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On the Structure of Normalized Models of Circular-Arc Graphs I. Hsu’s Approach

Author Tomasz Krawczyk

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Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Graph algorithms analysis
Keywords
  • intersection graphs and models
  • circular-arc graphs and circular-arc intersection models

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Abstract

In the work [𝒪(m⋅ n) algorithms for the recognition and isomorphism problems on circular-arc graphs, SIAM J. Comput. 24(3), 411-439, (1995)], Wen-Lian Hsu claims three results concerning the class of circular-arc graphs:  
- the design of so-called decomposition trees that represent the structure of all normalized intersection models of circular-arc graphs, 
- an 𝒪(nm)-time recognition algorithm for circular-arc graphs, 
- an 𝒪(nm)-time isomorphism algorithm for circular-arc graphs.  In [Discrete Math. Theor. Comput. Sci., 15(1), 157-182, 2013] Curtis, Lin, McConnell, Nussbaum, Soulignac, Spinrad, and Szwarcfiter showed that Hsu’s isomorphism algorithm is incorrect. In this note, we show that the other two results - namely, the construction of decomposition trees and the recognition algorithm - are also flawed.
We also present the main ideas that made it possible to construct a data structure that maintains normalized models of circular-arc graphs.

Cite As Get BibTex

Tomasz Krawczyk. On the Structure of Normalized Models of Circular-Arc Graphs I. Hsu’s Approach. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.GD.2025.3

Author Details

Tomasz Krawczyk
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland

Funding

  • Krawczyk, Tomasz: Partially supported by grant no. 2024/53/B/ST6/02558 from National Science Centre, Poland.

References

  1. Andrew R. Curtis, Min Chih Lin, Ross M. McConnell, Yahav Nussbaum, Francisco J. Soulignac, Jeremy P. Spinrad, and Jayme Luiz Szwarcfiter. Isomorphism of graph classes related to the circular-ones property. Discret. Math. Theor. Comput. Sci., 15(1):157-182, 2013. URL: https://doi.org/10.46298/DMTCS.625.
  2. Ben Dushnik and E. W. Miller. Partially ordered sets. Amer. J. Math., 63:600-610, 1941. URL: https://doi.org/10.2307/2371374.
  3. Csaba P. Gabor, Kenneth J. Supowit, and Wen-Lian Hsu. Recognizing circle graphs in polynomial time. J. ACM, 36(3):435-473, 1989. URL: https://doi.org/10.1145/65950.65951.
  4. Tibor Gallai. Transitiv orientierbare Graphen. Acta Math. Acad. Sci. Hung., 18(1-2):25-66, 1967. Google Scholar
  5. Wen-Lian Hsu. O(M*N) algorithms for the recognition and isomorphism problems on circular-arc graphs. SIAM J. Comput., 24(3):411-439, 1995. URL: https://doi.org/10.1137/S0097539793260726.
  6. Tomasz Krawczyk. On the structure of normalized models of circular-arc graphs - Hsu’s approach revisited. CoRR, abs/2411.13374, 2024. URL: https://doi.org/10.48550/arXiv.2411.13374.
  7. Jeremy P. Spinrad. Circular-arc graphs with clique cover number two. J. Comb. Theory B, 44(3):300-306, 1988. URL: https://doi.org/10.1016/0095-8956(88)90038-X.
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