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Recognizing H-Graphs - Beyond Circular-Arc Graphs
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Tim A. Hartmann 
,
Petr Hliněný ,
Tomasz Krawczyk ,
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48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Mathematical Foundations of Computer Science (MFCS) - License:
Creative Commons Attribution 4.0 International license
- Publication date: 2023-08-21
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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)
https://doi.org/10.4230/LIPIcs.MFCS.2023.8
https://doi.org/10.4230/LIPIcs.MFCS.2023.8
Abstract
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
Keywords
- H-graphs
- Intersection Graphs
- Helly Property
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References
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