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On Strict (Outer-)Confluent Graphs

  • Henry Förster
  • Robert Ganian
  • Fabian KluteEmail author
  • Martin Nöllenburg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11904)

Abstract

A strict confluent (SC) graph drawing is a drawing of a graph with vertices as points in the plane, where vertex adjacencies are represented not by individual curves but rather by unique smooth paths through a planar system of junctions and arcs. If all vertices of the graph lie in the outer face of the drawing, the drawing is called a strict outerconfluent (SOC) drawing. SC and SOC graphs were first considered by Eppstein et al. in Graph Drawing 2013. Here, we establish several new relationships between the class of SC graphs and other graph classes, in particular string graphs and unit-interval graphs. Further, we extend earlier results about special bipartite graph classes to the notion of strict outerconfluency, show that SOC graphs have cop number two, and establish that tree-like (\(\varDelta \)-)SOC graphs have bounded cliquewidth.

References

  1. 1.
    Aigner, M., Fromme, M.: A game of cops and robbers. Discrete Appl. Math. 8(1), 1–12 (1984).  https://doi.org/10.1016/0166-218X(84)90073-8MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bach, B., Riche, N.H., Hurter, C., Marriott, K., Dwyer, T.: Towards unambiguous edge bundling: investigating confluent drawings for network visualization. IEEE Trans. Vis. Comput. Graph. 23(1), 541–550 (2017).  https://doi.org/10.1109/TVCG.2016.2598958CrossRefGoogle Scholar
  3. 3.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994).  https://doi.org/10.1145/174644.174650MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benzaken, C., Crama, Y., Duchet, P., Hammer, P.L., Maffray, F.: More characterizations of triangulated graphs. J. Graph Theory 14(4), 413–422 (1990).  https://doi.org/10.1002/jgt.3190140404MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brandstädt, A., Spinrad, J., Stewart, L.: Bipartite permutation graphs are bipartite tolerance graphs. Congressus Numerantium 58, 165–174 (1987)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000).  https://doi.org/10.1007/s002249910009MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Appl. Math. 101(1–3), 77–114 (2000).  https://doi.org/10.1016/S0166-218X(99)00184-5MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dickerson, M., Eppstein, D., Goodrich, M.T., Meng, J.Y.: Confluent drawings: visualizing non-planar diagrams in a planar way. J. Graph Algorithms Appl. 9(1), 31–52 (2005).  https://doi.org/10.7155/jgaa.00099MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ehrlich, G., Even, S., Tarjan, R.E.: Intersection graphs of curves in the plane. J. Comb. Theory Ser. B 21(1), 8–20 (1976).  https://doi.org/10.1016/0095-8956(76)90022-8MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Eppstein, D., Goodrich, M.T., Meng, J.Y.: Delta-confluent drawings. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 165–176. Springer, Heidelberg (2006).  https://doi.org/10.1007/11618058_16CrossRefGoogle Scholar
  11. 11.
    Eppstein, D., Goodrich, M.T., Meng, J.Y.: Confluent layered drawings. Algorithmica 47, 439–452 (2007).  https://doi.org/10.1007/s00453-006-0159-8MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Eppstein, D., Holten, D., Löffler, M., Nöllenburg, M., Speckmann, B., Verbeek, K.: Strict confluent drawing. J. Comput. Geom. 7(1), 22–46 (2016).  https://doi.org/10.20382/jocg.v7i1a2MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Eppstein, D., Simons, J.A.: Confluent Hasse diagrams. J. Graph Algorithms Appl. 17(7), 689–710 (2013).  https://doi.org/10.1007/978-3-642-25878-7_2MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Förster, H., Ganian, R., Klute, F., Nöllenburg, M.: On strict (outer-)confluent graphs. CoRR abs/1908.05345 (2019). http://arxiv.org/abs/1908.05345
  15. 15.
    Gabor, C.P., Supowit, K.J., Hsu, W.L.: Recognizing circle graphs in polynomial time. J. ACM 36(3), 435–473 (1989).  https://doi.org/10.1145/65950.65951MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gavenčiak, T., Jelínek, V., Klavík, P., Kratochvíl, J.: Cops and robbers on intersection graphs. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) ISAAC 2013. LNCS, vol. 8283, pp. 174–184. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-45030-3_17,  https://doi.org/10.1016/j.ejc.2018.04.009
  17. 17.
    Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J. Comput. 1(2), 180–187 (1972).  https://doi.org/10.1137/0201013MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gavril, F.: Maximum weight independent sets and cliques in intersection graphs of filaments. Inf. Process. Lett. 73(5–6), 181–188 (2000).  https://doi.org/10.1016/S0020-0190(00)00025-9MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gioan, E., Paul, C.: Split decomposition and graph-labelled trees: characterizations and fully dynamic algorithms for totally decomposable graphs. Discrete Appl. Math. 160(6), 708–733 (2012).  https://doi.org/10.1016/j.dam.2011.05.007MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, vol. 57. Elsevier, Amsterdam (2004).  https://doi.org/10.1002/net.3230130214CrossRefzbMATHGoogle Scholar
  21. 21.
    Golumbic, M.C., Monma, C.L., Trotter Jr., W.T.: Tolerance graphs. Discrete Appl. Math. 9(2), 157–170 (1984).  https://doi.org/10.1016/0166-218X(84)90016-7MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Golumbic, M.C., Rotem, D., Urrutia, J.: Comparability graphs and intersection graphs. Discrete Math. 43(1), 37–46 (1983).  https://doi.org/10.1016/0012-365X(83)90019-5MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Golumbic, M.C., Rotics, U.: On the clique-width of some perfect graph classes. Int. J. Found. Comput. Sci. 11(3), 423–443 (2000).  https://doi.org/10.1142/S0129054100000260MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Halldórsson, M.M., Kitaev, S., Pyatkin, A.: Alternation graphs. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 191–202. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-25870-1_18CrossRefGoogle Scholar
  25. 25.
    Holten, D.: Hierarchical edge bundles: visualization of adjacency relations in hierarchical data. IEEE Trans. Vis. Comput. Graph. 12(5), 741–748 (2006).  https://doi.org/10.1109/TVCG.2006.147CrossRefGoogle Scholar
  26. 26.
    Hsu, W.L.: Maximum weight clique algorithms for circular-arc graphs and circle graphs. SIAM J. Comput. 14(1), 224–231 (1985).  https://doi.org/10.1137/0214018MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hui, P., Pelsmajer, M.J., Schaefer, M., Stefankovic, D.: Train tracks and confluent drawings. Algorithmica 47(4), 465–479 (2007).  https://doi.org/10.1007/s00453-006-0165-xMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kratochvíl, J.: String graphs. I. The number of critical nonstring graphs is infinite. J. Comb. Theory Ser. B 52(1), 53–66 (1991).  https://doi.org/10.1016/0095-8956(91)90090-7MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pnueli, A., Lempel, A., Even, S.: Transitive orientation of graphs and identification of permutation graphs. Can. J. Math. 23(1), 160–175 (1971).  https://doi.org/10.4153/CJM-1971-016-5MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Roberts, F.S.: Indifference graphs. In: Proof Techniques in Graph Theory, pp. 139–146 (1969)Google Scholar
  31. 31.
    Takamizawa, K., Nishizeki, T., Saito, N.: Linear-time computability of combinatorial problems on series-parallel graphs. J. ACM 29(3), 623–641 (1982).  https://doi.org/10.1145/322326.322328MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory, vol. 6. JHU Press, Baltimore (2001).  https://doi.org/10.1137/1035116CrossRefzbMATHGoogle Scholar
  33. 33.
    Wegner, G.: Eigenschaften der Nerven homologisch-einfacher Familien im Rn. Ph.D. thesis, Universität Göttingen (1967)Google Scholar
  34. 34.
    Yu, C.W., Chen, G.H.: Efficient parallel algorithms for doubly convex-bipartite graphs. Theoret. Comput. Sci. 147(1–2), 249–265 (1995).  https://doi.org/10.1016/0304-3975(94)00220-DMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of TübingenTübingenGermany
  2. 2.Algorithms and Complexity GroupTU WienViennaAustria

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