Abstract
B\(_{0}\)-VPG graphs are intersection graphs of axis-parallel line segments in the plane. We show that all AT-free outerplanar graphs are B\(_{0}\)-VPG. In the course of the argument, we show that any AT-free outerplanar graph can be identified as an induced subgraph of a 2-connected outerplanar graph whose weak dual is a path. Our B\(_{0}\)-VPG drawing procedure works for such graphs and has the potential to be extended to larger classes of outerplanar graphs.
S. Jain—A part of this work was done while at Indian Institute of Technology Palakkad.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alcón, L., Bonomo, F., Mazzoleni, M.P.: Vertex intersection graphs of paths on a grid: characterization within block graphs. Graphs and Combinatorics 33(4), 653–664 (2017)
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(2), 129–150 (2012)
Biedl, T., Derka, M.: 1-string B\(_{2}\)-VPG representation of planar graphs. J. Comput. Geom. (Old Web Site) 7(2), 191–215 (2016)
de Castro, N., Cobos, F.J., Dana, J.C., Márquez, A., Noy, M.: Triangle-free planar graphs and segment intersection graphs. J. Graph Algorithms Appl. 6(1), 7–26 (2002)
Catanzaro, D., et al.: Max point-tolerance graphs. Discret. Appl. Math. 216, 84–97 (2017)
Chakraborty, D., Das, S., Mukherjee, J., Sahoo, U.K.: Bounds on the bend number of split and cocomparability graphs. Theory Comput. Syst. 63(6), 1336–1357 (2019). https://doi.org/10.1007/s00224-019-09912-4
Chalopin, J., Gonçalves, D.: Every planar graph is the intersection graph of segments in the plane. In: Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, pp. 631–638 (2009)
Chaplick, S., Cohen, E., Stacho, J.: Recognizing some subclasses of vertex intersection graphs of 0-bend paths in a grid. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 319–330. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25870-1_29
Chaplick, S., Jelínek, V., Kratochvíl, J., Vyskočil, T.: Bend-bounded path intersection graphs: sausages, noodles, and waffles on a grill. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds.) WG 2012. LNCS, vol. 7551, pp. 274–285. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-34611-8_28
Chaplick, S., Ueckerdt, T.: Planar graphs as VPG-graphs. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 174–186. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36763-2_16
Cohen, E., Golumbic, M.C., Trotter, W.T., Wang, R.: Posets and VPG graphs. Order 33(1), 39–49 (2016)
Czyzowicz, J., Kranakis, E., Urrutia, J.: A simple proof of the representation of bipartite planar graphs as the contact graphs of orthogonal straight line segments. Inf. Process. Lett. 66(3), 125–126 (1998)
De Fraysseix, H., Ossona de Mendez, P., Pach, J.: Representation of planar graphs by segments. Intuitive Geom. 63, 109–117 (1991)
Fleischner, H.J., Geller, D.P., Harary, F.: Outerplanar graphs and weak duals. J. Indian Math. Soc. 38(1–4), 215–219 (1974)
Francis, M.C., Lahiri, A.: VPG and EPG bend-numbers of Halin graphs. Discret. Appl. Math. 215, 95–105 (2016)
Golumbic, M.C., Ries, B.: On the intersection graphs of orthogonal line segments in the plane: characterizations of some subclasses of chordal graphs. Graphs and Combinatorics 29(3), 499–517 (2013)
Gonçalves, D., Isenmann, L., Pennarun, C.: Planar graphs as L-intersection or L-contact graphs. In: Proceedings of the Twenty-Ninth Annual ACMSIAM Symposium on Discrete Algorithms, pp. 172–184. SIAM (2018)
Hartman, I.B.-A., Newman, I., Ziv, R.: On grid intersection graphs. Discrete Math. 87(1), 41–52 (1991)
Kratochvíl, J.: String graphs. II. Recognizing string graphs is NP-hard. J. Comb. Theory Ser. B 52(1), 67–78 (1991)
Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. J. Comb. Theory, Series B 62(2), 289–315 (1994)
Mehrabi, S.: Approximation algorithms for independence and domination on B\({_{1}}\)-VPG and B\({_{1}}\)-EPG graphs. arXiv preprint arXiv:1702.05633 (2017)
Pallathumadam, S.K., Rajendraprasad, D.: Characterization and a 2D visualization of B\(_{0}\)-VPG cocomparability graphs. In: GD 2020. LNCS, vol. 12590, pp. 191–204. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-68766-3_16
Schaefer, M., Sedgwick, E., Štefankovič, D.: Recognizing string graphs in NP. J. Comput. Syst. Sci. 67(2), 365–380 (2003)
Scheinerman, E.R.: Intersection classes and multiple intersection parameters of graphs. Princeton University (1984)
West, D.: Open problems. SIAM J. Discrete Math. Newslett. 2, 10–12 (1991)
Acknowledgments
We thank K. Muralikrishnan for posing the question of characterizing B\(_{0}\)-VPG outerplanar graphs.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Jain, S., Pallathumadam, S.K., Rajendraprasad, D. (2022). \(B_0\)-VPG Representation of AT-free Outerplanar Graphs. In: Balachandran, N., Inkulu, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2022. Lecture Notes in Computer Science(), vol 13179. Springer, Cham. https://doi.org/10.1007/978-3-030-95018-7_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-95018-7_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-95017-0
Online ISBN: 978-3-030-95018-7
eBook Packages: Computer ScienceComputer Science (R0)