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Recognizing H-Graphs - Beyond Circular-Arc Graphs Ağaoğlu Çağırıcı, Deniz Çağırıcı, Onur Derbisz, Jan Hartmann, Tim A. Hliněný, Petr Kratochvíl, Jan Krawczyk, Tomasz Zeman, Peter H-graphs Intersection Graphs Helly Property 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Deniz Ağaoğlu Çağırıcı and Onur Çağırıcı and Jan Derbisz and Tim A. Hartmann and Petr Hliněný and Jan Kratochvíl and Tomasz Krawczyk and Peter Zeman 2023 Is Part Of LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.MFCS.2023.8 urn:nbn:de:0030-drops-185420 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.8 eng https://creativecommons.org/licenses/by/4.0/legalcode