Vertex Contact Graphs of Paths on a Grid

  • Nieke AertsEmail author
  • Stefan Felsner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)


We study Vertex Contact representations of Paths on a Grid (VCPG). In such a representation the vertices of \(G\) are represented by a family of interiorly disjoint grid-paths. Adjacencies are represented by contacts between an endpoint of one grid-path and an interior point of another grid-path. Defining \(u \rightarrow v\) if the path of \(u\) ends on path of \(v\) we obtain an orientation on \(G\) from a VCPG. To get hand on the bends of the grid path the orientation is not enough. We therefore consider pairs (\(\alpha ,\psi \)): a 2-orientation \(\alpha \) and a flow \(\psi \) in the angle graph. The 2-orientation describes the contacts of the ends of a grid-path and the flow describes the behavior of a grid-path between its two ends. We give a necessary and sufficient condition for such a pair \((\alpha ,\psi \)) to be realizable as a VCPG.

Using realizable pairs we show that every planar (2, 2)-tight graph admits a VCPG with at most 2 bends per path and that this is tight. Using the same we show that simple planar (2, 1)-sparse graphs have a 4-bend representation and simple planar (2, 0)-sparse graphs have 6-bend representation. We do not believe that the latter two are tight, we conjecture that simple planar (2, 0)-sparse graphs have a 3-bend representation.


  1. 1.
    Aerts, N., Felsner, S.: Vertex Contact graphs of Paths on a Grid.
  2. 2.
    Asinowski, A., Cohen, E., Golumbic, M.C., Limouzy, V., Lipshteyn, M., Stern, M.: Vertex intersection graphs of paths on a grid. J. Graph Algorithms Appl. 16, 129–150 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chaplick, S., Ueckerdt, T.: Planar graphs as VPG-graphs. J. Graph Algorithms Appl. 17, 475–494 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    de Fraysseix, H., de Mendez, P.O.: On topological aspects of orientations. Discrete Math. 229, 57–72 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    de Fraysseix, H., de Mendez, P.O., Pach, J.: A left-first search algorithm for planar graphs. Discrete Comput. Geom. 13, 459–468 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Felsner, S.: Rectangle and square representations of planar graphs. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 213–248. Springer, New York (2013)CrossRefGoogle Scholar
  7. 7.
    Fößmeier, U., Kant, G., Kaufmann, M.: 2-visibility drawings of planar graphs. In: North, Stephen C. (ed.) GD 1996. LNCS, vol. 1190, pp. 155–168. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  8. 8.
    Hartman, I.B.-A., Newman, I., Ziv, R.: On grid intersection graphs. Discrete Math. 87, 41–52 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kobourov, S.G., Ueckerdt, T., Verbeek, K.: Combinatorial and geometric properties of planar laman graphs. In: Khanna, S. (ed.) SODA, pp. 1668–1678. SIAM (2013)Google Scholar
  10. 10.
    Schäffter, M.W.: Drawing graphs on rectangular grids. Discrete Appl. Math. 63, 75–89 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16, 421–444 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institut Für MathematikTechnische Universität BerlinBerlinGermany

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