Monotone Drawings of k-Inner Planar Graphs

  • Anargyros Oikonomou
  • Antonios SymvonisEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)


A k-inner planar graph is a planar graph that has a plane drawing with at most k internal vertices, i.e., vertices that do not lie on the boundary of the outer face of its drawing. An outerplanar graph is a 0-inner planar graph. In this paper, we show how to construct a monotone drawing of a k-inner planar graph on a \(2(k+1)n \times 2(k+1)n\) grid. In the special case of an outerplanar graph, we can produce a planar monotone drawing on a \(n \times n\) grid, improving the results in [2, 11].


  1. 1.
    Angelini, P., Colasante, E., Di Battista, G., Frati, F., Patrignani, M.: Monotone drawings of graphs. J. Graph Algorithms Appl. 16(1), 5–35 (2012). Scholar
  2. 2.
    Angelini, P., et al.: Monotone drawings of graphs with fixed embedding. Algorithmica 71(2), 233–257 (2015). Scholar
  3. 3.
    Brocot, A.: Calcul des rouages par approximation, nouvelle methode. Revue Chronometrique 6, 186–194 (1860)Google Scholar
  4. 4.
    Chiang, Y., Lin, C., Lu, H.: Orderly spanning trees with applications. SIAM J. Comput. 34(4), 924–945 (2005). Scholar
  5. 5.
    Felsner, S.: Convex drawings of planar graphs and the order dimension of 3-polytopes. Order 18(1), 19–37 (2001). Scholar
  6. 6.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison-Wesley Longman Publishing Co., Inc., Boston (1994)zbMATHGoogle Scholar
  7. 7.
    He, D., He, X.: Nearly optimal monotone drawing of trees. Theor. Comput. Sci. 654, 26–32 (2016). Scholar
  8. 8.
    He, D., He, X.: Optimal monotone drawings of trees. SIAM J. Discret. Math. 31(3), 1867–1877 (2017). Scholar
  9. 9.
    He, X., He, D.: Compact monotone drawing of trees. In: Xu, D., Du, D., Du, D. (eds.) COCOON 2015. LNCS, vol. 9198, pp. 457–468. Springer, Cham (2015). Scholar
  10. 10.
    He, X., He, D.: Monotone drawings of 3-connected plane graphs. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 729–741. Springer, Heidelberg (2015). Scholar
  11. 11.
    Hossain, M.I., Rahman, M.S.: Good spanning trees in graph drawing. Theor. Comput. Sci. 607, 149–165 (2015). Scholar
  12. 12.
    Kindermann, P., Schulz, A., Spoerhase, J., Wolff, A.: On monotone drawings of trees. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 488–500. Springer, Heidelberg (2014). Scholar
  13. 13.
    Oikonomou, A., Symvonis, A.: Monotone drawings of \(k\)-inner planar graphs. CoRR abs/1808.06892v1 (2017).
  14. 14.
    Oikonomou, A., Symvonis, A.: Simple compact monotone tree drawings. In: Frati, F., Ma, K.L. (eds.) GD 2017. LNCS, vol. 10692, pp. 326–333. Springer, Cham (2018). Scholar
  15. 15.
    Stern, M.: Ueber eine zahlentheoretische funktion. Journal fur die reine und angewandte Mathematik 55, 193–220 (1858)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Applied Mathematical and Physical SciencesNational Technical University of AthensAthensGreece

Personalised recommendations