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On H-Topological Intersection Graphs

  • Steven Chaplick
  • Martin Töpfer
  • Jan Voborník
  • Peter ZemanEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10520)

Abstract

Biró, Hujter, and Tuza introduced the concept of H-graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph H. They naturally generalize many important classes of graphs, e.g., interval graphs and circular-arc graphs. Our paper is the first study of the recognition and dominating set problems of this large collection of intersection classes of graphs.

We negatively answer the question of Biró, Hujter, and Tuza who asked whether H-graphs can be recognized in polynomial time, for a fixed graph H. Namely, we show that recognizing H-graphs is Open image in new window -complete if H contains the diamond graph as a minor. On the other hand, for each tree T, we give a polynomial-time algorithm for recognizing T-graphs and an \(\mathcal {O}(n^4)\)-time algorithm for recognizing \(K_{1,d}\)-graphs. For the dominating set problem (parameterized by the size of H), we give Open image in new window - and Open image in new window -time algorithms on \(K_{1,d}\)-graphs and H-graphs, respectively. Our dominating set algorithm for H-graphs also provides Open image in new window -time algorithms for the independent set and independent dominating set problems on H-graphs.

References

  1. 1.
    Biro, M., Hujter, M., Tuza, Z.: Precoloring extension I. Interval graphs. Discret. Math. 100(1), 267–279 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Bonomo, F., Mattia, S., Oriolo, G.: Bounded coloring of co-comparability graphs and the pickup and delivery tour combination problem. Theor. Comput. Sci. 412(45), 6261–6268 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and planarity using PQ-tree algorithms. J. Comput. System Sci. 13, 335–379 (1976)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Booth, K.S., Johnson, J.H.: Dominating sets in chordal graphs. SIAM J. Comput. 11(1), 191–199 (1982)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Chang, M.S.: Efficient algorithms for the domination problems on interval and circular-arc graphs. SIAM J. Comput. 27(6), 1671–1694 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms, vol. 4. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21275-3 CrossRefzbMATHGoogle Scholar
  7. 7.
    Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Comb. Theory, Ser. B 16(1), 47–56 (1974)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, vol. 57. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
  10. 10.
    Golumbic, M.C.: The complexity of comparability graph recognition and coloring. Computing 18(3), 199–208 (1977)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Klavík, P., Kratochvíl, J., Otachi, Y., Saitoh, T.: Extending partial representations of subclasses of chordal graphs. Theor. Comput. Sci. 576, 85–101 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Lueker, G.S., Booth, K.S.: A linear time algorithm for deciding interval graph isomorphism. J. ACM (JACM) 26(2), 183–195 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Algebr. Discrete Methods 3(3), 351–358 (1982)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Steven Chaplick
    • 1
  • Martin Töpfer
    • 2
  • Jan Voborník
    • 3
  • Peter Zeman
    • 3
    Email author
  1. 1.Lehrstuhl für Informatik IUniversität WürzburgWürzburgGermany
  2. 2.Faculty of Mathematics and Physics, Computer Science InstituteCharles University in PraguePragueCzech Republic
  3. 3.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic

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