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Well-Partitioned Chordal Graphs: Obstruction Set and Disjoint Paths

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12301)

Abstract

We introduce a new subclass of chordal graphs that generalizes split graphs, which we call well-partitioned chordal graphs. Split graphs are graphs that admit a partition of the vertex set into cliques that can be arranged in a star structure, the leaves of which are of size one. Well-partitioned chordal graphs are a generalization of this concept in the following two ways. First, the cliques in the partition can be arranged in a tree structure, and second, each clique is of arbitrary size. We provide a characterization of well-partitioned chordal graphs by forbidden induced subgraphs, and give a polynomial-time algorithm that given any graph, either finds an obstruction, or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal. We demonstrate the algorithmic use of this graph class by showing that two variants of the problem of finding pairwise disjoint paths between k given pairs of vertices is in FPT parameterized by k on well-partitioned chordal graphs, while on chordal graphs, these problems are only known to be in XP. From the other end, we observe that there are problems that are polynomial-time solvable on split graphs, but become NP-complete on well-partitioned chordal graphs.

Keywords

Well-partitioned chordal graph Chordal graph Split graph Disjoint paths Forbidden induced subgraphs 

References

  1. 1.
    Belmonte, R., Golovach, P.A., Heggernes, P., Hof, P.V., Kamiński, M., Paulusma, D.: Detecting fixed patterns in chordal graphs in polynomial time. Algorithmica 69(3), 501–521 (2014)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theoret. Comput. Sci. 412(35), 4570–4578 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Brandstädt, A., Dragan, F.F., Le, H.O., Le, V.B.: Tree spanners on chordal graphs: complexity and algorithms. Theoret. Comput. Sci. 310(1–3), 329–354 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chang, Y.W., Jacobson, M.S., Monma, C.L., West, D.B.: Subtree and substar intersection numbers. Discret. Appl. Math. 44(1–3), 205–220 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Corneil, D.G., Perl, Y.: Clustering and domination in perfect graphs. Discret. Appl. Math. 9(1), 27–39 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cygan, M.: Parameterized Algorithms. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21275-3zbMATHCrossRefGoogle Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. TCS. Springer, London (2013).  https://doi.org/10.1007/978-1-4471-5559-1zbMATHCrossRefGoogle Scholar
  8. 8.
    Fomin, F.V., Lokshtanov, D., Saurabh, S., Zehavi, M.: Kernelization. Cambridge University Press, Cambridge (2019)Google Scholar
  9. 9.
    Fulkerson, D., Gross, O.: Incidence matrices and interval graphs. Pac. J. Math. 15(3), 835–855 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    George, A., Gilbert, J.R., Liu, J.W.: Graph Theory and Sparse Matrix Computation. IMA, vol. 56. Springer, New York (2012).  https://doi.org/10.1007/978-1-4613-8369-7CrossRefGoogle Scholar
  11. 11.
    Gustedt, J.: On the pathwidth of chordal graphs. Discret. Appl. Math. 45(3), 233–248 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Havet, F., Sales, C.L., Sampaio, L.: b-coloring of tight graphs. Discret. Appl. Math. 160(18), 2709–2715 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Heggernes, P., van’t Hof, P.V., van Leeuwen, E.J., Saei, R.: Finding disjoint paths in split graphs. Theor. Comput. Syst. 57(1), 140–159 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kammer, F., Tholey, T.: The k-disjoint paths problem on chordal graphs. In: Paul, C., Habib, M. (eds.) WG 2009. LNCS, vol. 5911, pp. 190–201. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-11409-0_17CrossRefGoogle Scholar
  15. 15.
    Karp, R.M.: On the computational complexity of combinatorial problems. Networks 5(1), 45–68 (1975)zbMATHCrossRefGoogle Scholar
  16. 16.
    Kawarabayashi, K.I., Kobayashi, Y., Reed, B.: The disjoint paths problem in quadratic time. J. Combin. Theor. Ser. B 102(2), 424–435 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Mengel, S.: Lower bounds on the mim-width of some graph classes. Discret. Appl. Math. 248, 28–32 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Nikolopoulos, S.D., Palios, L.: Detecting holes and antiholes in graphs. Algorithmica 47(2), 119–138 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Combin. Theor. Ser. B 63(1), 65–110 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Semple, C., Steel, M.: Phylogenetics. Oxford Lecture Series in Mathematics and its Applications, vol. 24. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  22. 22.
    Silva, A.: Graphs with small fall-spectrum. Discret. Appl. Math. 254, 183–188 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Vatshelle, M.: New Width Parameters of Graphs. Ph.D. thesis, University of Bergen, Norway (2012)Google Scholar
  24. 24.
    Watrigant, R., Bougeret, M., Giroudeau, R.: Approximating the Sparsest \(k\)-subgraph in chordal graphs. Theor. Comput. Syst. 58(1), 111–132 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKAISTDaejeonSouth Korea
  2. 2.Discrete Mathematics Group, IBSDaejeonSouth Korea
  3. 3.Department of InformaticsUniversity of BergenBergenNorway
  4. 4.Department of MathematicsIncheon National UniversityIncheonSouth Korea

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