Recognizing Proper Tree-Graphs

Authors Steven Chaplick , Petr A. Golovach , Tim A. Hartmann , Dušan Knop

Thumbnail PDF


  • Filesize: 0.59 MB
  • 15 pages

Document Identifiers

Author Details

Steven Chaplick
  • Maastricht University, The Netherlands
Petr A. Golovach
  • Department of Informatics, University of Bergen, Norway
Tim A. Hartmann
  • RWTH Aachen, Germany
Dušan Knop
  • Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Czech Republic


We thank Peter Zeman for initial discussions on this work, particularly relating to the NP-completeness of proper H-graph recognition. We also thank an anonymous reviewer to point to a mistake in an earlier version.

Cite AsGet BibTex

Steven Chaplick, Petr A. Golovach, Tim A. Hartmann, and Dušan Knop. Recognizing Proper Tree-Graphs. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We investigate the parameterized complexity of the recognition problem for the proper H-graphs. The H-graphs are the intersection graphs of connected subgraphs of a subdivision of a multigraph H, and the properness means that the containment relationship between the representations of the vertices is forbidden. The class of H-graphs was introduced as a natural (parameterized) generalization of interval and circular-arc graphs by Biró, Hujter, and Tuza in 1992, and the proper H-graphs were introduced by Chaplick et al. in WADS 2019 as a generalization of proper interval and circular-arc graphs. For these graph classes, H may be seen as a structural parameter reflecting the distance of a graph to a (proper) interval graph, and as such gained attention as a structural parameter in the design of efficient algorithms. We show the following results. - For a tree T with t nodes, it can be decided in 2^{𝒪(t² log t)} ⋅ n³ time, whether an n-vertex graph G is a proper T-graph. For yes-instances, our algorithm outputs a proper T-representation. This proves that the recognition problem for proper H-graphs, where H required to be a tree, is fixed-parameter tractable when parameterized by the size of T. Previously only NP-completeness was known. - Contrasting to the first result, we prove that if H is not constrained to be a tree, then the recognition problem becomes much harder. Namely, we show that there is a multigraph H with 4 vertices and 5 edges such that it is NP-complete to decide whether G is a proper H-graph.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Fixed parameter tractability
  • intersection graphs
  • H-graphs
  • recognition
  • fixed-parameter tractability


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Deniz Ağaoğlu and Petr Hliněný. Isomorphism problem for S_d-graphs, 2019. URL:
  2. Miklós Biró, Mihály Hujter, and Zsolt Tuza. Precoloring extension. I. Interval graphs. Discrete Mathematics, 100(1):267-279, 1992. Google Scholar
  3. Jean R. S. Blair and Barry Peyton. An introduction to chordal graphs and clique trees. In Graph Theory and Sparse Matrix Computation, pages 1-29, New York, NY, 1993. Springer New York. Google Scholar
  4. M. L. Brady and M. Sarrafzadeh. Stretching a knock-knee layout for multilayer wiring. IEEE Transactions on Computers, 39(1):148-151, January 1990. URL:
  5. Andreas Brandstädt, Van Bang Le, and Jeremy P. Spinrad. Graph classes: a survey. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. URL:
  6. Peter Buneman. A characterisation of rigid circuit graphs. Discrete Mathematics, 9(3):205-212, 1974. URL:
  7. Steven Chaplick, Fedor V. Fomin, Petr A. Golovach, Dusan Knop, and Peter Zeman. Kernelization of graph hamiltonicity: Proper H-graphs. In Zachary Friggstad, Jörg-Rüdiger Sack, and Mohammad R. Salavatipour, editors, WADS 2019, volume 11646 of LNCS, pages 296-310. Springer, 2019. URL:
  8. Steven Chaplick, Martin Toepfer, Jan Voborník, and Peter Zeman. On H-topological intersection graphs. In WG 2017, pages 167-179, 2017. URL:
  9. Steven Chaplick and Peter Zeman. Combinatorial problems on H-graphs. In EUROCOMB'17, volume 61 of ENDM, pages 223-229. Elsevier, 2017. URL:
  10. Derek G. Corneil. A simple 3-sweep LBFS algorithm for the recognition of unit interval graphs. Discrete Applied Mathematics, 138(3):371-379, 2004. URL:
  11. Xiaotie Deng, Pavol Hell, and Jing Huang. Linear-time representation algorithms for proper circular-arc graphs and proper interval graphs. SIAM Journal on Computing, 25(2):390-403, 1996. URL:
  12. Fedor V. Fomin, Petr A. Golovach, and Jean-Florent Raymond. On the tractability of optimization problems on H-graphs. Algorithmica, 2020. URL:
  13. Philippe Galinier, Michel Habib, and Christophe Paul. Chordal graphs and their clique graphs. In WG 1995, volume 1017 of LNCS, pages 358-371. Springer, 1995. URL:
  14. Fǎnicǎ Gavril. The intersection graphs of subtrees of trees are exactly the chordal graphs. Journal of Combinatorial Theory Series B, 16:47-56, 1974. Google Scholar
  15. Martin Charles Golumbic. Algorithmic graph theory and perfect graphs, volume 57 of Annals of Discrete Mathematics. Elsevier Science B.V., Amsterdam, second edition, 2004. With a foreword by Claude Berge. Google Scholar
  16. M. L. Huson and A. Sen. Broadcast scheduling algorithms for radio networks. In Military Communications Conference, 1995. MILCOM '95, Conference Record, IEEE, volume 2, pages 647-651 vol.2, November 1995. URL:
  17. Lars Jaffke, O-joung Kwon, Torstein J. F. Strømme, and Jan Arne Telle. Mim-width III. graph powers and generalized distance domination problems. Theoretical Computer Science, 796:216-236, 2019. URL:
  18. Lars Jaffke, O-joung Kwon, and Jan Arne Telle. Mim-width II. the feedback vertex set problem. Algorithmica, 82(1):118-145, 2020. URL:
  19. Deborah Joseph, Joao Meidanis, and Prasoon Tiwari. Determining DNA sequence similarity using maximum independent set algorithms for interval graphs. In Otto Nurmi and Esko Ukkonen, editors, SWAT '92, volume 621 of LNCS, pages 326-337. Springer, 1992. URL:
  20. Pavel Klavík, Jan Kratochvíl, Yota Otachi, and Toshiki Saitoh. Extending partial representations of subclasses of chordal graphs. Theoretical Computer Science, 576:85-101, 2015. Google Scholar
  21. In-Jen Lin, Terry A. McKee, and Douglas B. West. The leafage of a chordal graph. Discuss. Math. Graph Theory, 18(1):23-48, 1998. URL:
  22. Fred S Roberts. Graph theory and its applications to problems of society. SIAM, 1978. Google Scholar
  23. Donald J. Rose, Robert Endre Tarjan, and George S. Lueker. Algorithmic aspects of vertex elimination on graphs. SIAM Journal on Computing, 5(2):266-283, 1976. URL:
  24. F. W. Sinden. Topology of thin film rc circuits. Bell System Technical Journal, 45(9):1639-1662, 1966. URL:
  25. James R. Walter. Representations of chordal graphs as subtrees of a tree. Journal of Graph Theory, 2(3):265-267, 1978. URL: