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Recognizing H-Graphs - Beyond Circular-Arc Graphs

Authors Deniz Ağaoğlu Çağırıcı, Onur Çağırıcı, Jan Derbisz, Tim A. Hartmann , Petr Hliněný , Jan Kratochvíl, Tomasz Krawczyk , Peter Zeman

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Deniz Ağaoğlu Çağırıcı
  • Faculty of Informatics, Masaryk University, Brno, Czech Republic
Onur Çağırıcı
  • Toronto Metropolitan University, Canada
Jan Derbisz
  • Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
  • Doctoral School of Exact and Natural Sciences, Jagiellonian University, Kraków, Poland
Tim A. Hartmann
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Petr Hliněný
  • Faculty of Informatics, Masaryk University, Brno, Czech Republic
Jan Kratochvíl
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Tomasz Krawczyk
  • Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Peter Zeman
  • Technical University of Denmark, Lyngby, Denmark

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Deniz Ağaoğlu Çağırıcı, Onur Çağırıcı, Jan Derbisz, Tim A. Hartmann, Petr Hliněný, Jan Kratochvíl, Tomasz Krawczyk, and Peter Zeman. Recognizing H-Graphs - Beyond Circular-Arc Graphs. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 8:1-8:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


In 1992 Biró, Hujter and Tuza introduced, for every fixed connected graph H, the class of H-graphs, defined as the intersection graphs of connected subgraphs of some subdivision of H. Such classes of graphs are related to many known graph classes: for example, K₂-graphs coincide with interval graphs, K₃-graphs with circular-arc graphs, the union of T-graphs, where T ranges over all trees, coincides with chordal graphs. Recently, quite a lot of research has been devoted to understanding the tractability border for various computational problems, such as recognition or isomorphism testing, in classes of H-graphs for different graphs H. In this work we undertake this research topic, focusing on the recognition problem. Chaplick, Töpfer, Voborník, and Zeman showed an XP-algorithm testing whether a given graph is a T-graph, where the parameter is the size of the tree T. In particular, for every fixed tree T the recognition of T-graphs can be solved in polynomial time. Tucker showed a polynomial time algorithm recognizing K₃-graphs (circular-arc graphs). On the other hand, Chaplick et al. showed also that for every fixed graph H containing two distinct cycles sharing an edge, the recognition of H-graphs is NP-hard. The main two results of this work narrow the gap between the NP-hard and 𝖯 cases of H-graph recognition. First, we show that the recognition of H-graphs is NP-hard when H contains two distinct cycles. On the other hand, we show a polynomial-time algorithm recognizing L-graphs, where L is a graph containing a cycle and an edge attached to it (which we call lollipop graphs). Our work leaves open the recognition problems of M-graphs for every unicyclic graph M different from a cycle and a lollipop.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • H-graphs
  • Intersection Graphs
  • Helly Property


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  1. Deniz Ağaoğlu Çağırıcı and Petr Hliněný. Efficient isomorphism for S_d-graphs and T-graphs. Algorithmica, 2022. URL:
  2. Deniz Ağaoğlu Çağırıcı and Petr Hliněný. Isomorphism testing for T-graphs in FPT. WALCOM: Algorithms and Computation, pages 239-250, 2022. Google Scholar
  3. Deniz Ağaoğlu Çağırıcı and Peter Zeman. Recognition and isomorphism of proper U-graphs in FPT-time. CoRR, abs/2206.13372, 2022. URL:
  4. Vikraman Arvind, Roman Nedela, Ilia Ponomarenko, and Peter Zeman. Testing isomorphism of chordal graphs of bounded leafage is fixed-parameter tractable (extended abstract). In Michael A. Bekos and Michael Kaufmann, editors, Graph-Theoretic Concepts in Computer Science - 48th International Workshop, WG 2022, Tübingen, Germany, June 22-24, 2022, Revised Selected Papers, volume 13453 of Lecture Notes in Computer Science, pages 29-42. Springer, 2022. URL:
  5. Miklós Biró, Mihály Hujter, and Zsolt Tuza. Precoloring extension. i. interval graphs. Discret. Math., 100(1-3):267-279, 1992. URL:
  6. K. Booth and G. Lueker. Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. System Sci., 13(3):335-379, 1976. URL:
  7. K. S. Booth. Pq-tree algorithms, November 1975. URL:
  8. Steven Chaplick, Fedor V. Fomin, Petr A. Golovach, Dusan Knop, and Peter Zeman. Kernelization of graph hamiltonicity: Proper H-graphs. SIAM J. Discret. Math., 35(2):840-892, 2021. URL:
  9. Steven Chaplick, Petr A. Golovach, Tim A. Hartmann, and Dusan Knop. Recognizing proper tree-graphs. In Yixin Cao and Marcin Pilipczuk, editors, 15th International Symposium on Parameterized and Exact Computation, IPEC 2020, December 14-18, 2020, Hong Kong, China (Virtual Conference), volume 180 of LIPIcs, pages 8:1-8:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  10. Steven Chaplick, Martin Töpfer, Jan Voborník, and Peter Zeman. On H-topological intersection graphs. Algorithmica, 83(11):3281-3318, 2021. URL:
  11. Steven Chaplick and Peter Zeman. Combinatorial problems on h-graphs. Electron. Notes Discret. Math., 61:223-229, 2017. URL:
  12. Jan Derbisz and Tomasz Krawczyk. Circular-arc graphs and the Helly property. Google Scholar
  13. Fedor V. Fomin, Petr A. Golovach, and Jean-Florent Raymond. On the tractability of optimization problems on H-graphs. In Yossi Azar, Hannah Bast, and Grzegorz Herman, editors, 26th Annual European Symposium on Algorithms, ESA 2018, August 20-22, 2018, Helsinki, Finland, volume 112 of LIPIcs, pages 30:1-30:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL:
  14. Fănică Gavril. The intersection graphs of subtrees in trees are exactly the chordal graphs. Journal of Combinatorial Theory, Series B, 16(1):47-56, 1974. URL:
  15. Daniel Gonçalves and Pascal Ochem. On star and caterpillar arboricity. Discrete Mathematics, 309(11):3694-3702, 2009. URL:
  16. Lars Jaffke, O-joung Kwon, and Jan Arne Telle. Mim-width II. the feedback vertex set problem. Algorithmica, 82(1):118-145, 2020. URL:
  17. Haim Kaplan and Yahav Nussbaum. A simpler linear-time recognition of circular-arc graphs. Algorithmica, 61(3):694-737, 2011. URL:
  18. Pavel Klavík, Jan Kratochvíl, Yota Otachi, and Toshiki Saitoh. Extending partial representations of subclasses of chordal graphs. Theor. Comput. Sci., 576:85-101, 2015. URL:
  19. Min Chih Lin and Jayme L. Szwarcfiter. Characterizations and linear time recognition of Helly circular-arc graphs. In Computing and combinatorics, volume 4112 of Lecture Notes in Comput. Sci., pages 73-82. Springer, Berlin, 2006. URL:
  20. Ross M. McConnell. Linear-time recognition of circular-arc graphs. Algorithmica, 37(2):93-147, 2003. URL:
  21. Donald J. Rose, Robert Endre Tarjan, and George S. Lueker. Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput., 5(2):266-283, 1976. URL:
  22. Alan Tucker. An efficient test for circular-arc graphs. SIAM J. Comput., 9(1):1-24, 1980. URL:
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