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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