R. Tamassia, On embedding a graph in the grid with the minimum number of bends, SIAM J. Comput., 16 (1987) 421–444.MATHCrossRefMathSciNetGoogle Scholar
  10. [T84]
    C. Thomassen, Plane representations of graphs, (Eds.) J.A. Bondy and U.S.R. Murty, Progress in Graph Theory, Academic Press Canada (1984) 43–69.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Shin-ichi Nakano
    • 1
  • Makiko Yoshikawa
    • 1
  1. 1.Gunma UniversityKiryuJapan

Personalised recommendations