Straight-Line Drawability of a Planar Graph Plus an Edge

  • Peter Eades
  • Seok-Hee HongEmail author
  • Giuseppe Liotta
  • Naoki Katoh
  • Sheung-Hung Poon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)


We investigate straight-line drawings of topological graphs that consist of a planar graph plus one edge, also called almost-planar graphs. We present a characterization of such graphs that admit a straight-line drawing. The characterization enables a linear-time testing algorithm to determine whether an almost-planar graph admits a straight-line drawing, and a linear-time drawing algorithm that constructs such a drawing, if it exists. We also show that some almost-planar graphs require exponential area for a straight-line drawing.


Hamilton Path External Face Clockwise Order Edge Crossing Closed Walk 
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  1. 1.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall (1999)Google Scholar
  2. 2.
    Di Battista, G., Tamassia, R., Tollis, I.G.: Area requirement and symmetry display of planar upward drawings. Discrete & Computational Geometry 7, 381–401 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cabello, S., Mohar, B.: Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM J. Comput. 42(5), 1803–1829 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chiba, N., Yamanouchi, T., Nishizeki, T.: Linear algorithms for convex drawings of planar graphs. In: Progress in Graph Theory, pp. 153–173. Academic Press, London (1984)Google Scholar
  5. 5.
    Chrobak, M., Eppstein, D.: Planar orientations with low out-degree and compaction of adjacency matrices. Theor. Comput. Sci. 86(2), 243–266 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Eades, P., Hong, S.H., Liotta, G., Katoh, N., Poon, S.H.: Straight-line drawability of a planar graph plus an edge (2015). ArXiv, 1504.06540Google Scholar
  7. 7.
    Eades, P., Feng, Q.-W., Lin, X., Nagamochi, H.: Straight-line drawing algorithms for hierarchical graphs and clustered graphs. Algorithmica 44(1), 1–32 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gutwenger, C., Mutzel, P., Weiskircher, R.: Inserting an edge into a planar graph. Algorithmica 41(4), 289–308 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hong, S.H., Eades, P., Liotta, G., Poon, S.H.: Fáry’s theorem for 1-planar graphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 335–346. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  10. 10.
    Hong, S.H., Nagamochi, H.: Convex drawings of graphs with non-convex boundary constraints. Discrete Applied Mathematics 156(12), 2368–2380 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jünger, M., Leipert, S.: Level planar embedding in linear time. J. Graph Algorithms Appl. 6(1), 67–113 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nagamochi, H.: Straight-line drawability of embedded graphs. Technical Report 2013–005, Graduate School of Informatics, Kyoto University (2013)Google Scholar
  13. 13.
    Thomassen, C.: Rectilinear drawings of graphs. Journal of Graph Theory 12(3), 335–341 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Peter Eades
    • 1
  • Seok-Hee Hong
    • 1
    Email author
  • Giuseppe Liotta
    • 2
  • Naoki Katoh
    • 3
  • Sheung-Hung Poon
    • 4
  1. 1.University of SydneySydneyAustralia
  2. 2.University of PerugiaPerugiaItaly
  3. 3.Kwansei Gakuin UniversityNishinomiyaJapan
  4. 4.Institut Teknologi BruneiGadongBrunei

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