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Efficient Isomorphism for \(S_d\)-Graphs and T-Graphs

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Abstract

An H-graph is one representable as the intersection graph of connected subgraphs of a suitable subdivision of a fixed graph H, introduced by Biró et al. (Discrete Mathematics 100:267–279, 1992). An H-graph is proper if the representing subgraphs of H can be chosen incomparable by the inclusion. In this paper, we focus on the isomorphism problem for \(S_d\)-graphs and T-graphs, where \(S_d\) is the star with d rays and T is an arbitrary fixed tree. Answering an open problem of Chaplick et al. (2016, personal communication), we provide an FPT-time algorithm for testing isomorphism and computing the automorphism group of \(S_d\)-graphs when parameterized by d, which involves the classical group-computing machinery by Furst et al. (in Proceedings of 11th southeastern conference on combinatorics, graph theory, and computing, congressum numerantium 3, 1980). We also show that the isomorphism problem of \(S_d\)-graphs is at least as hard as the isomorphism problem of posets of bounded width, for which no efficient combinatorial-only algorithm is known to date. Then we extend our approach to an XP-time algorithm for isomorphism of T-graphs when parameterized by the size of T. Lastly, we contribute an FPT-time combinatorial algorithm for isomorphism testing in the special case of proper \(S_d\)- and T-graphs.

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Acknowledgements

We would like to thank to Pascal Schweitzer for pointing us to the paper [17], and to Onur Çağırıcı for comments on this manuscript.

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  1. Masaryk University, Brno, Czech Republic

    Deniz Ağaoğlu Çağırıcı & Petr Hliněný

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  1. Deniz Ağaoğlu Çağırıcı
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Correspondence to Petr Hliněný.

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This paper is the full extended version of the conference paper that appeared at MFCS 2020. It contains the detailed algorithms and full proofs, and few additional small results.

Supported by research grant GAČR 20-04567S of the Czech Science Foundation.

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Ağaoğlu Çağırıcı, D., Hliněný, P. Efficient Isomorphism for \(S_d\)-Graphs and T-Graphs. Algorithmica 85, 352–383 (2023). https://doi.org/10.1007/s00453-022-01033-8

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  • DOI: https://doi.org/10.1007/s00453-022-01033-8

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