Well-Partitioned Chordal Graphs: Obstruction Set and Disjoint Paths

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


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.


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


  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). Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. TCS. Springer, London (2013). 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). 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). 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

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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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