EPG-representations with Small Grid-Size

  • Therese BiedlEmail author
  • Martin Derka
  • Vida Dujmović
  • Pat Morin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10692)


In an EPG-representation of a graph G, each vertex is represented by a path in the rectangular grid, and (vw) is an edge in G if and only if the paths representing v an w share a grid-edge. Requiring paths representing edges to be x-monotone or, even stronger, both x- and y-monotone gives rise to three natural variants of EPG-representations, one where edges have no monotonicity requirements and two with the aforementioned monotonicity requirements. The focus of this paper is understanding how small a grid can be achieved for such EPG-representations with respect to various graph parameters.

We show that there are m-edge graphs that require a grid of area \(\varOmega (m)\) in any variant of EPG-representations. Similarly there are pathwidth-k graphs that require height \(\varOmega (k)\) and area \(\varOmega (kn)\) in any variant of EPG-representations. We prove a matching upper bound of O(kn) area for all pathwidth-k graphs in the strongest model, the one where edges are required to be both x- and y-monotone. Thus in this strongest model, the result implies, for example, O(n), \(O(n \log n)\) and \(O(n^{3/2})\) area bounds for bounded pathwidth graphs, bounded treewidth graphs and all classes of graphs that exclude a fixed minor, respectively. For the model with no restrictions on the monotonicity of the edges, stronger results can be achieved for some graph classes, for example an O(n) area bound for bounded treewidth graphs and \(O(n \log ^2 n)\) bound for graphs of bounded genus.


  1. 1.
    Alam, M.J., Bläsius, T., Rutter, I., Ueckerdt, T., Wolff, A.: Pixel and Voxel Representations of Graphs. In: Di Giacomo, E., Lubiw, A. (eds.) GD 2015. LNCS, vol. 9411, pp. 472–486. Springer, Cham (2015). CrossRefGoogle Scholar
  2. 2.
    Alon, N., Seymour, P., Thomas, R.: A separator theorem for graphs with an excluded minor and its applications. In: Proceedings of the Twenty-second Annual ACM Symposium on Theory of Computing, STOC 1990, pp. 293–299. ACM, New York (1990)Google Scholar
  3. 3.
    Asinowski, A., Suk, A.: Edge intersection graphs of systems of paths on a grid with a bounded number of bends. Discrete Appl. Math. 157(14), 3174–3180 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Biedl, T., Stern, M.: Edge-intersection graphs of \(k\)-bend paths in grids. Discrete Math. Theor. Comput. Sci. 12(1), 1–12 (2010). (electronic journal)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bodlaender, H.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209(1–2), 1–45 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Diestel, R.: Graph Theory. GTM, vol. 173. Springer, Heidelberg (2017). zbMATHGoogle Scholar
  7. 7.
    Dvorak, Z., Norin, S.: Treewidth of graphs with balanced separations. CoRR abs/1408.3869 (2014)Google Scholar
  8. 8.
    Epstein, D., Golumbic, M.C., Morgenstern, G.: Approximation Algorithms for \(B_1\)-EPG Graphs. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 328–340. Springer, Heidelberg (2013). CrossRefGoogle Scholar
  9. 9.
    Francis, M., Lahiri, A.: VPG and EPG bend-numbers of Halin graphs. Discrete Appl. Math. 215, 95–105 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, 2nd edn. Academic Press, New York (2004)zbMATHGoogle Scholar
  11. 11.
    Golumbic, M., Lipshteyn, M., Stern, M.: Edge intersection graphs of single bend paths on a grid. Networks 54(3), 130–138 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grohe, M., Marx, D.: On tree width, bramble size, and expansion. J. Comb. Theory, Ser. B 99(1), 218–228 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Heldt, D., Knauer, K., Ueckerdt, T.: Edge-intersection graphs of grid paths: the bend-number. Discrete Appl. Math. 167, 144–162 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Leiserson, C.: Area-efficient graph layouts (for VLSI). In: IEEE Symposium on Foundations of Computer Science (FOCS 1980), pp. 270–281 (1980)Google Scholar
  15. 15.
    Marcus, A., Spielman, D., Srivastava, N.: Interlacing families I: bipartite Ramanujan graphs of all degrees. In: Symposium on Foundations of Computer Science, FOCS. pp. 529–537. IEEE Computer Society (2013)Google Scholar
  16. 16.
    Markov, I., Shi, Y.: Constant-degree graph expansions that preserve treewidth. Algorithmica 59(4), 461–470 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mehrabi, S.: Approximation algorithms for independence and domination on \(B_1\)-VPG and \(B_1\)-EPG-graphs. CoRR abs/1702.05633 (2017)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Therese Biedl
    • 1
    Email author
  • Martin Derka
    • 1
  • Vida Dujmović
    • 2
  • Pat Morin
    • 3
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.School of Computer Science and Electrical EngineeringUniversity of OttawaOttawaCanada
  3. 3.School of Computer ScienceCarleton UniversityOttawaCanada

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